0 Prolog
↳1 PrologToPiTRSProof (⇒, 74 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 178 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 PiDP
↳8 UsableRulesProof (⇔, 0 ms)
↳9 PiDP
↳10 PiDPToQDPProof (⇔, 0 ms)
↳11 QDP
↳12 QDPSizeChangeProof (⇔, 0 ms)
↳13 YES
↳14 PiDP
↳15 UsableRulesProof (⇔, 0 ms)
↳16 PiDP
↳17 PiDPToQDPProof (⇔, 0 ms)
↳18 QDP
↳19 QDPSizeChangeProof (⇔, 0 ms)
↳20 YES
↳21 PiDP
↳22 UsableRulesProof (⇔, 0 ms)
↳23 PiDP
↳24 PiDPToQDPProof (⇒, 8 ms)
↳25 QDP
↳26 QDPOrderProof (⇔, 14 ms)
↳27 QDP
↳28 DependencyGraphProof (⇔, 0 ms)
↳29 QDP
↳30 UsableRulesProof (⇔, 0 ms)
↳31 QDP
↳32 QReductionProof (⇔, 4 ms)
↳33 QDP
↳34 QDPSizeChangeProof (⇔, 0 ms)
↳35 YES
↳36 PiDP
↳37 UsableRulesProof (⇔, 0 ms)
↳38 PiDP
↳39 PiDPToQDPProof (⇒, 0 ms)
↳40 QDP
↳41 QDPSizeChangeProof (⇔, 0 ms)
↳42 YES
↳43 PiDP
↳44 PiDPToQDPProof (⇒, 0 ms)
↳45 QDP
↳46 Rewriting (⇔, 29 ms)
↳47 QDP
↳48 Rewriting (⇔, 0 ms)
↳49 QDP
↳50 QDPQMonotonicMRRProof (⇔, 860 ms)
↳51 QDP
↳52 QDPOrderProof (⇔, 146 ms)
↳53 QDP
↳54 DependencyGraphProof (⇔, 0 ms)
↳55 QDP
↳56 UsableRulesProof (⇔, 0 ms)
↳57 QDP
↳58 QReductionProof (⇔, 0 ms)
↳59 QDP
↳60 QDPOrderProof (⇔, 173 ms)
↳61 QDP
↳62 DependencyGraphProof (⇔, 0 ms)
↳63 TRUE
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
MERGESORT_IN_GAA(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_GAA(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
MERGESORT_IN_GAA(.(X, .(Y, Xs)), Ys, .(H, Ls)) → SPLIT_IN_GAAA(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_GAAA(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → MERGESORT_IN_GAA(X1s, Y1s, Ls)
U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → MERGESORT_IN_GAA(X2s, Y2s, Ls)
U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_GAA(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → MERGE_IN_GGAA(Y1s, Y2s, Ys, .(H, Ls))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_GGAA(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_GGAA(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys, Zs, Ls)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
MERGESORT_IN_GAA(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_GAA(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
MERGESORT_IN_GAA(.(X, .(Y, Xs)), Ys, .(H, Ls)) → SPLIT_IN_GAAA(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_GAAA(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → MERGESORT_IN_GAA(X1s, Y1s, Ls)
U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → MERGESORT_IN_GAA(X2s, Y2s, Ls)
U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_GAA(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
U3_GAA(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → MERGE_IN_GGAA(Y1s, Y2s, Ys, .(H, Ls))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_GGAA(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_GGAA(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys, Zs, Ls)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
From the DPs we obtained the following set of size-change graphs:
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
From the DPs we obtained the following set of size-change graphs:
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys, Zs, Ls)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys, Zs, Ls)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U6_GGAA(X, Xs, Y, Ys, le_in_gg(X, Y))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, Y))
U8_GGAA(X, Xs, Y, Ys, gt_out_gg) → MERGE_IN_GGAA(.(X, Xs), Ys)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, Y))
POL(.(x1, x2)) = 1 + x1 + x2
POL(0) = 0
POL(MERGE_IN_GGAA(x1, x2)) = x2
POL(U10_gg(x1)) = 0
POL(U11_gg(x1)) = 1
POL(U6_GGAA(x1, x2, x3, x4, x5)) = 1 + x3 + x4
POL(U8_GGAA(x1, x2, x3, x4, x5)) = x4
POL(gt_in_gg(x1, x2)) = 0
POL(gt_out_gg) = 0
POL(le_in_gg(x1, x2)) = 1
POL(le_out_gg) = 0
POL(s(x1)) = 0
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U6_GGAA(X, Xs, Y, Ys, le_in_gg(X, Y))
U8_GGAA(X, Xs, Y, Ys, gt_out_gg) → MERGE_IN_GGAA(.(X, Xs), Ys)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U6_GGAA(X, Xs, Y, Ys, le_in_gg(X, Y))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U6_GGAA(X, Xs, Y, Ys, le_in_gg(X, Y))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)
gt_in_gg(x0, x1)
U10_gg(x0)
MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U6_GGAA(X, Xs, Y, Ys, le_in_gg(X, Y))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
le_in_gg(x0, x1)
U11_gg(x0)
From the DPs we obtained the following set of size-change graphs:
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)
SPLIT_IN_GAAA(.(X, Xs)) → SPLIT_IN_GAAA(Xs)
From the DPs we obtained the following set of size-change graphs:
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_GAA(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → MERGESORT_IN_GAA(X2s, Y2s, Ls)
MERGESORT_IN_GAA(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_GAA(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → MERGESORT_IN_GAA(X1s, Y1s, Ls)
mergesort_in_gaa([], [], Ls) → mergesort_out_gaa([], [], Ls)
mergesort_in_gaa(.(X, []), .(X, []), Ls) → mergesort_out_gaa(.(X, []), .(X, []), Ls)
mergesort_in_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls)) → U1_gaa(X, Y, Xs, Ys, H, Ls, split_in_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)))
split_in_gaaa([], [], [], Ls) → split_out_gaaa([], [], [], Ls)
split_in_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_in_gaaa(Xs, Zs, Ys, Ls))
U5_gaaa(X, Xs, Ys, Zs, H, Ls, split_out_gaaa(Xs, Zs, Ys, Ls)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs, .(H, Ls))
U1_gaa(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_in_gaa(X1s, Y1s, Ls))
U2_gaa(X, Y, Xs, Ys, H, Ls, X2s, mergesort_out_gaa(X1s, Y1s, Ls)) → U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_in_gaa(X2s, Y2s, Ls))
U3_gaa(X, Y, Xs, Ys, H, Ls, Y1s, mergesort_out_gaa(X2s, Y2s, Ls)) → U4_gaa(X, Y, Xs, Ys, H, Ls, merge_in_ggaa(Y1s, Y2s, Ys, .(H, Ls)))
merge_in_ggaa([], Xs, Xs, Ls) → merge_out_ggaa([], Xs, Xs, Ls)
merge_in_ggaa(Xs, [], Xs, Ls) → merge_out_ggaa(Xs, [], Xs, Ls)
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) → U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg(0, s(0))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U6_ggaa(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(Xs, .(Y, Ys), Zs, Ls))
merge_in_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) → U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_ggaa(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_in_ggaa(.(X, Xs), Ys, Zs, Ls))
U9_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(.(X, Xs), Ys, Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls))
U7_ggaa(X, Xs, Y, Ys, Zs, H, Ls, merge_out_ggaa(Xs, .(Y, Ys), Zs, Ls)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls))
U4_gaa(X, Y, Xs, Ys, H, Ls, merge_out_ggaa(Y1s, Y2s, Ys, .(H, Ls))) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys, .(H, Ls))
U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(split_in_gaaa(.(X, .(Y, Xs))))
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []))
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, split_in_gaaa(.(Y, Xs))))
U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, split_in_gaaa(.(Y, Xs))))
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []))
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []))
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []))
POL(.(x1, x2)) = 1 + x1 + 2·x2
POL(0) = 2
POL(MERGESORT_IN_GAA(x1)) = x1
POL(U10_gg(x1)) = 1
POL(U11_gg(x1)) = 2
POL(U1_GAA(x1)) = x1
POL(U1_gaa(x1)) = x1
POL(U2_GAA(x1, x2)) = 2·x1 + x2
POL(U2_gaa(x1, x2)) = 2·x1 + x2
POL(U3_gaa(x1, x2)) = x2
POL(U4_gaa(x1)) = 0
POL(U5_gaaa(x1, x2)) = 1 + x1 + 2·x2
POL(U6_ggaa(x1, x2, x3, x4, x5)) = 0
POL(U7_ggaa(x1, x2)) = 0
POL(U8_ggaa(x1, x2, x3, x4, x5)) = 0
POL(U9_ggaa(x1, x2)) = 0
POL([]) = 0
POL(gt_in_gg(x1, x2)) = x2
POL(gt_out_gg) = 0
POL(le_in_gg(x1, x2)) = 2
POL(le_out_gg) = 0
POL(merge_in_ggaa(x1, x2)) = 0
POL(merge_out_ggaa(x1)) = 0
POL(mergesort_in_gaa(x1)) = x1
POL(mergesort_out_gaa(x1)) = 0
POL(s(x1)) = 2
POL(split_in_gaaa(x1)) = x1
POL(split_out_gaaa(x1, x2)) = x1 + 2·x2
U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)
POL(.(x1, x2)) = 0
POL(0) = 1
POL(MERGESORT_IN_GAA(x1)) = 0
POL(U10_gg(x1)) = 0
POL(U11_gg(x1)) = 1
POL(U1_GAA(x1)) = x1
POL(U1_gaa(x1)) = x1
POL(U2_GAA(x1, x2)) = x2
POL(U2_gaa(x1, x2)) = x2
POL(U3_gaa(x1, x2)) = 1
POL(U4_gaa(x1)) = 1
POL(U5_gaaa(x1, x2)) = 0
POL(U6_ggaa(x1, x2, x3, x4, x5)) = 0
POL(U7_ggaa(x1, x2)) = 0
POL(U8_ggaa(x1, x2, x3, x4, x5)) = 0
POL(U9_ggaa(x1, x2)) = 0
POL([]) = 1
POL(gt_in_gg(x1, x2)) = 0
POL(gt_out_gg) = 0
POL(le_in_gg(x1, x2)) = 1 + x1
POL(le_out_gg) = 1
POL(merge_in_ggaa(x1, x2)) = 0
POL(merge_out_ggaa(x1)) = 0
POL(mergesort_in_gaa(x1)) = x1
POL(mergesort_out_gaa(x1)) = 1
POL(s(x1)) = 1 + x1
POL(split_in_gaaa(x1)) = 0
POL(split_out_gaaa(x1, x2)) = x1
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
mergesort_in_gaa([]) → mergesort_out_gaa([])
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa(Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs)
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U6_ggaa(X, Xs, Y, Ys, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(0)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U11_gg(le_out_gg) → le_out_gg
U6_ggaa(X, Xs, Y, Ys, le_out_gg) → U7_ggaa(X, merge_in_ggaa(Xs, .(Y, Ys)))
merge_in_ggaa(.(X, Xs), .(Y, Ys)) → U8_ggaa(X, Xs, Y, Ys, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U8_ggaa(X, Xs, Y, Ys, gt_out_gg) → U9_ggaa(Y, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(Y, merge_out_ggaa(Zs)) → merge_out_ggaa(.(Y, Zs))
U7_ggaa(X, merge_out_ggaa(Zs)) → merge_out_ggaa(.(X, Zs))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
mergesort_in_gaa(x0)
U1_gaa(x0)
U2_gaa(x0, x1)
U3_gaa(x0, x1)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U4_gaa(x0)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
split_in_gaaa(x0)
U5_gaaa(x0, x1)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))
POL(U1_GAA(x1)) = -I + 0A · x1
POL(split_out_gaaa(x1, x2)) = 0A + 0A · x1 + 3A · x2
POL(MERGESORT_IN_GAA(x1)) = 0A + 0A · x1
POL(.(x1, x2)) = -I + -I · x1 + 5A · x2
POL(U5_gaaa(x1, x2)) = -I + -I · x1 + 3A · x2
POL(split_in_gaaa(x1)) = -I + 3A · x1
POL([]) = 0A
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
split_in_gaaa(x0)
U5_gaaa(x0, x1)