Left Termination proof of /tmp/tmpLjziW1/mergesort

Left Termination of the query pattern mergesort_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

mergesort([], [], Ls).
mergesort(.(X, []), .(X, []), Ls).
mergesort(.(X, .(Y, Xs)), Ys, .(H, Ls)) :- ','(split(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)), ','(mergesort(X1s, Y1s, Ls), ','(mergesort(X2s, Y2s, Ls), merge(Y1s, Y2s, Ys, .(H, Ls))))).
split([], [], [], Ls).
split(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) :- split(Xs, Zs, Ys, Ls).
merge([], Xs, Xs, Ls).
merge(Xs, [], Xs, Ls).
merge(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) :- ','(le(X, Y), merge(Xs, .(Y, Ys), Zs, Ls)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) :- ','(gt(X, Y), merge(.(X, Xs), Ys, Zs, Ls)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(0), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(0)).
le(0, 0).

Queries:

mergesort(g,a,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f,f)
split_in: (b,f,f,f)
merge_in: (b,b,f,f)
le_in: (b,b)
gt_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

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)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
mergesort_in_gaa(x1, x2, x3) = mergesort_in_gaa(x1)
[] = []
mergesort_out_gaa(x1, x2, x3) = mergesort_out_gaa(x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x7)
split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1)
split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x2, x3)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7) = U5_gaaa(x1, x7)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x7, x8)
U3_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaa(x7, x8)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x7)
merge_in_ggaa(x1, x2, x3, x4) = merge_in_ggaa(x1, x2)
merge_out_ggaa(x1, x2, x3, x4) = merge_out_ggaa(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaa(x1, x2, x3, x4, x8)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x1, x8)
U8_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U8_ggaa(x1, x2, x3, x4, x8)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U9_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_ggaa(x3, x8)
MERGE_IN_GGAA(x1, x2, x3, x4) = MERGE_IN_GGAA(x1, x2)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAA(x1, x2, x3, x4, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x7, x8)
U1_GAA(x1, x2, x3, x4, x5, x6, x7) = U1_GAA(x7)
SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U3_GAA(x7, x8)
U7_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGAA(x1, x8)
U9_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U9_GGAA(x3, x8)
U10_GG(x1, x2, x3) = U10_GG(x3)
MERGESORT_IN_GAA(x1, x2, x3) = MERGESORT_IN_GAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7) = U5_GAAA(x1, x7)
U11_GG(x1, x2, x3) = U11_GG(x3)
U8_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U8_GGAA(x1, x2, x3, x4, x8)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U4_GAA(x1, x2, x3, x4, x5, x6, x7) = U4_GAA(x7)

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

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)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_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:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_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

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_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)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), 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))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
mergesort_in_gaa(x1, x2, x3) = mergesort_in_gaa(x1)
[] = []
mergesort_out_gaa(x1, x2, x3) = mergesort_out_gaa(x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x7)
split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1)
split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x2, x3)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7) = U5_gaaa(x1, x7)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x7, x8)
U3_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaa(x7, x8)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x7)
merge_in_ggaa(x1, x2, x3, x4) = merge_in_ggaa(x1, x2)
merge_out_ggaa(x1, x2, x3, x4) = merge_out_ggaa(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaa(x1, x2, x3, x4, x8)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x1, x8)
U8_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U8_ggaa(x1, x2, x3, x4, x8)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U9_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_ggaa(x3, x8)
MERGE_IN_GGAA(x1, x2, x3, x4) = MERGE_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAA(x1, x2, x3, x4, x8)
U8_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U8_GGAA(x1, x2, x3, x4, x8)

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

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), 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))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
MERGE_IN_GGAA(x1, x2, x3, x4) = MERGE_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAA(x1, x2, x3, x4, x8)
U8_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U8_GGAA(x1, x2, x3, x4, x8)

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

                        ↳ QDPOrderProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, 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))
U8_GGAA(X, Xs, Y, Ys, gt_out_gg) → MERGE_IN_GGAA(.(X, Xs), Ys)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(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.

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)

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))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(MERGE_IN_GGAA(x₁, x₂)) = x₁ + x₂
POL(U10_gg(x₁)) = 1
POL(U11_gg(x₁)) = 0
POL(U6_GGAA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂ + x₄
POL(U8_GGAA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂ + x₄ + x₅
POL(gt_in_gg(x₁, x₂)) = 1
POL(gt_out_gg) = 1
POL(le_in_gg(x₁, x₂)) = 0
POL(le_out_gg) = 0
POL(s(x₁)) = 0

The following usable rules [17] were oriented:

gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
U10_gg(gt_out_gg) → gt_out_gg
gt_in_gg(s(0), 0) → gt_out_gg

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, Y))
U6_GGAA(X, Xs, Y, Ys, le_out_gg) → MERGE_IN_GGAA(Xs, .(Y, Ys))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0)
U10_gg(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs)) → SPLIT_IN_GAAA(Xs)

From the DPs we obtained the following set of size-change graphs:

SPLIT_IN_GAAA(.(X, Xs)) → SPLIT_IN_GAAA(Xs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

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))) → 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)
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → MERGESORT_IN_GAA(X1s, Y1s, Ls)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
mergesort_in_gaa(x1, x2, x3) = mergesort_in_gaa(x1)
[] = []
mergesort_out_gaa(x1, x2, x3) = mergesort_out_gaa(x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x7)
split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1)
split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x2, x3)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7) = U5_gaaa(x1, x7)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x7, x8)
U3_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaa(x7, x8)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x7)
merge_in_ggaa(x1, x2, x3, x4) = merge_in_ggaa(x1, x2)
merge_out_ggaa(x1, x2, x3, x4) = merge_out_ggaa(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaa(x1, x2, x3, x4, x8)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x1, x8)
U8_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U8_ggaa(x1, x2, x3, x4, x8)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U9_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_ggaa(x3, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x7, x8)
U1_GAA(x1, x2, x3, x4, x5, x6, x7) = U1_GAA(x7)
MERGESORT_IN_GAA(x1, x2, x3) = MERGESORT_IN_GAA(x1)

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

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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(split_in_gaaa(.(X, .(Y, Xs))))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(split_in_gaaa(.(X, .(Y, Xs)))) at position [0] we obtained the following new rules:

MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, split_in_gaaa(.(Y, Xs))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

U1_GAA(split_out_gaaa(X1s, X2s)) → U2_GAA(X2s, mergesort_in_gaa(X1s))
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, split_in_gaaa(.(Y, Xs))))
U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, split_in_gaaa(.(Y, Xs)))) at position [0,1] we obtained the following new rules:

MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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))))
U2_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

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_GAA(X2s, mergesort_out_gaa(Y1s)) → MERGESORT_IN_GAA(X2s)

The remaining pairs can at least be oriented weakly.

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))))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( le_in_gg(x₁, x₂) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( merge_out_ggaa(x₁) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

M( U5_gaaa(x₁, x₂) ) =

/	1
	0	/

/	1	0
	0	0	/

x₁

/	1	0
	0	0	/

x₂

M( U4_gaa(x₁) ) =

/	1
	0	/

/	0	0
	0	0	/

x₁

M( [] ) =

/	0
	1	/

M( 0 ) =

/	0
	0	/

M( U9_ggaa(x₁, x₂) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( U3_gaa(x₁, x₂) ) =

/	1
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( s(x₁) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

M( U10_gg(x₁) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

M( U11_gg(x₁) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

M( mergesort_out_gaa(x₁) ) =

/	1
	0	/

/	0	0
	0	0	/

x₁

M( gt_in_gg(x₁, x₂) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( merge_in_ggaa(x₁, x₂) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( le_out_gg ) =

/	0
	0	/

M( U1_gaa(x₁) ) =

/	1
	0	/

/	0	0
	0	0	/

x₁

M( split_out_gaaa(x₁, x₂) ) =

/	0
	0	/

/	1	0
	0	1	/

x₁

/	1	0
	0	0	/

x₂

M( .(x₁, x₂) ) =

/	1
	0	/

/	1	0
	0	0	/

x₁

/	1	0
	0	0	/

x₂

M( U6_ggaa(x₁, ..., x₅) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

/	0	0
	0	0	/

x₃

/	0	0
	0	0	/

x₄

/	0	0
	0	0	/

x₅

M( U7_ggaa(x₁, x₂) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( mergesort_in_gaa(x₁) ) =

/	0
	0	/

/	1	1
	0	0	/

x₁

M( U8_ggaa(x₁, ..., x₅) ) =

/	0
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

/	0	0
	0	0	/

x₃

/	0	0
	0	0	/

x₄

/	0	0
	0	0	/

x₅

M( split_in_gaaa(x₁) ) =

/	0
	0	/

/	1	0
	0	1	/

x₁

M( U2_gaa(x₁, x₂) ) =

/	1
	0	/

/	0	0
	0	0	/

x₁

/	0	0
	0	0	/

x₂

M( gt_out_gg ) =

/	0
	0	/

Tuple symbols:

M( U1_GAA(x₁) ) =

[

]

x₁

M( MERGESORT_IN_GAA(x₁) ) =

[

]

x₁

M( U2_GAA(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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:

split_in_gaaa([]) → split_out_gaaa([], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []))
U1_gaa(split_out_gaaa(X1s, X2s)) → U2_gaa(X2s, mergesort_in_gaa(X1s))
U3_gaa(Y1s, mergesort_out_gaa(Y2s)) → U4_gaa(merge_in_ggaa(Y1s, Y2s))
U2_gaa(X2s, mergesort_out_gaa(Y1s)) → U3_gaa(Y1s, mergesort_in_gaa(X2s))
U4_gaa(merge_out_ggaa(Ys)) → mergesort_out_gaa(Ys)
mergesort_in_gaa([]) → mergesort_out_gaa([])
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(split_in_gaaa(.(X, .(Y, Xs))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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))))

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

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

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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))))

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

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

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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))))

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

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)

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.

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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

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))))

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

split_in_gaaa(x0)
U5_gaaa(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.

MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(U5_gaaa(X, U5_gaaa(Y, split_in_gaaa(Xs))))

The remaining pairs can at least be oriented weakly.

U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U5_gaaa(x₁, x₂) ) =

/	1
	0	/

/	1	0
	1	0	/

x₁

/	0	1
	1	0	/

x₂

M( split_out_gaaa(x₁, x₂) ) =

/	0
	0	/

/	0	1
	0	0	/

x₁

/	0	0
	0	1	/

x₂

M( [] ) =

/	0
	0	/

M( split_in_gaaa(x₁) ) =

/	0
	0	/

/	0	1
	0	1	/

x₁

M( .(x₁, x₂) ) =

/	0
	1	/

/	0	0
	1	0	/

x₁

/	0	0
	0	1	/

x₂

Tuple symbols:

M( U1_GAA(x₁) ) =

[

]

x₁

M( MERGESORT_IN_GAA(x₁) ) =

[

]

x₁

split_in_gaaa(.(X, Xs)) → U5_gaaa(X, split_in_gaaa(Xs))
split_in_gaaa([]) → split_out_gaaa([], [])
U5_gaaa(X, split_out_gaaa(Zs, Ys)) → split_out_gaaa(.(X, Ys), Zs)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ Rewriting

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ QReductionProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

U1_GAA(split_out_gaaa(X1s, X2s)) → MERGESORT_IN_GAA(X1s)

The TRS R consists of the following rules:

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)

The set Q consists of the following terms:

split_in_gaaa(x0)
U5_gaaa(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.
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f,f)
split_in: (b,f,f,f)
merge_in: (b,b,f,f)
le_in: (b,b)
gt_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

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)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
mergesort_in_gaa(x1, x2, x3) = mergesort_in_gaa(x1)
[] = []
mergesort_out_gaa(x1, x2, x3) = mergesort_out_gaa(x1, x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x1, x2, x3, x7)
split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1)
split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x2, x3)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7) = U5_gaaa(x1, x2, x7)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x1, x2, x3, x7, x8)
U3_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaa(x1, x2, x3, x7, x8)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x1, x2, x3, x7)
merge_in_ggaa(x1, x2, x3, x4) = merge_in_ggaa(x1, x2)
merge_out_ggaa(x1, x2, x3, x4) = merge_out_ggaa(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaa(x1, x2, x3, x4, x8)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x1, x2, x3, x4, x8)
U8_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U8_ggaa(x1, x2, x3, x4, x8)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_ggaa(x1, x2, x3, x4, x8)
MERGE_IN_GGAA(x1, x2, x3, x4) = MERGE_IN_GGAA(x1, x2)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAA(x1, x2, x3, x4, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x1, x2, x3, x7, x8)
U1_GAA(x1, x2, x3, x4, x5, x6, x7) = U1_GAA(x1, x2, x3, x7)
SPLIT_IN_GAAA(x1, x2, x3, x4) = SPLIT_IN_GAAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U3_GAA(x1, x2, x3, x7, x8)
U7_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGAA(x1, x2, x3, x4, x8)
U9_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U9_GGAA(x1, x2, x3, x4, x8)
U10_GG(x1, x2, x3) = U10_GG(x1, x2, x3)
MERGESORT_IN_GAA(x1, x2, x3) = MERGESORT_IN_GAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6, x7) = U5_GAAA(x1, x2, x7)
U11_GG(x1, x2, x3) = U11_GG(x1, x2, x3)
U8_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U8_GGAA(x1, x2, x3, x4, x8)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U4_GAA(x1, x2, x3, x4, x5, x6, x7) = U4_GAA(x1, x2, x3, x7)

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

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)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

From the DPs we obtained the following set of size-change graphs:

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

LE_IN_GG(s(X), s(Y)) → LE_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)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), 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))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

U8_GGAA(X, Xs, Y, Ys, Zs, H, Ls, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), 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))
U6_GGAA(X, Xs, Y, Ys, Zs, H, Ls, le_out_gg(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys), Zs, Ls)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
MERGE_IN_GGAA(x1, x2, x3, x4) = MERGE_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAA(x1, x2, x3, x4, x8)
U8_GGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U8_GGAA(x1, x2, x3, x4, x8)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, Y))
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(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys))
U8_GGAA(X, Xs, Y, Ys, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys)

The TRS R consists of the following rules:

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))

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MERGE_IN_GGAA(.(X, Xs), .(Y, Ys)) → U8_GGAA(X, Xs, Y, Ys, gt_in_gg(X, Y))
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(X, Y)) → MERGE_IN_GGAA(Xs, .(Y, Ys))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(0) = 0
POL(MERGE_IN_GGAA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U10_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U11_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U6_GGAA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅
POL(U8_GGAA(x₁, x₂, x₃, x₄, x₅)) = 2 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅
POL(gt_in_gg(x₁, x₂)) = x₁ + x₂
POL(gt_out_gg(x₁, x₂)) = x₁ + x₂
POL(le_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(le_out_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

U8_GGAA(X, Xs, Y, Ys, gt_out_gg(X, Y)) → MERGE_IN_GGAA(.(X, Xs), Ys)

The TRS R consists of the following rules:

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))

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U10_gg(x0, x1, x2)

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.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)

The TRS R consists of the following rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) → SPLIT_IN_GAAA(Xs, Zs, Ys, Ls)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

SPLIT_IN_GAAA(.(X, Xs)) → SPLIT_IN_GAAA(Xs)

From the DPs we obtained the following set of size-change graphs:

SPLIT_IN_GAAA(.(X, Xs)) → SPLIT_IN_GAAA(Xs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

Pi DP problem:
The TRS P consists of the following rules:

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))) → 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)
U1_GAA(X, Y, Xs, Ys, H, Ls, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls))) → MERGESORT_IN_GAA(X1s, Y1s, Ls)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
mergesort_in_gaa(x1, x2, x3) = mergesort_in_gaa(x1)
[] = []
mergesort_out_gaa(x1, x2, x3) = mergesort_out_gaa(x1, x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x1, x2, x3, x7)
split_in_gaaa(x1, x2, x3, x4) = split_in_gaaa(x1)
split_out_gaaa(x1, x2, x3, x4) = split_out_gaaa(x1, x2, x3)
U5_gaaa(x1, x2, x3, x4, x5, x6, x7) = U5_gaaa(x1, x2, x7)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x1, x2, x3, x7, x8)
U3_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U3_gaa(x1, x2, x3, x7, x8)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x1, x2, x3, x7)
merge_in_ggaa(x1, x2, x3, x4) = merge_in_ggaa(x1, x2)
merge_out_ggaa(x1, x2, x3, x4) = merge_out_ggaa(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaa(x1, x2, x3, x4, x8)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U7_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U7_ggaa(x1, x2, x3, x4, x8)
U8_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U8_ggaa(x1, x2, x3, x4, x8)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_ggaa(x1, x2, x3, x4, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x1, x2, x3, x7, x8)
U1_GAA(x1, x2, x3, x4, x5, x6, x7) = U1_GAA(x1, x2, x3, x7)
MERGESORT_IN_GAA(x1, x2, x3) = MERGESORT_IN_GAA(x1)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U2_GAA(X, Y, Xs, X2s, mergesort_out_gaa(X1s, Y1s)) → MERGESORT_IN_GAA(X2s)
U1_GAA(X, Y, Xs, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GAA(X, Y, Xs, X2s, mergesort_in_gaa(X1s))
MERGESORT_IN_GAA(.(X, .(Y, Xs))) → U1_GAA(X, Y, Xs, split_in_gaaa(.(X, .(Y, Xs))))
U1_GAA(X, Y, Xs, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GAA(X1s)

The TRS R consists of the following rules:

mergesort_in_gaa([]) → mergesort_out_gaa([], [])
mergesort_in_gaa(.(X, [])) → mergesort_out_gaa(.(X, []), .(X, []))
mergesort_in_gaa(.(X, .(Y, Xs))) → U1_gaa(X, Y, Xs, split_in_gaaa(.(X, .(Y, Xs))))
split_in_gaaa([]) → split_out_gaaa([], [], [])
split_in_gaaa(.(X, Xs)) → U5_gaaa(X, Xs, split_in_gaaa(Xs))
U5_gaaa(X, Xs, split_out_gaaa(Xs, Zs, Ys)) → split_out_gaaa(.(X, Xs), .(X, Ys), Zs)
U1_gaa(X, Y, Xs, split_out_gaaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_gaa(X, Y, Xs, X2s, mergesort_in_gaa(X1s))
U2_gaa(X, Y, Xs, X2s, mergesort_out_gaa(X1s, Y1s)) → U3_gaa(X, Y, Xs, Y1s, mergesort_in_gaa(X2s))
U3_gaa(X, Y, Xs, Y1s, mergesort_out_gaa(X2s, Y2s)) → U4_gaa(X, Y, Xs, merge_in_ggaa(Y1s, Y2s))
merge_in_ggaa([], Xs) → merge_out_ggaa([], Xs, Xs)
merge_in_ggaa(Xs, []) → merge_out_ggaa(Xs, [], 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(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, le_out_gg(X, Y)) → U7_ggaa(X, Xs, Y, Ys, 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(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, gt_out_gg(X, Y)) → U9_ggaa(X, Xs, Y, Ys, merge_in_ggaa(.(X, Xs), Ys))
U9_ggaa(X, Xs, Y, Ys, merge_out_ggaa(.(X, Xs), Ys, Zs)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(Y, Zs))
U7_ggaa(X, Xs, Y, Ys, merge_out_ggaa(Xs, .(Y, Ys), Zs)) → merge_out_ggaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_gaa(X, Y, Xs, merge_out_ggaa(Y1s, Y2s, Ys)) → mergesort_out_gaa(.(X, .(Y, Xs)), Ys)

The set Q consists of the following terms:

mergesort_in_gaa(x0)
split_in_gaaa(x0)
U5_gaaa(x0, x1, x2)
U1_gaa(x0, x1, x2, x3)
U2_gaa(x0, x1, x2, x3, x4)
U3_gaa(x0, x1, x2, x3, x4)
merge_in_ggaa(x0, x1)
le_in_gg(x0, x1)
U11_gg(x0, x1, x2)
U6_ggaa(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U10_gg(x0, x1, x2)
U8_ggaa(x0, x1, x2, x3, x4)
U9_ggaa(x0, x1, x2, x3, x4)
U7_ggaa(x0, x1, x2, x3, x4)
U4_gaa(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.