Left Termination proof of /tmp/tmpUP8F3O/mergesort

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

0 Prolog

↳1 PrologToPiTRSProof (⇒, 59 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 186 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 38 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 MRRProof (⇔, 203 ms)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔, 0 ms)

        ↳29 QDP

        ↳30 UsableRulesProof (⇔, 0 ms)

        ↳31 QDP

        ↳32 QReductionProof (⇔, 8 ms)

        ↳33 QDP

        ↳34 QDPSizeChangeProof (⇔, 0 ms)

        ↳35 YES

    ↳36 PiDP

        ↳37 UsableRulesProof (⇔, 0 ms)

        ↳38 PiDP

        ↳39 PiDPToQDPProof (⇒, 0 ms)

        ↳40 QDP

        ↳41 QDPSizeChangeProof (⇔, 0 ms)

        ↳42 YES

    ↳43 PiDP

        ↳44 PiDPToQDPProof (⇒, 0 ms)

        ↳45 QDP

        ↳46 QDPOrderProof (⇔, 1300 ms)

        ↳47 QDP

        ↳48 DependencyGraphProof (⇔, 0 ms)

        ↳49 TRUE

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2)))).
split(L1, L2, L3) :- split0(L1, L2, L3).
split(L1, L2, L3) :- split1(L1, L2, L3).
split(L1, L2, L3) :- split2(L1, L2, L3).
split0([], [], []).
split1(.(X, []), .(X, []), []).
split2(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) :- split(L1, L2, L3).
merge([], L1, L1).
merge(L1, [], L1).
merge(.(X, L1), .(Y, L2), .(X, L3)) :- ','(le(X, Y), merge(L1, .(Y, L2), L3)).
merge(.(X, L1), .(Y, L2), .(Y, L3)) :- ','(gt(X, Y), merge(.(X, L1), L2, L3)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Query: mergesort(g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f)
split2_in: (b,f,f)
split_in: (b,f,f)
merge_in: (b,b,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_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5)
split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4)
split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1)
split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4)
split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1)
split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6)
U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5)
SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x6)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4)
SPLIT0_IN_GAA(x1, x2, x3) = SPLIT0_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4)
SPLIT1_IN_GAA(x1, x2, x3) = SPLIT1_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x1, x2, x3, x5, x6)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U14_GG(x1, x2, x3) = U14_GG(x1, x2, x3)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2)
U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3)
U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4)
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT0_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT1_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3))
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → MERGE_IN_GGA(L5, L6, L2)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U14_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U13_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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

(13) YES

(14) Obligation:

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_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

(20) YES

(21) Obligation:

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

U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2), L3)
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) → U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) → U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2, L3)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x1, x2, x3) = U14_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)
U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3)
gt_out_gg(x1, x2) = gt_out_gg(x1, x2)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))
U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2)

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(26) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) → MERGE_IN_GGA(.(X, L1), L2)

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂
POL(0) = 2
POL(MERGE_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U11_GGA(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U13_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U14_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U9_GGA(x₁, x₂, x₃, x₄, x₅)) = x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅
POL(gt_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(gt_out_gg(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(le_in_gg(x₁, x₂)) = x₁ + x₂
POL(le_out_gg(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 2·x₁

(27) Obligation:

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

U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(29) Obligation:

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

MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U13_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)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(30) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(31) Obligation:

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

MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U14_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(32) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)

(33) Obligation:

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

MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

le_in_gg(x0, x1)
U14_gg(x0, x1, x2)

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

(34) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) → MERGE_IN_GGA(L1, .(Y, L2))
The graph contains the following edges 2 >= 1
MERGE_IN_GGA(.(X, L1), .(Y, L2)) → U9_GGA(X, L1, Y, L2, le_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4

(35) YES

(36) Obligation:

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

SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

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

SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → SPLIT_IN_GAA(L1, L2, L3)
SPLIT_IN_GAA(L1, L2, L3) → SPLIT2_IN_GAA(L1, L2, L3)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)

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

(39) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(40) Obligation:

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

SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
SPLIT_IN_GAA(L1) → SPLIT2_IN_GAA(L1)

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

(41) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

SPLIT_IN_GAA(L1) → SPLIT2_IN_GAA(L1)
The graph contains the following edges 1 >= 1
SPLIT2_IN_GAA(.(X, .(Y, L1))) → SPLIT_IN_GAA(L1)
The graph contains the following edges 1 > 1

(42) YES

(43) Obligation:

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

U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4, L6)
MERGESORT_IN_GA(.(X, .(Y, L1)), L2) → U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3, L5)

The TRS R consists of the following rules:

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1)), L2) → U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4))
split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) → U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3))
split_in_gaa(L1, L2, L3) → U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3))
split0_in_gaa([], [], []) → split0_out_gaa([], [], [])
U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3))
split1_in_gaa(.(X, []), .(X, []), []) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1, L2, L3) → U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3))
U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5))
U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6))
U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2))
merge_in_gga([], L1, L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, [], L1) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) → U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3))
merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) → U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3))
U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

(44) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(45) Obligation:

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

U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L4, mergesort_in_ga(L3))
U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4)
MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3)

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L4, mergesort_in_ga(L3))
U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L5, mergesort_in_ga(L4))
U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, []) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2))
U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0, x1)
split1_in_gaa(x0)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1, x2, x3)
U1_ga(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3, x4)
U3_ga(x0, x1, x2, x3, x4)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0, x1, x2)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1, x2, x3, x4)
U10_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)

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

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MERGESORT_IN_GA(.(X, .(Y, L1))) → U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U2_GA(x₁, ..., x₅) ) = 2x₄ + 2

POL( mergesort_in_ga(x₁) ) = 0

POL( [] ) = 0

POL( mergesort_out_ga(x₁, x₂) ) = max{0, -2}

POL( .(x₁, x₂) ) = 2x₂ + 1

POL( U1_ga(x₁, ..., x₄) ) = 2x₁ + 2x₂ + x₄ + 1

POL( split2_in_gaa(x₁) ) = max{0, x₁ - 1}

POL( U1_GA(x₁, ..., x₄) ) = 2x₄ + 2

POL( U8_gaa(x₁, ..., x₄) ) = x₄ + 1

POL( split_in_gaa(x₁) ) = 2x₁ + 1

POL( split2_out_gaa(x₁, ..., x₃) ) = x₂ + x₃

POL( U2_ga(x₁, ..., x₅) ) = x₁ + x₂ + 2x₃ + 2x₄ + 1

POL( U3_ga(x₁, ..., x₅) ) = max{0, x₄ + x₅ - 2}

POL( U4_ga(x₁, ..., x₄) ) = max{0, 2x₁ + 2x₂ + 2x₃ + 2x₄ - 1}

POL( merge_in_gga(x₁, x₂) ) = x₁ + 2

POL( U7_gaa(x₁, x₂) ) = 2x₂ + 1

POL( split_out_gaa(x₁, ..., x₃) ) = 2x₂ + 2x₃ + 1

POL( U5_gaa(x₁, x₂) ) = x₂ + 1

POL( split0_in_gaa(x₁) ) = 0

POL( U6_gaa(x₁, x₂) ) = 2x₂ + 1

POL( split1_in_gaa(x₁) ) = x₁

POL( split0_out_gaa(x₁, ..., x₃) ) = x₁ + 2x₂ + 2x₃

POL( split1_out_gaa(x₁, ..., x₃) ) = x₂ + 2x₃

POL( merge_out_gga(x₁, ..., x₃) ) = x₃ + 2

POL( U9_gga(x₁, ..., x₅) ) = max{0, -1}

POL( le_in_gg(x₁, x₂) ) = 0

POL( U11_gga(x₁, ..., x₅) ) = max{0, x₁ + 2x₃ + 2x₄ - 2}

POL( gt_in_gg(x₁, x₂) ) = 2x₁ + 2

POL( s(x₁) ) = 0

POL( U14_gg(x₁, ..., x₃) ) = max{0, 2x₁ + x₂ - 2}

POL( 0 ) = 2

POL( le_out_gg(x₁, x₂) ) = max{0, 2x₁ + 2x₂ - 1}

POL( U10_gga(x₁, ..., x₅) ) = 2x₁ + x₂ + 2x₅ + 2

POL( U13_gg(x₁, ..., x₃) ) = max{0, 2x₁ + 2x₂ - 2}

POL( gt_out_gg(x₁, x₂) ) = max{0, x₁ - 2}

POL( U12_gga(x₁, ..., x₅) ) = 2x₁ + x₃ + 2

POL( MERGESORT_IN_GA(x₁) ) = 2x₁ + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)

(47) Obligation:

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

U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_GA(X, Y, L1, L4, mergesort_in_ga(L3))
U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → MERGESORT_IN_GA(L4)
U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → MERGESORT_IN_GA(L3)

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, L1))) → U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1))))
split2_in_gaa(.(X, .(Y, L1))) → U8_gaa(X, Y, L1, split_in_gaa(L1))
split_in_gaa(L1) → U5_gaa(L1, split0_in_gaa(L1))
split0_in_gaa([]) → split0_out_gaa([], [], [])
U5_gaa(L1, split0_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U6_gaa(L1, split1_in_gaa(L1))
split1_in_gaa(.(X, [])) → split1_out_gaa(.(X, []), .(X, []), [])
U6_gaa(L1, split1_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
split_in_gaa(L1) → U7_gaa(L1, split2_in_gaa(L1))
U7_gaa(L1, split2_out_gaa(L1, L2, L3)) → split_out_gaa(L1, L2, L3)
U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) → split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3))
U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) → U2_ga(X, Y, L1, L4, mergesort_in_ga(L3))
U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) → U3_ga(X, Y, L1, L5, mergesort_in_ga(L4))
U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) → U4_ga(X, Y, L1, merge_in_gga(L5, L6))
merge_in_gga([], L1) → merge_out_gga([], L1, L1)
merge_in_gga(L1, []) → merge_out_gga(L1, [], L1)
merge_in_gga(.(X, L1), .(Y, L2)) → U9_gga(X, L1, Y, L2, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U14_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U14_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) → U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2)))
merge_in_gga(.(X, L1), .(Y, L2)) → U11_gga(X, L1, Y, L2, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U13_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U13_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) → U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2))
U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3))
U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) → merge_out_gga(.(X, L1), .(Y, L2), .(X, L3))
U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) → mergesort_out_ga(.(X, .(Y, L1)), L2)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split2_in_gaa(x0)
split_in_gaa(x0)
split0_in_gaa(x0)
U5_gaa(x0, x1)
split1_in_gaa(x0)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1, x2, x3)
U1_ga(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3, x4)
U3_ga(x0, x1, x2, x3, x4)
merge_in_gga(x0, x1)
le_in_gg(x0, x1)
U14_gg(x0, x1, x2)
U9_gga(x0, x1, x2, x3, x4)
gt_in_gg(x0, x1)
U13_gg(x0, x1, x2)
U11_gga(x0, x1, x2, x3, x4)
U12_gga(x0, x1, x2, x3, x4)
U10_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(0) Obligation:

(1) PrologToPiTRSProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) MRRProof (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) Obligation:

(30) UsableRulesProof (EQUIVALENT transformation)

(31) Obligation:

(32) QReductionProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPSizeChangeProof (EQUIVALENT transformation)

(35) YES

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) YES

(43) Obligation:

(44) PiDPToQDPProof (SOUND transformation)

(45) Obligation:

(46) QDPOrderProof (EQUIVALENT transformation)

(47) Obligation:

(48) DependencyGraphProof (EQUIVALENT transformation)

(49) TRUE