Left Termination proof of /tmp/tmpaShAQj/mergesort.pl

Left Termination of the query pattern mergesort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PredefinedPredicateTransformerProof (⇐)

↳2 Prolog

  ↳3 PrologToPiTRSProof (⇐)

    ↳4 PiTRS

    ↳5 DependencyPairsProof (⇔)

    ↳6 PiDP

    ↳7 DependencyGraphProof (⇔)

    ↳8 AND

      ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 TRUE

      ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 TRUE

      ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔)

        ↳29 QDP

        ↳30 UsableRulesProof (⇔)

        ↳31 QDP

        ↳32 QReductionProof (⇔)

        ↳33 QDP

        ↳34 QDPOrderProof (⇔)

        ↳35 QDP

        ↳36 DependencyGraphProof (⇔)

        ↳37 TRUE

  ↳38 PrologToPiTRSProof (⇐)

    ↳39 PiTRS

    ↳40 DependencyPairsProof (⇔)

    ↳41 PiDP

    ↳42 DependencyGraphProof (⇔)

    ↳43 AND

      ↳44 PiDP

        ↳45 UsableRulesProof (⇔)

        ↳46 PiDP

        ↳47 PiDPToQDPProof (⇐)

        ↳48 QDP

        ↳49 QDPSizeChangeProof (⇔)

        ↳50 TRUE

      ↳51 PiDP

        ↳52 UsableRulesProof (⇔)

        ↳53 PiDP

        ↳54 PiDPToQDPProof (⇐)

        ↳55 QDP

        ↳56 QDPSizeChangeProof (⇔)

        ↳57 TRUE

      ↳58 PiDP

        ↳59 PiDPToQDPProof (⇐)

        ↳60 QDP

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)).

Queries:

mergesort(g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f)
split_in: (b,f,f)
merge_in: (b,b,f)
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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(5) 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, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GG(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

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(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x5, x6)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x4, x6)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x6)

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

(6) Obligation:

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

MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GG(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg(X, X)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x4, x6)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

U6_GGA(X, Xs, Ys, =_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Ys, =_in_gg(X, Y))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg

The set Q consists of the following terms:

=_in_gg(x0, x1)

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

(14) 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:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Ys, =_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3
U6_GGA(X, Xs, Ys, =_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
The graph contains the following edges 3 >= 2

(15) TRUE

(16) Obligation:

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

SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

SPLIT_IN_GAA(.(X, Xs)) → SPLIT_IN_GAA(Xs)

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

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

(22) TRUE

(23) Obligation:

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

U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

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(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, 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:

U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Ys, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg
U6_gga(X, Xs, Ys, =_out_gg) → U7_gga(X, merge_in_gga(.(X, Xs), Ys))
U7_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
U4_ga(x0)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U1_GA(x₁)) = 0 +
[ 0, 1 ]

· x₁

POL(split_out_gaa(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(U2_GA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 1, 0 ]

· x₂

POL(mergesort_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(mergesort_out_ga(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(MERGESORT_IN_GA(x₁)) = 0 +
[ 1, 0 ]

· x₁

POL(.(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(split_in_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁

POL([]) =
/ 0 \

\ 1 /

POL(U1_ga(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U5_gaa(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(U2_ga(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U3_ga(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U4_ga(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(merge_in_gga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(merge_out_gga(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(U6_gga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 1 /

· x₃ +
/ 0 1 \

\ 1 0 /

· x₄

POL(=_in_gg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(=_out_gg) =
/ 0 \

\ 1 /

POL(U7_gga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂

The following usable rules [FROCOS05] were oriented:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)

(27) Obligation:

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

U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Ys, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg
U6_gga(X, Xs, Ys, =_out_gg) → U7_gga(X, merge_in_gga(.(X, Xs), Ys))
U7_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
U4_ga(x0)

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:

U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))

The TRS R consists of the following rules:

mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Ys, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg
U6_gga(X, Xs, Ys, =_out_gg) → U7_gga(X, merge_in_gga(.(X, Xs), Ys))
U7_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
U4_ga(x0)

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:

U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))

The TRS R consists of the following rules:

split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
U4_ga(x0)

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].

mergesort_in_ga(x0)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
U4_ga(x0)

(33) Obligation:

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

U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))

The TRS R consists of the following rules:

split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)

The set Q consists of the following terms:

split_in_gaa(x0)
U5_gaa(x0, x1)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U1_GA(x₁)) = 1 +
[ 0, 1 ]

· x₁

POL(split_out_gaa(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂

POL(MERGESORT_IN_GA(x₁)) = 0 +
[ 1, 1 ]

· x₁

POL(.(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂

POL(split_in_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(U5_gaa(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 1 /

· x₂

POL([]) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)

(35) Obligation:

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

MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))

The TRS R consists of the following rules:

split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)

The set Q consists of the following terms:

split_in_gaa(x0)
U5_gaa(x0, x1)

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

(36) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(37) TRUE

(38) PrologToPiTRSProof (SOUND transformation)

mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(39) 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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(40) 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, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GG(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

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)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x2, x5)
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)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg(x1, x2)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_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)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x2, x5)
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)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)

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

(41) Obligation:

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

MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GG(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(42) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(43) Complex Obligation (AND)

(44) Obligation:

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

U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

U6_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg(X, X)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg(x1, x2)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

U6_GGA(X, Xs, Y, Ys, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_gg(X, Y))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg(X, X)

The set Q consists of the following terms:

=_in_gg(x0, x1)

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

(49) 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:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
U6_GGA(X, Xs, Y, Ys, =_out_gg(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys)
The graph contains the following edges 4 >= 2

(50) TRUE

(51) Obligation:

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

SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)

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

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) Obligation:

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

SPLIT_IN_GAA(.(X, Xs)) → SPLIT_IN_GAA(Xs)

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

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

(57) TRUE

(58) Obligation:

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

U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)

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, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

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)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x2, x5)
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)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg(x1, x2)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_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)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6)

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

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

U1_GA(X, Y, Xs, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, X2s, mergesort_in_ga(X1s))
U2_GA(X, Y, Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(X, Y, Xs, split_in_gaa(.(X, .(Y, Xs))))
U1_GA(X, Y, Xs, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)

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, Xs))) → U1_ga(X, Y, Xs, split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, Xs, split_in_gaa(Xs))
U5_gaa(X, Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, X2s, mergesort_in_ga(X1s))
U2_ga(X, Y, Xs, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Y1s, mergesort_in_ga(X2s))
U3_ga(X, Y, Xs, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_gg(X, Y))
=_in_gg(X, X) → =_out_gg(X, X)
U6_gga(X, Xs, Y, Ys, =_out_gg(X, Y)) → U7_gga(X, Xs, Y, Ys, merge_in_gga(.(X, Xs), Ys))
U7_gga(X, Xs, Y, Ys, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)

The set Q consists of the following terms:

mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1, x2)
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)
=_in_gg(x0, x1)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4)
U4_ga(x0, x1, x2, x3)

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