Left Termination proof of /tmp/tmp1

Left Termination of the query pattern mergesort_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

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)).
=(X, X).

Queries:

mergesort(a,g).

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

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

      ↳ PrologToPiTRSProof

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

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag
.(x1, x2) = .(x1, x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6)
mergesort_in_aa(x1, x2) = mergesort_in_aa
mergesort_out_aa(x1, x2) = mergesort_out_aa
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5)
merge_in_aaa(x1, x2, x3) = merge_in_aaa
merge_out_aaa(x1, x2, x3) = merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5)
merge_in_aag(x1, x2, x3) = merge_in_aag(x3)
merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x2)
U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x3, x6)
MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
=_IN_AA(x1, x2) = =_IN_AA
MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3)
U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6)
MERGESORT_IN_AA(x1, x2) = MERGESORT_IN_AA
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6)
U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6)
U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x3, x6)
U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5)
=_IN_GA(x1, x2) = =_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6)
U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5)

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

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

      ↳ PrologToPiTRSProof

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

MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

=_in_ga(X, X) → =_out_ga(X, X)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x2)
MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, =_in_ga(X))
U6_AAG(X, Zs, =_out_ga(Y)) → MERGE_IN_AAG(Zs)

The TRS R consists of the following rules:

=_in_ga(X) → =_out_ga(X)

The set Q consists of the following terms:

=_in_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

U6_AAG(X, Zs, =_out_ga(Y)) → MERGE_IN_AAG(Zs)
The graph contains the following edges 2 >= 1
MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, =_in_ga(X))
The graph contains the following edges 1 > 1, 1 > 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

=_in_aa(X, X) → =_out_aa(X, X)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA
U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_in_aa)

The TRS R consists of the following rules:

=_in_aa → =_out_aa

The set Q consists of the following terms:

=_in_aa

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGE_IN_AAA → U6_AAA(=_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U6_AAA(=_out_aa)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

The TRS R consists of the following rules:

=_in_aa → =_out_aa

The set Q consists of the following terms:

=_in_aa

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

R is empty.
The set Q consists of the following terms:

=_in_aa

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

=_in_aa

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                                        ↳ NonTerminationProof

                  ↳ PiDP

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

The TRS R consists of the following rules:none

s = MERGE_IN_AAA evaluates to t =MERGE_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

MERGE_IN_AAA → U6_AAA(=_out_aa)
with rule MERGE_IN_AAA → U6_AAA(=_out_aa) at position [] and matcher [ ]

U6_AAA(=_out_aa) → MERGE_IN_AAA
with rule U6_AAA(=_out_aa) → MERGE_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ NonTerminationProof

                  ↳ PiDP

      ↳ PrologToPiTRSProof

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

SPLIT_IN_AAA → SPLIT_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

SPLIT_IN_AAA → SPLIT_IN_AAA

The TRS R consists of the following rules:none

s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SPLIT_IN_AAA to SPLIT_IN_AAA.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

      ↳ PrologToPiTRSProof

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

MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

      ↳ PrologToPiTRSProof

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

MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)

The TRS R consists of the following rules:

split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
=_in_aa(X, X) → =_out_aa(X, X)
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
mergesort_in_aa(x1, x2) = mergesort_in_aa
mergesort_out_aa(x1, x2) = mergesort_out_aa
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5)
merge_in_aaa(x1, x2, x3) = merge_in_aaa
merge_out_aaa(x1, x2, x3) = merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6)
MERGESORT_IN_AA(x1, x2) = MERGESORT_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

      ↳ PrologToPiTRSProof

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

U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa)
U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_in_aaa)

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGESORT_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules:

MERGESORT_IN_AA → U1_AA(split_out_aaa)
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

      ↳ PrologToPiTRSProof

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

U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa)
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa) at position [0] we obtained the following new rules:

U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ NonTerminationProof

      ↳ PrologToPiTRSProof

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

U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

s = U1_AA(split_out_aaa) evaluates to t =U1_AA(split_out_aaa)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U1_AA(split_out_aaa) → MERGESORT_IN_AA
with rule U1_AA(split_out_aaa) → MERGESORT_IN_AA at position [] and matcher [ ]

MERGESORT_IN_AA → U1_AA(split_out_aaa)
with rule MERGESORT_IN_AA → U1_AA(split_out_aaa)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

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

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

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

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x2)
.(x1, x2) = .(x1, x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6)
mergesort_in_aa(x1, x2) = mergesort_in_aa
mergesort_out_aa(x1, x2) = mergesort_out_aa
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6)
U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5)
merge_in_aaa(x1, x2, x3) = merge_in_aaa
merge_out_aaa(x1, x2, x3) = merge_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5)
merge_in_aag(x1, x2, x3) = merge_in_aag(x3)
merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x1, x2)
U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x3, x5, x6)
MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
=_IN_AA(x1, x2) = =_IN_AA
MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3)
U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6)
MERGESORT_IN_AA(x1, x2) = MERGESORT_IN_AA
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6)
U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6)
U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x3, x5, x6)
U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x4, x5)
=_IN_GA(x1, x2) = =_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6)
U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5)

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

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

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

MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AA(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U3_AA(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAA(Y1s, Y2s, Ys)
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → MERGE_IN_AAG(Y1s, Y2s, Ys)
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_GA(X, Y)
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAG(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
U6_AAG(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

=_in_ga(X, X) → =_out_ga(X, X)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x1, x2)
MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

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

U6_AAG(X, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(Zs)
MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, =_in_ga(X))

The TRS R consists of the following rules:

=_in_ga(X) → =_out_ga(X, X)

The set Q consists of the following terms:

=_in_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

U6_AAG(X, Zs, =_out_ga(X, Y)) → MERGE_IN_AAG(Zs)
The graph contains the following edges 2 >= 1
MERGE_IN_AAG(.(X, Zs)) → U6_AAG(X, Zs, =_in_ga(X))
The graph contains the following edges 1 > 1, 1 > 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

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

MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

                  ↳ PiDP

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

MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_AAA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_AAA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_AAA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

=_in_aa(X, X) → =_out_aa(X, X)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                  ↳ PiDP

                  ↳ PiDP

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_in_aa)

The TRS R consists of the following rules:

=_in_aa → =_out_aa

The set Q consists of the following terms:

=_in_aa

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGE_IN_AAA → U6_AAA(=_in_aa) at position [0] we obtained the following new rules:

MERGE_IN_AAA → U6_AAA(=_out_aa)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                  ↳ PiDP

                  ↳ PiDP

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

The TRS R consists of the following rules:

=_in_aa → =_out_aa

The set Q consists of the following terms:

=_in_aa

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                  ↳ PiDP

                  ↳ PiDP

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

R is empty.
The set Q consists of the following terms:

=_in_aa

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

=_in_aa

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                                        ↳ NonTerminationProof

                  ↳ PiDP

                  ↳ PiDP

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

U6_AAA(=_out_aa) → MERGE_IN_AAA
MERGE_IN_AAA → U6_AAA(=_out_aa)

The TRS R consists of the following rules:none

s = MERGE_IN_AAA evaluates to t =MERGE_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                  ↳ PiDP

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

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

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                  ↳ PiDP

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

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

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ NonTerminationProof

                  ↳ PiDP

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

SPLIT_IN_AAA → SPLIT_IN_AAA

The TRS R consists of the following rules:none

s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SPLIT_IN_AAA to SPLIT_IN_AAA.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

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

MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)

The TRS R consists of the following rules:

mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys))
merge_in_aag([], Xs, Xs) → merge_out_aag([], Xs, Xs)
merge_in_aag(Xs, [], Xs) → merge_out_aag(Xs, [], Xs)
merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aag(X, Xs, Y, Ys, Zs, =_in_ga(X, Y))
=_in_ga(X, X) → =_out_ga(X, X)
U6_aag(X, Xs, Y, Ys, Zs, =_out_ga(X, Y)) → U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs))
U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) → merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

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

MERGESORT_IN_AA(.(X, .(Y, Xs)), Ys) → U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AA(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_AA(X1s, Y1s)
U2_AA(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → MERGESORT_IN_AA(X2s, Y2s)

The TRS R consists of the following rules:

split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
mergesort_in_aa([], []) → mergesort_out_aa([], [])
mergesort_in_aa(.(X, []), .(X, [])) → mergesort_out_aa(.(X, []), .(X, []))
mergesort_in_aa(.(X, .(Y, Xs)), Ys) → U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_aa(X, Y, Xs, Ys, X2s, mergesort_in_aa(X1s, Y1s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
U2_aa(X, Y, Xs, Ys, X2s, mergesort_out_aa(X1s, Y1s)) → U3_aa(X, Y, Xs, Ys, Y1s, mergesort_in_aa(X2s, Y2s))
U3_aa(X, Y, Xs, Ys, Y1s, mergesort_out_aa(X2s, Y2s)) → U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys))
U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) → mergesort_out_aa(.(X, .(Y, Xs)), Ys)
merge_in_aaa([], Xs, Xs) → merge_out_aaa([], Xs, Xs)
merge_in_aaa(Xs, [], Xs) → merge_out_aaa(Xs, [], Xs)
merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_aaa(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_aaa(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs))
=_in_aa(X, X) → =_out_aa(X, X)
U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) → merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs))

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

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

U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa)
U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_in_aaa)

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGESORT_IN_AA → U1_AA(split_in_aaa) at position [0] we obtained the following new rules:

MERGESORT_IN_AA → U1_AA(split_out_aaa)
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

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

U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa)
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(split_out_aaa) → U2_AA(mergesort_in_aa) at position [0] we obtained the following new rules:

U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ PrologToPiTRSProof

      ↳ PrologToPiTRSProof

        ↳ PiTRS

          ↳ DependencyPairsProof

            ↳ PiDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                  ↳ PiDP

                    ↳ UsableRulesProof

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ NonTerminationProof

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

U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

The set Q consists of the following terms:

split_in_aaa
mergesort_in_aa
U5_aaa(x0)
U1_aa(x0)
U2_aa(x0)
U3_aa(x0)
U4_aa(x0)
merge_in_aaa
U6_aaa(x0)
=_in_aa
U7_aaa(x0)

U2_AA(mergesort_out_aa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(split_out_aaa)
U1_AA(split_out_aaa) → U2_AA(mergesort_out_aa)
U1_AA(split_out_aaa) → U2_AA(U1_aa(split_in_aaa))
U1_AA(split_out_aaa) → MERGESORT_IN_AA
MERGESORT_IN_AA → U1_AA(U5_aaa(split_in_aaa))

The TRS R consists of the following rules:

split_in_aaa → U5_aaa(split_in_aaa)
mergesort_in_aa → mergesort_out_aa
mergesort_in_aa → U1_aa(split_in_aaa)
U5_aaa(split_out_aaa) → split_out_aaa
U1_aa(split_out_aaa) → U2_aa(mergesort_in_aa)
split_in_aaa → split_out_aaa
U2_aa(mergesort_out_aa) → U3_aa(mergesort_in_aa)
U3_aa(mergesort_out_aa) → U4_aa(merge_in_aaa)
U4_aa(merge_out_aaa) → mergesort_out_aa
merge_in_aaa → merge_out_aaa
merge_in_aaa → U6_aaa(=_in_aa)
U6_aaa(=_out_aa) → U7_aaa(merge_in_aaa)
=_in_aa → =_out_aa
U7_aaa(merge_out_aaa) → merge_out_aaa

s = U1_AA(split_out_aaa) evaluates to t =U1_AA(split_out_aaa)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]