(0) Obligation:

Clauses:

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

Queries:

mergesort(g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

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(g,a).

(3) BuiltinConflictTransformerProof (SOUND transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(4) Obligation:

Clauses:

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

Queries:

mergesort(g,a).

(5) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(6) Obligation:

Clauses:

split21([], [], []).
split21(.(T42, T43), .(T42, X99), X100) :- split21(T43, X100, X99).
split17(T36, T37, .(T36, X81), X82) :- split21(T37, X82, X81).
mergesort32([], []).
mergesort32(.(T54, []), .(T54, [])).
mergesort32(.(T61, .(T62, T63)), X137) :- split17(T61, .(T62, T63), X133, X134).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), mergesort32(T67, X135)).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, T72), mergesort32(T68, X136))).
mergesort32(.(T61, .(T62, T63)), T80) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, []), mergesort32(T68, T80))).
mergesort32(.(T61, .(T62, T63)), T85) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, T85), mergesort32(T68, []))).
mergesort32(.(T61, .(T62, T63)), .(T104, X185)) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, .(T104, T95)), ','(mergesort32(T68, .(T104, T97)), merge33(.(T104, T95), T97, X185)))).
merge33([], T115, T115).
merge33(T120, [], T120).
merge33(.(T143, T132), .(T143, T134), .(T143, T136)) :- merge33(.(T143, T132), T134, T136).
mergesort1([], []).
mergesort1(.(T4, []), .(T4, [])).
mergesort1(.(T23, .(T24, T25)), T14) :- split17(T24, T25, X52, X51).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), mergesort1(.(T23, T29), X23)).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), ','(mergesort1(.(T23, T29), T46), mergesort32(T28, X24))).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), ','(mergesort1(.(T23, T29), T46), ','(mergesort32(T28, T49), merge33(T46, T49, T14)))).

Queries:

mergesort1(g,a).

(7) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (f,b,f,f)
split21_in: (b,f,f)
mergesort32_in: (b,f) (b,b)
merge33_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:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(8) Obligation:

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

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)

(9) DependencyPairsProof (EQUIVALENT transformation)

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U3_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GG(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GA(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAA(.(T104, T95), T97, X185)
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAA(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_GA(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GG(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAG(.(T104, T95), T97, X185)
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAG(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_GG(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GG(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GG(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GA(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x5)
SPLIT17_IN_AGAA(x1, x2, x3, x4)  =  SPLIT17_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5)  =  U2_AGAA(x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x5)
U22_GA(x1, x2, x3, x4, x5, x6)  =  U22_GA(x5, x6)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x6, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x6)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x5, x6, x7)
U16_GG(x1, x2, x3, x4, x5, x6, x7)  =  U16_GG(x5, x7)
U17_GG(x1, x2, x3, x4, x5, x6)  =  U17_GG(x6)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)
U18_AAG(x1, x2, x3, x4, x5)  =  U18_AAG(x5)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x5, x6)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U24_GA(x1, x2, x3, x4, x5, x6)  =  U24_GA(x6)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)

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

(10) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U3_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GG(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GA(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAA(.(T104, T95), T97, X185)
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAA(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_GA(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GG(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAG(.(T104, T95), T97, X185)
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAG(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_GG(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GG(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GG(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GA(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x5)
SPLIT17_IN_AGAA(x1, x2, x3, x4)  =  SPLIT17_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5)  =  U2_AGAA(x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x5)
U22_GA(x1, x2, x3, x4, x5, x6)  =  U22_GA(x5, x6)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x6, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x6)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x5, x6, x7)
U16_GG(x1, x2, x3, x4, x5, x6, x7)  =  U16_GG(x5, x7)
U17_GG(x1, x2, x3, x4, x5, x6)  =  U17_GG(x6)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)
U18_AAG(x1, x2, x3, x4, x5)  =  U18_AAG(x5)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x5, x6)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U24_GA(x1, x2, x3, x4, x5, x6)  =  U24_GA(x6)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 34 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

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

MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)

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

(14) UsableRulesProof (EQUIVALENT transformation)

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

(15) Obligation:

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

MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)

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

(16) PiDPToQDPProof (SOUND transformation)

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

(17) Obligation:

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

MERGE33_IN_AAG(.(T136)) → MERGE33_IN_AAG(T136)

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

(18) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • MERGE33_IN_AAG(.(T136)) → MERGE33_IN_AAG(T136)
    The graph contains the following edges 1 > 1

(19) YES

(20) Obligation:

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

MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

MERGE33_IN_AAAMERGE33_IN_AAA

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

(25) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MERGE33_IN_AAA evaluates to t =MERGE33_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MERGE33_IN_AAA to MERGE33_IN_AAA.



(26) NO

(27) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

SPLIT21_IN_GAA(.(T43)) → SPLIT21_IN_GAA(T43)

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

(32) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • SPLIT21_IN_GAA(.(T43)) → SPLIT21_IN_GAA(T43)
    The graph contains the following edges 1 > 1

(33) YES

(34) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x6, x7)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x5, x6, x7)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x5, x6)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x6, x7)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x5, x6, x7)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x5, x6)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(X185, split17_in_agaa(.(T63)))
U14_GG(X185, split17_out_agaa(T67, T68)) → U15_GG(X185, T68, mergesort32_in_ga(T67))
U15_GG(X185, T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GG(X185, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.([])) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U3_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(X185, split17_in_agaa(.(T63)))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U3_ga(split17_out_agaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U14_gg(X185, split17_out_agaa(T67, T68)) → U15_gg(X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U15_gg(X185, T68, mergesort32_out_ga) → U16_gg(X185, mergesort32_in_ga(T68))
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
U16_gg(X185, mergesort32_out_ga) → U17_gg(merge33_in_aag(X185))
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U17_gg(merge33_out_aag(.(T95), T97)) → mergesort32_out_gg
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [])
merge33_in_aag(.(T136)) → U18_aag(merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(merge33_out_aag(.(T132), T134)) → merge33_out_aag(.(T132), .(T134))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U3_ga(x0)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
U14_gg(x0, x1)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U15_gg(x0, x1, x2)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U16_gg(x0, x1)
U17_ga(x0)
U17_gg(x0)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0)

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

(39) MRRProof (EQUIVALENT transformation)

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

MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(X185, split17_in_agaa(.(T63)))
U14_GG(X185, split17_out_agaa(T67, T68)) → U15_GG(X185, T68, mergesort32_in_ga(T67))
U15_GG(X185, T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GG(X185, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)

Strictly oriented rules of the TRS R:

mergesort32_in_ga(.([])) → mergesort32_out_ga
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(X185, split17_in_agaa(.(T63)))
U14_gg(X185, split17_out_agaa(T67, T68)) → U15_gg(X185, T68, mergesort32_in_ga(T67))
U15_gg(X185, T68, mergesort32_out_ga) → U16_gg(X185, mergesort32_in_ga(T68))
U16_gg(X185, mergesort32_out_ga) → U17_gg(merge33_in_aag(X185))
U17_gg(merge33_out_aag(.(T95), T97)) → mergesort32_out_gg
merge33_in_aag(T120) → merge33_out_aag(T120, [])

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 5 + x1   
POL(MERGESORT32_IN_GA(x1)) = x1   
POL(MERGESORT32_IN_GG(x1, x2)) = x1 + x2   
POL(U10_ga(x1)) = x1   
POL(U10_gg(x1)) = x1   
POL(U11_GA(x1)) = x1   
POL(U11_GG(x1, x2)) = x1 + x2   
POL(U11_ga(x1)) = x1   
POL(U11_gg(x1, x2)) = x1 + x2   
POL(U12_GA(x1, x2)) = x1 + x2   
POL(U12_GG(x1, x2)) = x1 + x2   
POL(U12_ga(x1, x2)) = x1 + x2   
POL(U12_gg(x1, x2)) = x1 + x2   
POL(U13_ga(x1)) = x1   
POL(U13_gg(x1)) = x1   
POL(U14_GA(x1)) = x1   
POL(U14_GG(x1, x2)) = 2 + x1 + x2   
POL(U14_ga(x1)) = x1   
POL(U14_gg(x1, x2)) = 4 + x1 + x2   
POL(U15_GA(x1, x2)) = x1 + x2   
POL(U15_GG(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U15_ga(x1, x2)) = x1 + x2   
POL(U15_gg(x1, x2, x3)) = 3 + x1 + x2 + x3   
POL(U16_ga(x1)) = x1   
POL(U16_gg(x1, x2)) = 2 + x1 + x2   
POL(U17_ga(x1)) = x1   
POL(U17_gg(x1)) = x1   
POL(U18_aaa(x1)) = x1   
POL(U18_aag(x1)) = 5 + x1   
POL(U1_gaa(x1)) = 5 + x1   
POL(U2_agaa(x1)) = 5 + x1   
POL(U3_ga(x1)) = x1   
POL(U3_gg(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL(U4_GG(x1)) = x1   
POL(U4_ga(x1)) = x1   
POL(U4_gg(x1)) = x1   
POL(U5_ga(x1)) = x1   
POL(U5_gg(x1)) = x1   
POL(U6_GA(x1, x2)) = x1 + x2   
POL(U6_GG(x1, x2)) = x1 + x2   
POL(U6_ga(x1, x2)) = x1 + x2   
POL(U6_gg(x1, x2)) = x1 + x2   
POL(U7_ga(x1)) = x1   
POL(U7_gg(x1)) = x1   
POL(U8_GA(x1)) = x1   
POL(U8_GG(x1, x2)) = x1 + x2   
POL(U8_ga(x1)) = x1   
POL(U8_gg(x1, x2)) = x1 + x2   
POL(U9_GA(x1, x2)) = x1 + x2   
POL(U9_GG(x1, x2, x3)) = x1 + x2 + x3   
POL(U9_ga(x1, x2)) = x1 + x2   
POL(U9_gg(x1, x2, x3)) = x1 + x2 + x3   
POL([]) = 0   
POL(merge33_in_aaa) = 0   
POL(merge33_in_aag(x1)) = 1 + x1   
POL(merge33_out_aaa) = 0   
POL(merge33_out_aag(x1, x2)) = x1 + x2   
POL(mergesort32_in_ga(x1)) = x1   
POL(mergesort32_in_gg(x1, x2)) = x1 + x2   
POL(mergesort32_out_ga) = 0   
POL(mergesort32_out_gg) = 0   
POL(split17_in_agaa(x1)) = 5 + x1   
POL(split17_out_agaa(x1, x2)) = x1 + x2   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2)) = x1 + x2   

(40) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U3_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U3_ga(split17_out_agaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(.(T136)) → U18_aag(merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(merge33_out_aag(.(T132), T134)) → merge33_out_aag(.(T132), .(T134))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U3_ga(x0)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
U14_gg(x0, x1)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U15_gg(x0, x1, x2)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U16_gg(x0, x1)
U17_ga(x0)
U17_gg(x0)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U3_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U18_aaa(merge33_out_aaa) → merge33_out_aaa
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U3_ga(split17_out_agaa(X133, X134)) → mergesort32_out_ga

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U3_ga(x0)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
U14_gg(x0, x1)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U15_gg(x0, x1, x2)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U16_gg(x0, x1)
U17_ga(x0)
U17_gg(x0)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0)

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

(43) QReductionProof (EQUIVALENT transformation)

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

U14_gg(x0, x1)
U15_gg(x0, x1, x2)
U16_gg(x0, x1)
U17_gg(x0)
merge33_in_aag(x0)
U18_aag(x0)

(44) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U3_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U18_aaa(merge33_out_aaa) → merge33_out_aaa
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U3_ga(split17_out_agaa(X133, X134)) → mergesort32_out_ga

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U3_ga(x0)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U17_ga(x0)
merge33_in_aaa
U18_aaa(x0)

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

(45) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

mergesort32_in_ga(.(.(T63))) → U3_ga(split17_in_agaa(.(T63)))
U3_ga(split17_out_agaa(X133, X134)) → mergesort32_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 2 + 2·x1   
POL(MERGESORT32_IN_GA(x1)) = 2·x1   
POL(MERGESORT32_IN_GG(x1, x2)) = 2·x1 + 2·x2   
POL(U10_ga(x1)) = x1   
POL(U10_gg(x1)) = x1   
POL(U11_GA(x1)) = 2·x1   
POL(U11_GG(x1, x2)) = 2·x1 + 2·x2   
POL(U11_ga(x1)) = 2·x1   
POL(U11_gg(x1, x2)) = x1 + 2·x2   
POL(U12_GA(x1, x2)) = 2·x1 + x2   
POL(U12_GG(x1, x2)) = 2·x1 + x2   
POL(U12_ga(x1, x2)) = 2·x1 + x2   
POL(U12_gg(x1, x2)) = 2·x1 + x2   
POL(U13_ga(x1)) = x1   
POL(U13_gg(x1)) = x1   
POL(U14_GA(x1)) = 2·x1   
POL(U14_ga(x1)) = 2·x1   
POL(U15_GA(x1, x2)) = 2·x1 + x2   
POL(U15_ga(x1, x2)) = 2·x1 + x2   
POL(U16_ga(x1)) = x1   
POL(U17_ga(x1)) = 2·x1   
POL(U18_aaa(x1)) = 2·x1   
POL(U1_gaa(x1)) = 2 + 2·x1   
POL(U2_agaa(x1)) = 2 + 2·x1   
POL(U3_ga(x1)) = 2 + x1   
POL(U3_gg(x1)) = 2·x1   
POL(U4_GA(x1)) = 2·x1   
POL(U4_GG(x1)) = 2·x1   
POL(U4_ga(x1)) = 2·x1   
POL(U4_gg(x1)) = 2·x1   
POL(U5_ga(x1)) = x1   
POL(U5_gg(x1)) = x1   
POL(U6_GA(x1, x2)) = 2·x1 + x2   
POL(U6_GG(x1, x2)) = 2·x1 + x2   
POL(U6_ga(x1, x2)) = 2·x1 + x2   
POL(U6_gg(x1, x2)) = 2·x1 + x2   
POL(U7_ga(x1)) = x1   
POL(U7_gg(x1)) = x1   
POL(U8_GA(x1)) = 2·x1   
POL(U8_GG(x1, x2)) = 2·x1 + 2·x2   
POL(U8_ga(x1)) = 2·x1   
POL(U8_gg(x1, x2)) = x1 + 2·x2   
POL(U9_GA(x1, x2)) = 2·x1 + x2   
POL(U9_GG(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(U9_ga(x1, x2)) = 2·x1 + x2   
POL(U9_gg(x1, x2, x3)) = x1 + 2·x2 + x3   
POL([]) = 0   
POL(merge33_in_aaa) = 0   
POL(merge33_out_aaa) = 0   
POL(mergesort32_in_ga(x1)) = 2·x1   
POL(mergesort32_in_gg(x1, x2)) = 2·x1 + x2   
POL(mergesort32_out_ga) = 0   
POL(mergesort32_out_gg) = 0   
POL(split17_in_agaa(x1)) = 2 + 2·x1   
POL(split17_out_agaa(x1, x2)) = x1 + x2   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2)) = x1 + 2·x2   

(46) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U18_aaa(merge33_out_aaa) → merge33_out_aaa
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U5_ga(mergesort32_out_ga) → mergesort32_out_ga

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U3_ga(x0)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U17_ga(x0)
merge33_in_aaa
U18_aaa(x0)

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

(47) QReductionProof (EQUIVALENT transformation)

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

U3_ga(x0)

(48) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U18_aaa(merge33_out_aaa) → merge33_out_aaa
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U5_ga(mergesort32_out_ga) → mergesort32_out_ga

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U17_ga(x0)
merge33_in_aaa
U18_aaa(x0)

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

(49) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 1 + 2·x1   
POL(MERGESORT32_IN_GA(x1)) = 2 + x1   
POL(MERGESORT32_IN_GG(x1, x2)) = 2 + x1 + 2·x2   
POL(U10_ga(x1)) = x1   
POL(U10_gg(x1)) = x1   
POL(U11_GA(x1)) = 2 + x1   
POL(U11_GG(x1, x2)) = 2 + 2·x1 + x2   
POL(U11_ga(x1)) = 2 + x1   
POL(U11_gg(x1, x2)) = 2 + x1 + x2   
POL(U12_GA(x1, x2)) = x1 + x2   
POL(U12_GG(x1, x2)) = x1 + x2   
POL(U12_ga(x1, x2)) = x1 + x2   
POL(U12_gg(x1, x2)) = x1 + x2   
POL(U13_ga(x1)) = x1   
POL(U13_gg(x1)) = x1   
POL(U14_GA(x1)) = 2 + x1   
POL(U14_ga(x1)) = 2 + x1   
POL(U15_GA(x1, x2)) = x1 + x2   
POL(U15_ga(x1, x2)) = x1 + x2   
POL(U16_ga(x1)) = x1   
POL(U17_ga(x1)) = 2 + 2·x1   
POL(U18_aaa(x1)) = 2·x1   
POL(U1_gaa(x1)) = 1 + 2·x1   
POL(U2_agaa(x1)) = 1 + x1   
POL(U3_gg(x1)) = 2 + x1   
POL(U4_GA(x1)) = 2 + x1   
POL(U4_GG(x1)) = 2 + x1   
POL(U4_ga(x1)) = 2 + x1   
POL(U4_gg(x1)) = 2 + x1   
POL(U5_ga(x1)) = x1   
POL(U5_gg(x1)) = x1   
POL(U6_GA(x1, x2)) = x1 + x2   
POL(U6_GG(x1, x2)) = x1 + x2   
POL(U6_ga(x1, x2)) = x1 + x2   
POL(U6_gg(x1, x2)) = x1 + x2   
POL(U7_ga(x1)) = x1   
POL(U7_gg(x1)) = x1   
POL(U8_GA(x1)) = 2 + x1   
POL(U8_GG(x1, x2)) = 2 + 2·x1 + x2   
POL(U8_ga(x1)) = 2 + x1   
POL(U8_gg(x1, x2)) = 2 + x1 + x2   
POL(U9_GA(x1, x2)) = x1 + x2   
POL(U9_GG(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(U9_ga(x1, x2)) = x1 + x2   
POL(U9_gg(x1, x2, x3)) = x1 + x2 + x3   
POL([]) = 0   
POL(merge33_in_aaa) = 0   
POL(merge33_out_aaa) = 0   
POL(mergesort32_in_ga(x1)) = 2 + x1   
POL(mergesort32_in_gg(x1, x2)) = 2 + x1 + x2   
POL(mergesort32_out_ga) = 2   
POL(mergesort32_out_gg) = 2   
POL(split17_in_agaa(x1)) = 2 + x1   
POL(split17_out_agaa(x1, x2)) = x1 + x2   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2)) = x1 + 2·x2   

(50) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(split17_in_agaa(.(T63)))
U4_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(split17_in_agaa(.(T63)))
U8_GA(split17_out_agaa(T67, T68)) → U9_GA(T68, mergesort32_in_gg(T67, []))
U9_GA(T68, mergesort32_out_gg) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(split17_in_agaa(.(T63)))
U11_GA(split17_out_agaa(T67, T68)) → U12_GA(T68, mergesort32_in_ga(T67))
U12_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(split17_in_agaa(.(T63)))
U4_GG(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(split17_in_agaa(.(T63)))
U14_GA(split17_out_agaa(T67, T68)) → U15_GA(T68, mergesort32_in_ga(T67))
U15_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
U14_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(split17_out_agaa(T67, T68)) → U6_GG(T68, mergesort32_in_ga(T67))
U6_GG(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T80, split17_in_agaa(.(T63)))
U8_GG(T80, split17_out_agaa(T67, T68)) → U9_GG(T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T80, T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T85, split17_in_agaa(.(T63)))
U11_GG(T85, split17_out_agaa(T67, T68)) → U12_GG(T68, mergesort32_in_gg(T67, T85))
U12_GG(T68, mergesort32_out_gg) → MERGESORT32_IN_GG(T68, [])
U11_GG(T85, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T80, split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(split17_out_agaa(T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(split17_out_agaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(.(T63))) → U4_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(split17_in_agaa(.(T63)))
U14_ga(split17_out_agaa(T67, T68)) → U15_ga(T68, mergesort32_in_ga(T67))
U15_ga(T68, mergesort32_out_ga) → U16_ga(mergesort32_in_ga(T68))
U16_ga(mergesort32_out_ga) → U17_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
U17_ga(merge33_out_aaa) → mergesort32_out_ga
U18_aaa(merge33_out_aaa) → merge33_out_aaa
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)
U11_ga(split17_out_agaa(T67, T68)) → U12_ga(T68, mergesort32_in_ga(T67))
U12_ga(T68, mergesort32_out_ga) → U13_ga(mergesort32_in_gg(T68, []))
mergesort32_in_gg([], []) → mergesort32_out_gg
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T85, split17_in_agaa(.(T63)))
U13_ga(mergesort32_out_gg) → mergesort32_out_ga
U11_gg(T85, split17_out_agaa(T67, T68)) → U12_gg(T68, mergesort32_in_gg(T67, T85))
U12_gg(T68, mergesort32_out_gg) → U13_gg(mergesort32_in_gg(T68, []))
U13_gg(mergesort32_out_gg) → mergesort32_out_gg
U8_gg(T80, split17_out_agaa(T67, T68)) → U9_gg(T80, T68, mergesort32_in_gg(T67, []))
U9_gg(T80, T68, mergesort32_out_gg) → U10_gg(mergesort32_in_gg(T68, T80))
U10_gg(mergesort32_out_gg) → mergesort32_out_gg
U4_gg(split17_out_agaa(T67, T68)) → U5_gg(mergesort32_in_ga(T67))
U4_gg(split17_out_agaa(T67, T68)) → U6_gg(T68, mergesort32_in_ga(T67))
U6_gg(T68, mergesort32_out_ga) → U7_gg(mergesort32_in_ga(T68))
U7_gg(mergesort32_out_ga) → mergesort32_out_gg
U5_gg(mergesort32_out_ga) → mergesort32_out_gg
U3_gg(split17_out_agaa(X133, X134)) → mergesort32_out_gg
U8_ga(split17_out_agaa(T67, T68)) → U9_ga(T68, mergesort32_in_gg(T67, []))
U9_ga(T68, mergesort32_out_gg) → U10_ga(mergesort32_in_ga(T68))
U10_ga(mergesort32_out_ga) → mergesort32_out_ga
U4_ga(split17_out_agaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_agaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U5_ga(mergesort32_out_ga) → mergesort32_out_ga

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0)
U3_gg(x0)
U4_gg(x0)
U8_gg(x0, x1)
U11_gg(x0, x1)
U4_ga(x0)
U8_ga(x0)
U11_ga(x0)
U14_ga(x0)
split21_in_gaa(x0)
U5_gg(x0)
U6_gg(x0, x1)
U9_gg(x0, x1, x2)
U12_gg(x0, x1)
U5_ga(x0)
U6_ga(x0, x1)
U9_ga(x0, x1)
U12_ga(x0, x1)
U15_ga(x0, x1)
U1_gaa(x0)
U7_gg(x0)
U10_gg(x0)
U13_gg(x0)
U7_ga(x0)
U10_ga(x0)
U13_ga(x0)
U16_ga(x0)
U17_ga(x0)
merge33_in_aaa
U18_aaa(x0)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 28 less nodes.

(52) TRUE

(53) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x3, x4)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x5)

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

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(split17_in_agaa(T25))
U20_GA(split17_out_agaa(T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0)
split21_in_gaa(x0)
U1_gaa(x0)

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

(58) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

split21_in_gaa(.(T43)) → U1_gaa(split21_in_gaa(T43))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 1 + 2·x1   
POL(MERGESORT1_IN_GA(x1)) = x1   
POL(U1_gaa(x1)) = 1 + 2·x1   
POL(U20_GA(x1)) = 1 + 2·x1   
POL(U2_agaa(x1)) = 1 + x1   
POL([]) = 0   
POL(split17_in_agaa(x1)) = 1 + 2·x1   
POL(split17_out_agaa(x1, x2)) = x1 + x2   
POL(split21_in_gaa(x1)) = 2·x1   
POL(split21_out_gaa(x1, x2)) = x1 + 2·x2   

(59) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(split17_in_agaa(T25))
U20_GA(split17_out_agaa(T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [])
U1_gaa(split21_out_gaa(X100, X99)) → split21_out_gaa(.(X99), X100)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0)
split21_in_gaa(x0)
U1_gaa(x0)

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(split17_in_agaa(T25))
U20_GA(split17_out_agaa(T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
split21_in_gaa([]) → split21_out_gaa([], [])
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0)
split21_in_gaa(x0)
U1_gaa(x0)

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

(62) QReductionProof (EQUIVALENT transformation)

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

U1_gaa(x0)

(63) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(split17_in_agaa(T25))
U20_GA(split17_out_agaa(T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
split21_in_gaa([]) → split21_out_gaa([], [])
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0)
split21_in_gaa(x0)

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

(64) MRRProof (EQUIVALENT transformation)

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(split17_in_agaa(T25))


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 1 + 2·x1   
POL(MERGESORT1_IN_GA(x1)) = 2·x1   
POL(U20_GA(x1)) = 2·x1   
POL(U2_agaa(x1)) = 2 + x1   
POL([]) = 0   
POL(split17_in_agaa(x1)) = 2 + 2·x1   
POL(split17_out_agaa(x1, x2)) = 1 + x1 + 2·x2   
POL(split21_in_gaa(x1)) = 2·x1   
POL(split21_out_gaa(x1, x2)) = 2·x1 + 2·x2   

(65) Obligation:

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

U20_GA(split17_out_agaa(T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(split21_in_gaa(T37))
split21_in_gaa([]) → split21_out_gaa([], [])
U2_agaa(split21_out_gaa(X82, X81)) → split17_out_agaa(.(X81), X82)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0)
split21_in_gaa(x0)

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

(66) DependencyGraphProof (EQUIVALENT transformation)

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

(67) TRUE

(68) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (f,b,f,f)
split21_in: (b,f,f)
mergesort32_in: (b,f) (b,b)
merge33_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:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(69) Obligation:

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

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)

(70) DependencyPairsProof (EQUIVALENT transformation)

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U3_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GG(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GA(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAA(.(T104, T95), T97, X185)
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAA(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_GA(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GG(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAG(.(T104, T95), T97, X185)
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAG(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_GG(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GG(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GG(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GA(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x3, x5)
SPLIT17_IN_AGAA(x1, x2, x3, x4)  =  SPLIT17_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5)  =  U2_AGAA(x2, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x3, x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x3, x5)
U22_GA(x1, x2, x3, x4, x5, x6)  =  U22_GA(x3, x5, x6)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x3, x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x3, x4, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x3, x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x3, x6, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x3, x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x3, x6)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x3, x5)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x3, x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x3, x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x3, x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x3, x4, x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x3, x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x3, x5, x6, x7)
U16_GG(x1, x2, x3, x4, x5, x6, x7)  =  U16_GG(x3, x5, x7)
U17_GG(x1, x2, x3, x4, x5, x6)  =  U17_GG(x3, x5, x6)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)
U18_AAG(x1, x2, x3, x4, x5)  =  U18_AAG(x4, x5)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x3, x4, x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x3, x4, x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x3, x4, x5, x6)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x3, x4, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x3, x5)
U24_GA(x1, x2, x3, x4, x5, x6)  =  U24_GA(x3, x6)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x3, x5)

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

(71) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U3_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GG(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GA(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_GA(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAA(.(T104, T95), T97, X185)
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAA(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_GA(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → SPLIT17_IN_AGAA(T61, .(T62, T63), T67, T68)
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_GG(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U16_GG(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → MERGE33_IN_AAG(.(T104, T95), T97, X185)
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_AAG(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_GG(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GG(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GG(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_GA(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U22_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U24_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x3, x5)
SPLIT17_IN_AGAA(x1, x2, x3, x4)  =  SPLIT17_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5)  =  U2_AGAA(x2, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x3, x5)
U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x3, x5)
U22_GA(x1, x2, x3, x4, x5, x6)  =  U22_GA(x3, x5, x6)
U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x3, x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x3, x4, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x3, x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x3, x6, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x3, x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x3, x6)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x3, x5)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x3, x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x3, x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x3, x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x3, x4, x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x3, x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x3, x5, x6, x7)
U16_GG(x1, x2, x3, x4, x5, x6, x7)  =  U16_GG(x3, x5, x7)
U17_GG(x1, x2, x3, x4, x5, x6)  =  U17_GG(x3, x5, x6)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)
U18_AAG(x1, x2, x3, x4, x5)  =  U18_AAG(x4, x5)
U13_GG(x1, x2, x3, x4, x5)  =  U13_GG(x3, x4, x5)
U10_GG(x1, x2, x3, x4, x5)  =  U10_GG(x3, x4, x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x3, x4, x5, x6)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x3, x4, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x3, x5)
U24_GA(x1, x2, x3, x4, x5, x6)  =  U24_GA(x3, x6)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x3, x5)

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

(72) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 34 less nodes.

(73) Complex Obligation (AND)

(74) Obligation:

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

MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)

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

(75) UsableRulesProof (EQUIVALENT transformation)

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

(76) Obligation:

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

MERGE33_IN_AAG(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAG(.(T143, T132), T134, T136)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
MERGE33_IN_AAG(x1, x2, x3)  =  MERGE33_IN_AAG(x3)

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

(77) PiDPToQDPProof (SOUND transformation)

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

(78) Obligation:

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

MERGE33_IN_AAG(.(T136)) → MERGE33_IN_AAG(T136)

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

(79) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • MERGE33_IN_AAG(.(T136)) → MERGE33_IN_AAG(T136)
    The graph contains the following edges 1 > 1

(80) YES

(81) Obligation:

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

MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA

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

(82) UsableRulesProof (EQUIVALENT transformation)

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

(83) Obligation:

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

MERGE33_IN_AAA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_AAA(.(T143, T132), T134, T136)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA

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

(84) PiDPToQDPProof (SOUND transformation)

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

(85) Obligation:

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

MERGE33_IN_AAAMERGE33_IN_AAA

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

(86) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MERGE33_IN_AAA evaluates to t =MERGE33_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MERGE33_IN_AAA to MERGE33_IN_AAA.



(87) NO

(88) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(89) UsableRulesProof (EQUIVALENT transformation)

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

(90) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(91) PiDPToQDPProof (SOUND transformation)

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

(92) Obligation:

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

SPLIT21_IN_GAA(.(T43)) → SPLIT21_IN_GAA(T43)

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

(93) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • SPLIT21_IN_GAA(.(T43)) → SPLIT21_IN_GAA(T43)
    The graph contains the following edges 1 > 1

(94) YES

(95) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x3, x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x3, x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x3, x6, x7)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x3, x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x3, x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x3, x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x3, x4, x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x3, x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x3, x5, x6, x7)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x3, x4, x5, x6)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x3, x5, x6)

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

(96) UsableRulesProof (EQUIVALENT transformation)

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

(97) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T80) → U8_GA(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GA(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68, T80)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), T85) → U11_GA(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GA(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U12_GA(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(T61, .(T62, T63)), X137) → U4_GG(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U14_GA(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GA(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GA(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U4_GG(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GG(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GG(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T80) → U8_GG(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GG(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), T85) → U11_GG(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_GG(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(T61, .(T62, T63)), .(T104, X185)) → U14_GG(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_GG(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → MERGESORT32_IN_GA(T68, .(T104, T97))
U14_GG(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, .(T104, T95))
U8_GG(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, T85)
U8_GA(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5, x6)  =  U9_GA(x3, x5, x6)
MERGESORT32_IN_GG(x1, x2)  =  MERGESORT32_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x3, x5, x6)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x3, x6, x7)
U8_GG(x1, x2, x3, x4, x5)  =  U8_GG(x3, x4, x5)
U9_GG(x1, x2, x3, x4, x5, x6)  =  U9_GG(x3, x4, x5, x6)
U11_GG(x1, x2, x3, x4, x5)  =  U11_GG(x3, x4, x5)
U12_GG(x1, x2, x3, x4, x5, x6)  =  U12_GG(x3, x4, x5, x6)
U14_GG(x1, x2, x3, x4, x5, x6)  =  U14_GG(x3, x5, x6)
U15_GG(x1, x2, x3, x4, x5, x6, x7)  =  U15_GG(x3, x5, x6, x7)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x3, x4, x5, x6)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x3, x5, x6)

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

(98) PiDPToQDPProof (SOUND transformation)

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

(99) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, split17_in_agaa(.(T63)))
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, split17_in_agaa(.(T63)))
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, split17_in_agaa(.(T63)))
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, split17_in_agaa(.(T63)))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63)))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(100) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, split17_in_agaa(.(T63))) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(101) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, split17_in_agaa(.(T63)))
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, split17_in_agaa(.(T63)))
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, split17_in_agaa(.(T63)))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63)))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(102) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, split17_in_agaa(.(T63))) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(103) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, split17_in_agaa(.(T63)))
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, split17_in_agaa(.(T63)))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63)))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(104) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, split17_in_agaa(.(T63))) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(105) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, split17_in_agaa(.(T63)))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63)))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(106) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, split17_in_agaa(.(T63))) at position [2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(107) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63)))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(108) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, split17_in_agaa(.(T63))) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(109) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63)))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(110) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, split17_in_agaa(.(T63))) at position [2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(111) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63)))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(112) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, split17_in_agaa(.(T63))) at position [2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(113) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63)))
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(114) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, split17_in_agaa(.(T63))) at position [2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))

(115) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(116) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [1,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(117) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(118) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [1,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(119) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(120) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [1,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(121) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(122) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [2,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(123) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(124) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [1,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(125) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(126) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [2,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(127) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(128) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [2,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(129) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(130) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), split21_in_gaa(.(T63)))) at position [2,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

(131) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MERGESORT32_IN_GG(.(.(T63)), .(X185)) → U14_GG(T63, X185, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U4_GA(x1, x2)) = 0 +
[0,0]
·x1 +
[0,0]
·x2

POL(split17_out_agaa(x1, x2, x3)) =
/1\
\1/
+
/10\
\10/
·x1 +
/00\
\10/
·x2 +
/00\
\01/
·x3

POL(.(x1)) =
/1\
\1/
+
/00\
\01/
·x1

POL(MERGESORT32_IN_GA(x1)) = 0 +
[0,0]
·x1

POL(U8_GA(x1, x2)) = 0 +
[0,0]
·x1 +
[0,0]
·x2

POL(U9_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3

POL(mergesort32_in_gg(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/11\
\11/
·x2

POL([]) =
/0\
\0/

POL(mergesort32_out_gg(x1, x2)) =
/0\
\1/
+
/00\
\11/
·x1 +
/11\
\00/
·x2

POL(U11_GA(x1, x2)) = 0 +
[0,0]
·x1 +
[0,0]
·x2

POL(U12_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3

POL(mergesort32_in_ga(x1)) =
/0\
\1/
+
/10\
\11/
·x1

POL(mergesort32_out_ga(x1)) =
/1\
\0/
+
/11\
\00/
·x1

POL(MERGESORT32_IN_GG(x1, x2)) = 0 +
[0,0]
·x1 +
[1,1]
·x2

POL(U4_GG(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,0]
·x2 +
[0,0]
·x3

POL(U14_GA(x1, x2)) = 0 +
[0,0]
·x1 +
[0,0]
·x2

POL(U15_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3

POL(U6_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[1,0]
·x2 +
[0,0]
·x3 +
[0,0]
·x4

POL(U8_GG(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3

POL(U9_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3 +
[0,0]
·x4

POL(U11_GG(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3

POL(U12_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,1]
·x2 +
[0,0]
·x3 +
[0,0]
·x4

POL(U14_GG(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[0,1]
·x2 +
[0,0]
·x3

POL(U15_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3 +
[0,0]
·x4

POL(U6_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3

POL(U2_agaa(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U1_gaa(x1, x2)) =
/0\
\0/
+
/00\
\11/
·x1 +
/00\
\10/
·x2

POL(split21_in_gaa(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(U3_gg(x1, x2, x3)) =
/0\
\0/
+
/10\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(split17_in_agaa(x1)) =
/1\
\1/
+
/10\
\10/
·x1

POL(U4_gg(x1, x2, x3)) =
/0\
\1/
+
/01\
\10/
·x1 +
/00\
\00/
·x2 +
/00\
\11/
·x3

POL(U8_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\01/
·x2 +
/00\
\11/
·x3

POL(U11_gg(x1, x2, x3)) =
/0\
\1/
+
/00\
\00/
·x1 +
/01\
\10/
·x2 +
/00\
\00/
·x3

POL(U3_ga(x1, x2)) =
/0\
\1/
+
/01\
\10/
·x1 +
/00\
\00/
·x2

POL(U4_ga(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U8_ga(x1, x2)) =
/1\
\0/
+
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(U11_ga(x1, x2)) =
/1\
\0/
+
/00\
\00/
·x1 +
/01\
\00/
·x2

POL(U14_ga(x1, x2)) =
/0\
\0/
+
/00\
\01/
·x1 +
/11\
\00/
·x2

POL(U14_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\10/
·x1 +
/00\
\11/
·x2 +
/00\
\00/
·x3

POL(split21_out_gaa(x1, x2, x3)) =
/0\
\1/
+
/01\
\00/
·x1 +
/00\
\00/
·x2 +
/10\
\10/
·x3

POL(U5_gg(x1, x2, x3)) =
/1\
\1/
+
/00\
\01/
·x1 +
/00\
\01/
·x2 +
/01\
\10/
·x3

POL(U5_ga(x1, x2)) =
/0\
\0/
+
/10\
\10/
·x1 +
/00\
\00/
·x2

POL(U9_ga(x1, x2, x3)) =
/0\
\1/
+
/11\
\11/
·x1 +
/10\
\10/
·x2 +
/00\
\01/
·x3

POL(U9_gg(x1, x2, x3, x4)) =
/1\
\0/
+
/01\
\00/
·x1 +
/11\
\01/
·x2 +
/00\
\11/
·x3 +
/00\
\00/
·x4

POL(U12_gg(x1, x2, x3, x4)) =
/1\
\0/
+
/11\
\00/
·x1 +
/10\
\00/
·x2 +
/11\
\00/
·x3 +
/11\
\00/
·x4

POL(U15_gg(x1, x2, x3, x4)) =
/0\
\1/
+
/00\
\01/
·x1 +
/10\
\01/
·x2 +
/01\
\11/
·x3 +
/00\
\11/
·x4

POL(U12_ga(x1, x2, x3)) =
/0\
\1/
+
/10\
\10/
·x1 +
/01\
\00/
·x2 +
/11\
\00/
·x3

POL(U15_ga(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/01\
\00/
·x3

POL(U16_ga(x1, x2)) =
/0\
\0/
+
/10\
\10/
·x1 +
/00\
\00/
·x2

POL(U17_ga(x1, x2)) =
/1\
\0/
+
/10\
\01/
·x1 +
/11\
\00/
·x2

POL(merge33_in_aaa) =
/1\
\1/

POL(U13_ga(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(U16_gg(x1, x2, x3)) =
/1\
\1/
+
/00\
\10/
·x1 +
/01\
\10/
·x2 +
/10\
\01/
·x3

POL(U17_gg(x1, x2, x3)) =
/0\
\0/
+
/01\
\00/
·x1 +
/10\
\00/
·x2 +
/00\
\00/
·x3

POL(merge33_in_aag(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(U13_gg(x1, x2, x3)) =
/0\
\1/
+
/01\
\00/
·x1 +
/01\
\01/
·x2 +
/00\
\00/
·x3

POL(U10_gg(x1, x2, x3)) =
/0\
\1/
+
/00\
\00/
·x1 +
/01\
\00/
·x2 +
/11\
\11/
·x3

POL(U10_ga(x1, x2)) =
/0\
\0/
+
/00\
\11/
·x1 +
/00\
\00/
·x2

POL(U6_ga(x1, x2, x3)) =
/0\
\1/
+
/00\
\11/
·x1 +
/11\
\11/
·x2 +
/11\
\10/
·x3

POL(U7_ga(x1, x2)) =
/0\
\0/
+
/11\
\01/
·x1 +
/00\
\00/
·x2

POL(U6_gg(x1, x2, x3, x4)) =
/0\
\0/
+
/10\
\00/
·x1 +
/01\
\00/
·x2 +
/00\
\10/
·x3 +
/00\
\01/
·x4

POL(U7_gg(x1, x2, x3)) =
/0\
\0/
+
/10\
\11/
·x1 +
/10\
\11/
·x2 +
/00\
\00/
·x3

POL(merge33_out_aaa) =
/1\
\0/

POL(U18_aaa(x1)) =
/1\
\1/
+
/10\
\00/
·x1

POL(merge33_out_aag(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\10/
·x2 +
/00\
\00/
·x3

POL(U18_aag(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\01/
·x2

The following usable rules [FROCOS05] were oriented: none

(133) Obligation:

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

U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_GG(T63, X185, T68, mergesort32_in_ga(T67))
U15_GG(T63, X185, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GG(T63, X185, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(134) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(135) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
MERGESORT32_IN_GA(.(.(T63))) → U11_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U12_GA(T63, T68, mergesort32_in_ga(T67))
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U15_GA(T63, T68, mergesort32_in_ga(T67))
U14_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GG(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_GG(T63, X137, T68, mergesort32_in_ga(T67))
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_GG(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_GG(T63, T85, T68, mergesort32_in_gg(T67, T85))
U11_GG(T63, T85, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, T85)
U8_GG(T63, T80, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U11_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GG(T67, [])
U4_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesort32_in_ga(T67))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(MERGESORT32_IN_GA(x1)) = 0 +
[1,1]
·x1

POL(.(x1)) =
/1\
\1/
+
/11\
\11/
·x1

POL(U4_GA(x1, x2)) = 1 +
[0,0]
·x1 +
[1,1]
·x2

POL(U2_agaa(x1, x2)) =
/1\
\1/
+
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(U1_gaa(x1, x2)) =
/0\
\1/
+
/00\
\00/
·x1 +
/11\
\11/
·x2

POL(split21_in_gaa(x1)) =
/1\
\0/
+
/11\
\11/
·x1

POL(split17_out_agaa(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\11/
·x2 +
/00\
\11/
·x3

POL(U8_GA(x1, x2)) = 1 +
[0,0]
·x1 +
[1,1]
·x2

POL(U9_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[1,0]
·x3

POL(mergesort32_in_gg(x1, x2)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL([]) =
/0\
\0/

POL(mergesort32_out_gg(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U11_GA(x1, x2)) = 1 +
[0,0]
·x1 +
[0,1]
·x2

POL(U12_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3

POL(mergesort32_in_ga(x1)) =
/1\
\0/
+
/11\
\00/
·x1

POL(mergesort32_out_ga(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(MERGESORT32_IN_GG(x1, x2)) = 0 +
[1,1]
·x1 +
[0,0]
·x2

POL(U4_GG(x1, x2, x3)) = 1 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3

POL(U14_GA(x1, x2)) = 1 +
[0,0]
·x1 +
[1,1]
·x2

POL(U15_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3

POL(U6_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3 +
[0,0]
·x4

POL(U8_GG(x1, x2, x3)) = 1 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3

POL(U9_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3 +
[0,0]
·x4

POL(U11_GG(x1, x2, x3)) = 1 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3

POL(U12_GG(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[1,1]
·x3 +
[0,0]
·x4

POL(U6_GA(x1, x2, x3)) = 0 +
[0,0]
·x1 +
[1,1]
·x2 +
[0,0]
·x3

POL(split21_out_gaa(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\11/
·x2 +
/11\
\11/
·x3

POL(U3_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(split17_in_agaa(x1)) =
/0\
\0/
+
/11\
\00/
·x1

POL(U4_gg(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U8_gg(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U11_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U3_ga(x1, x2)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U4_ga(x1, x2)) =
/0\
\0/
+
/10\
\00/
·x1 +
/00\
\00/
·x2

POL(U8_ga(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U11_ga(x1, x2)) =
/0\
\0/
+
/01\
\00/
·x1 +
/00\
\00/
·x2

POL(U14_ga(x1, x2)) =
/1\
\0/
+
/11\
\00/
·x1 +
/00\
\00/
·x2

POL(U14_gg(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U5_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U5_ga(x1, x2)) =
/0\
\0/
+
/10\
\00/
·x1 +
/00\
\00/
·x2

POL(U9_ga(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U9_gg(x1, x2, x3, x4)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\00/
·x4

POL(U12_gg(x1, x2, x3, x4)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/01\
\00/
·x4

POL(U15_gg(x1, x2, x3, x4)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\00/
·x4

POL(U12_ga(x1, x2, x3)) =
/0\
\0/
+
/01\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U15_ga(x1, x2, x3)) =
/1\
\0/
+
/11\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U16_ga(x1, x2)) =
/1\
\0/
+
/10\
\00/
·x1 +
/00\
\00/
·x2

POL(U17_ga(x1, x2)) =
/1\
\0/
+
/10\
\00/
·x1 +
/00\
\00/
·x2

POL(merge33_in_aaa) =
/0\
\0/

POL(U13_ga(x1, x2)) =
/0\
\0/
+
/01\
\00/
·x1 +
/00\
\00/
·x2

POL(U16_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U17_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(merge33_in_aag(x1)) =
/0\
\0/
+
/10\
\00/
·x1

POL(U13_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U10_gg(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U10_ga(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(U6_ga(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U7_ga(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2

POL(U6_gg(x1, x2, x3, x4)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\00/
·x4

POL(U7_gg(x1, x2, x3)) =
/1\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(merge33_out_aaa) =
/1\
\1/

POL(U18_aaa(x1)) =
/0\
\0/
+
/00\
\11/
·x1

POL(merge33_out_aag(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U18_aag(x1, x2)) =
/0\
\1/
+
/10\
\10/
·x1 +
/10\
\10/
·x2

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))

(137) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)
U12_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GG(T68, [])
MERGESORT32_IN_GG(.(.(T63)), X137) → U4_GG(T63, X137, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
MERGESORT32_IN_GA(.(.(T63))) → U14_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U15_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
U6_GG(T63, X137, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GG(.(.(T63)), T80) → U8_GG(T63, T80, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U9_GG(T63, T80, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GG(T68, T80)
MERGESORT32_IN_GG(.(.(T63)), T85) → U11_GG(T63, T85, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U12_GG(T63, T85, T68, mergesort32_out_gg(T67, T85)) → MERGESORT32_IN_GG(T68, [])
U6_GA(T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(138) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.

(139) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MERGESORT32_IN_GA(.(.(T63))) → U8_GA(T63, U2_agaa(.(T63), U1_gaa(T63, split21_in_gaa(T63))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 1 + x1   
POL(MERGESORT32_IN_GA(x1)) = x1   
POL(U10_ga(x1, x2)) = 0   
POL(U10_gg(x1, x2, x3)) = 0   
POL(U11_ga(x1, x2)) = 0   
POL(U11_gg(x1, x2, x3)) = 0   
POL(U12_ga(x1, x2, x3)) = 0   
POL(U12_gg(x1, x2, x3, x4)) = 0   
POL(U13_ga(x1, x2)) = 0   
POL(U13_gg(x1, x2, x3)) = 0   
POL(U14_ga(x1, x2)) = 0   
POL(U14_gg(x1, x2, x3)) = 0   
POL(U15_ga(x1, x2, x3)) = 0   
POL(U15_gg(x1, x2, x3, x4)) = 0   
POL(U16_ga(x1, x2)) = 0   
POL(U16_gg(x1, x2, x3)) = 0   
POL(U17_ga(x1, x2)) = 0   
POL(U17_gg(x1, x2, x3)) = 0   
POL(U18_aaa(x1)) = 0   
POL(U18_aag(x1, x2)) = 0   
POL(U1_gaa(x1, x2)) = 1 + x2   
POL(U2_agaa(x1, x2)) = x2   
POL(U3_ga(x1, x2)) = 1   
POL(U3_gg(x1, x2, x3)) = 0   
POL(U4_ga(x1, x2)) = 0   
POL(U4_gg(x1, x2, x3)) = 0   
POL(U5_ga(x1, x2)) = 0   
POL(U5_gg(x1, x2, x3)) = 0   
POL(U6_ga(x1, x2, x3)) = 0   
POL(U6_gg(x1, x2, x3, x4)) = 0   
POL(U7_ga(x1, x2)) = 0   
POL(U7_gg(x1, x2, x3)) = 0   
POL(U8_GA(x1, x2)) = x2   
POL(U8_ga(x1, x2)) = 0   
POL(U8_gg(x1, x2, x3)) = 0   
POL(U9_GA(x1, x2, x3)) = x2   
POL(U9_ga(x1, x2, x3)) = 0   
POL(U9_gg(x1, x2, x3, x4)) = 0   
POL([]) = 0   
POL(merge33_in_aaa) = 0   
POL(merge33_in_aag(x1)) = 0   
POL(merge33_out_aaa) = 0   
POL(merge33_out_aag(x1, x2, x3)) = 0   
POL(mergesort32_in_ga(x1)) = 1 + x1   
POL(mergesort32_in_gg(x1, x2)) = 0   
POL(mergesort32_out_ga(x1)) = 1   
POL(mergesort32_out_gg(x1, x2)) = 0   
POL(split17_in_agaa(x1)) = 0   
POL(split17_out_agaa(x1, x2, x3)) = x3   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2, x3)) = x2 + x3   

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)

(141) Obligation:

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

U8_GA(T63, split17_out_agaa(.(T63), T67, T68)) → U9_GA(T63, T68, mergesort32_in_gg(T67, []))
U9_GA(T63, T68, mergesort32_out_gg(T67, [])) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(.(T63)), X137) → U3_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), X137) → U4_gg(T63, X137, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T80) → U8_gg(T63, T80, split17_in_agaa(.(T63)))
mergesort32_in_gg(.(.(T63)), T85) → U11_gg(T63, T85, split17_in_agaa(.(T63)))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.([])) → mergesort32_out_ga(.([]))
mergesort32_in_ga(.(.(T63))) → U3_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U4_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U8_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U11_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_ga(.(.(T63))) → U14_ga(T63, split17_in_agaa(.(T63)))
mergesort32_in_gg(.([]), .([])) → mergesort32_out_gg(.([]), .([]))
mergesort32_in_gg(.(.(T63)), .(X185)) → U14_gg(T63, X185, split17_in_agaa(.(T63)))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U3_gg(T63, X137, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_gg(.(.(T63)), X137)
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U5_gg(T63, X137, mergesort32_in_ga(T67))
U4_gg(T63, X137, split17_out_agaa(.(T63), T67, T68)) → U6_gg(T63, X137, T68, mergesort32_in_ga(T67))
U8_gg(T63, T80, split17_out_agaa(.(T63), T67, T68)) → U9_gg(T63, T80, T68, mergesort32_in_gg(T67, []))
U11_gg(T63, T85, split17_out_agaa(.(T63), T67, T68)) → U12_gg(T63, T85, T68, mergesort32_in_gg(T67, T85))
U3_ga(T63, split17_out_agaa(.(T63), X133, X134)) → mergesort32_out_ga(.(.(T63)))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U5_ga(T63, mergesort32_in_ga(T67))
U4_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U6_ga(T63, T68, mergesort32_in_ga(T67))
U8_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U9_ga(T63, T68, mergesort32_in_gg(T67, []))
U11_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U12_ga(T63, T68, mergesort32_in_ga(T67))
U14_ga(T63, split17_out_agaa(.(T63), T67, T68)) → U15_ga(T63, T68, mergesort32_in_ga(T67))
U14_gg(T63, X185, split17_out_agaa(.(T63), T67, T68)) → U15_gg(T63, X185, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U5_gg(T63, X137, mergesort32_out_ga(T67)) → mergesort32_out_gg(.(.(T63)), X137)
U6_gg(T63, X137, T68, mergesort32_out_ga(T67)) → U7_gg(T63, X137, mergesort32_in_ga(T68))
U9_gg(T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T63, T80, mergesort32_in_gg(T68, T80))
U12_gg(T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T63, T85, mergesort32_in_gg(T68, []))
U5_ga(T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(.(T63)))
U6_ga(T63, T68, mergesort32_out_ga(T67)) → U7_ga(T63, mergesort32_in_ga(T68))
U9_ga(T63, T68, mergesort32_out_gg(T67, [])) → U10_ga(T63, mergesort32_in_ga(T68))
U12_ga(T63, T68, mergesort32_out_ga(T67)) → U13_ga(T63, mergesort32_in_gg(T68, []))
U15_ga(T63, T68, mergesort32_out_ga(T67)) → U16_ga(T63, mergesort32_in_ga(T68))
U15_gg(T63, X185, T68, mergesort32_out_ga(T67)) → U16_gg(T63, X185, mergesort32_in_ga(T68))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)
U7_gg(T63, X137, mergesort32_out_ga(T68)) → mergesort32_out_gg(.(.(T63)), X137)
U10_gg(T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(.(T63)), T80)
U13_gg(T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(.(T63)), T85)
U7_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U10_ga(T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(.(T63)))
U13_ga(T63, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(.(T63)))
U16_ga(T63, mergesort32_out_ga(T68)) → U17_ga(T63, merge33_in_aaa)
U16_gg(T63, X185, mergesort32_out_ga(T68)) → U17_gg(T63, X185, merge33_in_aag(X185))
U17_ga(T63, merge33_out_aaa) → mergesort32_out_ga(.(.(T63)))
U17_gg(T63, X185, merge33_out_aag(.(T95), T97, X185)) → mergesort32_out_gg(.(.(T63)), .(X185))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU18_aaa(merge33_in_aaa)
merge33_in_aag(T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T136)) → U18_aag(T136, merge33_in_aag(T136))
U18_aaa(merge33_out_aaa) → merge33_out_aaa
U18_aag(T136, merge33_out_aag(.(T132), T134, T136)) → merge33_out_aag(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

split17_in_agaa(x0)
mergesort32_in_gg(x0, x1)
mergesort32_in_ga(x0)
U2_agaa(x0, x1)
U3_gg(x0, x1, x2)
U4_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1)
U4_ga(x0, x1)
U8_ga(x0, x1)
U11_ga(x0, x1)
U14_ga(x0, x1)
U14_gg(x0, x1, x2)
split21_in_gaa(x0)
U5_gg(x0, x1, x2)
U6_gg(x0, x1, x2, x3)
U9_gg(x0, x1, x2, x3)
U12_gg(x0, x1, x2, x3)
U5_ga(x0, x1)
U6_ga(x0, x1, x2)
U9_ga(x0, x1, x2)
U12_ga(x0, x1, x2)
U15_ga(x0, x1, x2)
U15_gg(x0, x1, x2, x3)
U1_gaa(x0, x1)
U7_gg(x0, x1, x2)
U10_gg(x0, x1, x2)
U13_gg(x0, x1, x2)
U7_ga(x0, x1)
U10_ga(x0, x1)
U13_ga(x0, x1)
U16_ga(x0, x1)
U16_gg(x0, x1, x2)
U17_ga(x0, x1)
U17_gg(x0, x1, x2)
merge33_in_aaa
merge33_in_aag(x0)
U18_aaa(x0)
U18_aag(x0, x1)

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

(142) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(143) TRUE

(144) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U19_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
U19_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U20_ga(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U21_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U21_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U20_ga(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → U22_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U23_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T80) → U8_ga(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_ga(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_ga(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg([], []) → mergesort32_out_gg([], [])
mergesort32_in_gg(.(T54, []), .(T54, [])) → mergesort32_out_gg(.(T54, []), .(T54, []))
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U3_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), X133, X134))
U3_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
mergesort32_in_gg(.(T61, .(T62, T63)), X137) → U4_gg(T61, T62, T63, X137, split17_in_agaa(T61, .(T62, T63), T67, T68))
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_gg(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
mergesort32_in_ga(.(T61, .(T62, T63)), T85) → U11_ga(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_ga(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_ga(T61, T62, T63, T85, T68, mergesort32_in_ga(T67, T85))
mergesort32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U14_ga(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_ga(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_ga(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_ga(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_ga(T61, T62, T63, T104, X185, merge33_in_aaa(.(T104, T95), T97, X185))
merge33_in_aaa([], T115, T115) → merge33_out_aaa([], T115, T115)
merge33_in_aaa(T120, [], T120) → merge33_out_aaa(T120, [], T120)
merge33_in_aaa(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aaa(T143, T132, T134, T136, merge33_in_aaa(.(T143, T132), T134, T136))
U18_aaa(T143, T132, T134, T136, merge33_out_aaa(.(T143, T132), T134, T136)) → merge33_out_aaa(.(T143, T132), .(T143, T134), .(T143, T136))
U17_ga(T61, T62, T63, T104, X185, merge33_out_aaa(.(T104, T95), T97, X185)) → mergesort32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U12_ga(T61, T62, T63, T85, T68, mergesort32_out_ga(T67, T85)) → U13_ga(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T80) → U8_gg(T61, T62, T63, T80, split17_in_agaa(T61, .(T62, T63), T67, T68))
U8_gg(T61, T62, T63, T80, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_gg(T61, T62, T63, T80, T68, mergesort32_in_gg(T67, []))
mergesort32_in_gg(.(T61, .(T62, T63)), T85) → U11_gg(T61, T62, T63, T85, split17_in_agaa(T61, .(T62, T63), T67, T68))
U11_gg(T61, T62, T63, T85, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U12_gg(T61, T62, T63, T85, T68, mergesort32_in_gg(T67, T85))
mergesort32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U14_gg(T61, T62, T63, T104, X185, split17_in_agaa(T61, .(T62, T63), T67, T68))
U14_gg(T61, T62, T63, T104, X185, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_in_ga(T67, .(T104, T95)))
U15_gg(T61, T62, T63, T104, X185, T68, mergesort32_out_ga(T67, .(T104, T95))) → U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_in_ga(T68, .(T104, T97)))
U16_gg(T61, T62, T63, T104, X185, T95, mergesort32_out_ga(T68, .(T104, T97))) → U17_gg(T61, T62, T63, T104, X185, merge33_in_aag(.(T104, T95), T97, X185))
merge33_in_aag([], T115, T115) → merge33_out_aag([], T115, T115)
merge33_in_aag(T120, [], T120) → merge33_out_aag(T120, [], T120)
merge33_in_aag(.(T143, T132), .(T143, T134), .(T143, T136)) → U18_aag(T143, T132, T134, T136, merge33_in_aag(.(T143, T132), T134, T136))
U18_aag(T143, T132, T134, T136, merge33_out_aag(.(T143, T132), T134, T136)) → merge33_out_aag(.(T143, T132), .(T143, T134), .(T143, T136))
U17_gg(T61, T62, T63, T104, X185, merge33_out_aag(.(T104, T95), T97, X185)) → mergesort32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U12_gg(T61, T62, T63, T85, T68, mergesort32_out_gg(T67, T85)) → U13_gg(T61, T62, T63, T85, mergesort32_in_gg(T68, []))
U13_gg(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_gg(.(T61, .(T62, T63)), T85)
U9_gg(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_gg(T61, T62, T63, T80, mergesort32_in_gg(T68, T80))
U10_gg(T61, T62, T63, T80, mergesort32_out_gg(T68, T80)) → mergesort32_out_gg(.(T61, .(T62, T63)), T80)
U13_ga(T61, T62, T63, T85, mergesort32_out_gg(T68, [])) → mergesort32_out_ga(.(T61, .(T62, T63)), T85)
U5_gg(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U4_gg(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_gg(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_gg(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_gg(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_gg(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_gg(.(T61, .(T62, T63)), X137)
U9_ga(T61, T62, T63, T80, T68, mergesort32_out_gg(T67, [])) → U10_ga(T61, T62, T63, T80, mergesort32_in_ga(T68, T80))
U10_ga(T61, T62, T63, T80, mergesort32_out_ga(T68, T80)) → mergesort32_out_ga(.(T61, .(T62, T63)), T80)
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U23_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U22_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U24_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U25_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2)  =  mergesort1_in_ga(x1)
[]  =  []
mergesort1_out_ga(x1, x2)  =  mergesort1_out_ga(x1)
.(x1, x2)  =  .(x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x3, x5)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x3, x5)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x3, x5)
U22_ga(x1, x2, x3, x4, x5, x6)  =  U22_ga(x3, x5, x6)
U23_ga(x1, x2, x3, x4, x5)  =  U23_ga(x3, x5)
mergesort32_in_ga(x1, x2)  =  mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2)  =  mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5, x6)  =  U9_ga(x3, x5, x6)
mergesort32_in_gg(x1, x2)  =  mergesort32_in_gg(x1, x2)
mergesort32_out_gg(x1, x2)  =  mergesort32_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U5_gg(x1, x2, x3, x4, x5)  =  U5_gg(x3, x4, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5, x6)  =  U12_ga(x3, x5, x6)
U14_ga(x1, x2, x3, x4, x5, x6)  =  U14_ga(x3, x6)
U15_ga(x1, x2, x3, x4, x5, x6, x7)  =  U15_ga(x3, x6, x7)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
U16_ga(x1, x2, x3, x4, x5, x6, x7)  =  U16_ga(x3, x7)
U17_ga(x1, x2, x3, x4, x5, x6)  =  U17_ga(x3, x6)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x3, x5)
U8_gg(x1, x2, x3, x4, x5)  =  U8_gg(x3, x4, x5)
U9_gg(x1, x2, x3, x4, x5, x6)  =  U9_gg(x3, x4, x5, x6)
U11_gg(x1, x2, x3, x4, x5)  =  U11_gg(x3, x4, x5)
U12_gg(x1, x2, x3, x4, x5, x6)  =  U12_gg(x3, x4, x5, x6)
U14_gg(x1, x2, x3, x4, x5, x6)  =  U14_gg(x3, x5, x6)
U15_gg(x1, x2, x3, x4, x5, x6, x7)  =  U15_gg(x3, x5, x6, x7)
U16_gg(x1, x2, x3, x4, x5, x6, x7)  =  U16_gg(x3, x5, x7)
U17_gg(x1, x2, x3, x4, x5, x6)  =  U17_gg(x3, x5, x6)
merge33_in_aag(x1, x2, x3)  =  merge33_in_aag(x3)
merge33_out_aag(x1, x2, x3)  =  merge33_out_aag(x1, x2, x3)
U18_aag(x1, x2, x3, x4, x5)  =  U18_aag(x4, x5)
U13_gg(x1, x2, x3, x4, x5)  =  U13_gg(x3, x4, x5)
U10_gg(x1, x2, x3, x4, x5)  =  U10_gg(x3, x4, x5)
U6_gg(x1, x2, x3, x4, x5, x6)  =  U6_gg(x3, x4, x5, x6)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x3, x4, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x3, x6)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x3, x5)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x3, x5)

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

(145) UsableRulesProof (EQUIVALENT transformation)

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

(146) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U20_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, T28, T29))
U20_GA(T23, T24, T25, T14, split17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_agaa(T36, T37, .(T36, X81), X82) → U2_agaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_agaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
split21_in_gaa(x1, x2, x3)  =  split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3)  =  split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
split17_in_agaa(x1, x2, x3, x4)  =  split17_in_agaa(x2)
U2_agaa(x1, x2, x3, x4, x5)  =  U2_agaa(x2, x5)
split17_out_agaa(x1, x2, x3, x4)  =  split17_out_agaa(x2, x3, x4)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x3, x5)

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

(147) PiDPToQDPProof (SOUND transformation)

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

(148) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, split17_in_agaa(T25))
U20_GA(T25, split17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(149) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, split17_in_agaa(T25)) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, U2_agaa(T25, split21_in_gaa(T25)))

(150) Obligation:

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

U20_GA(T25, split17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, U2_agaa(T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split17_in_agaa(T37) → U2_agaa(T37, split21_in_gaa(T37))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(151) UsableRulesProof (EQUIVALENT transformation)

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

(152) Obligation:

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

U20_GA(T25, split17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, U2_agaa(T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

split17_in_agaa(x0)
U2_agaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(153) QReductionProof (EQUIVALENT transformation)

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

split17_in_agaa(x0)

(154) Obligation:

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

U20_GA(T25, split17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, U2_agaa(T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

U2_agaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U20_GA(T25, split17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 1 + x1   
POL(MERGESORT1_IN_GA(x1)) = x1   
POL(U1_gaa(x1, x2)) = 1 + x2   
POL(U20_GA(x1, x2)) = 1 + x2   
POL(U2_agaa(x1, x2)) = 1 + x2   
POL([]) = 0   
POL(split17_out_agaa(x1, x2, x3)) = 1 + x3   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2, x3)) = x2 + x3   

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

(156) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U20_GA(T25, U2_agaa(T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T43)) → U1_gaa(T43, split21_in_gaa(T43))
U2_agaa(T37, split21_out_gaa(T37, X82, X81)) → split17_out_agaa(T37, .(X81), X82)
U1_gaa(T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

U2_agaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(157) DependencyGraphProof (EQUIVALENT transformation)

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

(158) TRUE