(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V)))).
merge(X, [], X).
merge([], X, X).
merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)).
merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)).
split([], [], []).
split(.(E, U), .(E, V), W) :- split(U, W, V).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

mergesort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) 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)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, T72), ','(mergesort32(T68, T73), merge33(T72, T73, X137)))).
merge33(T82, [], T82).
merge33([], T87, T87).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- le68(T108, T110).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- ','(le68(T108, T110), merge33(T109, .(T110, T111), T113)).
merge33(.(T151, T152), .(T153, T154), .(T153, T156)) :- gt85(T151, T153).
merge33(.(T151, T152), .(T153, T154), .(T153, T156)) :- ','(gt85(T151, T153), merge33(.(T151, T152), T154, T156)).
le68(s(T126), s(T127)) :- le68(T126, T127).
le68(0, s(T134)).
le68(0, 0).
gt85(s(T169), s(T170)) :- gt85(T169, T170).
gt85(s(T175), 0).
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).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (b,b,f,f)
split21_in: (b,f,f)
mergesort32_in: (b,f)
merge33_in: (f,f,f)
le68_in: (f,f)
gt85_in: (f,f)
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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(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) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_AA(T151, T153)
GT85_IN_AA(s(T169), s(T170)) → U15_AA(T169, T170, gt85_in_aa(T169, T170))
GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_AAA(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x3, x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4)  =  SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x2, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x3, x5)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x1, x2, x3, x5, x6)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x1, x2, x3, x6)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA
U14_AA(x1, x2, x3)  =  U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6)  =  U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA
U15_AA(x1, x2, x3)  =  U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x2, x3, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)

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

(6) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(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) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_AA(T151, T153)
GT85_IN_AA(s(T169), s(T170)) → U15_AA(T169, T170, gt85_in_aa(T169, T170))
GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_AAA(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x3, x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4)  =  SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x2, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x3, x5)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x1, x2, x3, x5, x6)
U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x1, x2, x3, x6)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA
U14_AA(x1, x2, x3)  =  U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6)  =  U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA
U15_AA(x1, x2, x3)  =  U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x1, x2, x3, x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

GT85_IN_AAGT85_IN_AA

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

(14) 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 = GT85_IN_AA evaluates to t =GT85_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT85_IN_AA to GT85_IN_AA.



(15) NO

(16) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LE68_IN_AALE68_IN_AA

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

(21) 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 = LE68_IN_AA evaluates to t =LE68_IN_AA

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 LE68_IN_AA to LE68_IN_AA.



(22) NO

(23) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)

The TRS R consists of the following rules:

le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
s(x1)  =  s(x1)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(le68_in_aa)
MERGE33_IN_AAAU12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAAU10_AAA(le68_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))

(29) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAAU12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAAU12_AAA(gt85_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAAU12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAAU12_AAA(gt85_out_aa(0))

(31) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))
MERGE33_IN_AAAU12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAAU12_AAA(gt85_out_aa(0))

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(32) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

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

MERGE33_IN_AAAU10_AAA(le68_out_aa(0))
with rule MERGE33_IN_AAAU10_AAA(le68_out_aa(0)) at position [] and matcher [ ]

U10_AAA(le68_out_aa(0))MERGE33_IN_AAA
with rule U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA

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


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



(33) NO

(34) 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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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:

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)  =  .(x1, x2)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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:

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

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

(39) 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(.(T42, T43)) → SPLIT21_IN_GAA(T43)
    The graph contains the following edges 1 > 1

(40) YES

(41) 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_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)

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

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) 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_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
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_ggaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
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_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))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
s(x1)  =  s(x1)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)

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

(44) PiDPToQDPProof (SOUND transformation)

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

(45) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(46) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63))))

(47) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63))))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(48) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))

(49) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U4_GA(x1, x2, x3, x4)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3 +
[1,0]
·x4

POL(split17_out_ggaa(x1, x2, x3, x4)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\11/
·x2 +
/01\
\00/
·x3 +
/01\
\00/
·x4

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

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

POL(U6_GA(x1, x2, x3, x4, x5)) = 0 +
[0,0]
·x1 +
[0,0]
·x2 +
[0,0]
·x3 +
[0,1]
·x4 +
[0,0]
·x5

POL(mergesort32_in_ga(x1)) =
/0\
\0/
+
/01\
\01/
·x1

POL(mergesort32_out_ga(x1)) =
/0\
\0/
+
/10\
\10/
·x1

POL(U2_ggaa(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\11/
·x2 +
/01\
\00/
·x3

POL(U1_gaa(x1, x2, x3)) =
/1\
\0/
+
/11\
\01/
·x1 +
/00\
\00/
·x2 +
/11\
\10/
·x3

POL(split21_in_gaa(x1)) =
/0\
\0/
+
/11\
\10/
·x1

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

POL(U3_ga(x1, x2, x3, x4)) =
/1\
\0/
+
/00\
\10/
·x1 +
/00\
\00/
·x2 +
/00\
\10/
·x3 +
/10\
\10/
·x4

POL(split17_in_ggaa(x1, x2)) =
/0\
\0/
+
/00\
\11/
·x1 +
/11\
\00/
·x2

POL(U4_ga(x1, x2, x3, x4)) =
/0\
\1/
+
/00\
\00/
·x1 +
/11\
\11/
·x2 +
/01\
\11/
·x3 +
/11\
\01/
·x4

POL(split21_out_gaa(x1, x2, x3)) =
/0\
\0/
+
/00\
\00/
·x1 +
/11\
\01/
·x2 +
/10\
\11/
·x3

POL(U5_ga(x1, x2, x3, x4)) =
/0\
\1/
+
/00\
\10/
·x1 +
/00\
\01/
·x2 +
/11\
\11/
·x3 +
/11\
\11/
·x4

POL(U6_ga(x1, x2, x3, x4, x5)) =
/0\
\1/
+
/11\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\11/
·x4 +
/11\
\10/
·x5

POL(U7_ga(x1, x2, x3, x4)) =
/1\
\1/
+
/00\
\00/
·x1 +
/01\
\00/
·x2 +
/11\
\10/
·x3 +
/11\
\10/
·x4

POL(U8_ga(x1, x2, x3, x4)) =
/0\
\0/
+
/00\
\00/
·x1 +
/00\
\11/
·x2 +
/11\
\00/
·x3 +
/00\
\11/
·x4

POL(U9_ga(x1, x2, x3, x4)) =
/1\
\0/
+
/01\
\11/
·x1 +
/11\
\00/
·x2 +
/11\
\10/
·x3 +
/00\
\11/
·x4

POL(merge33_in_aaa) =
/1\
\1/

POL(merge33_out_aaa) =
/0\
\0/

POL(U10_aaa(x1)) =
/1\
\0/
+
/00\
\00/
·x1

POL(le68_in_aa) =
/0\
\0/

POL(U12_aaa(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(gt85_in_aa) =
/0\
\0/

POL(U14_aa(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(le68_out_aa(x1)) =
/0\
\1/
+
/00\
\10/
·x1

POL(0) =
/0\
\1/

POL(U11_aaa(x1)) =
/0\
\0/
+
/11\
\11/
·x1

POL(U15_aa(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(gt85_out_aa(x1)) =
/1\
\0/
+
/01\
\00/
·x1

POL(U13_aaa(x1)) =
/0\
\0/
+
/11\
\00/
·x1

POL(s(x1)) =
/1\
\0/
+
/10\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)

(51) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

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

(53) TRUE

(54) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, 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(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)

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

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(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)  =  .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, x2, 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(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x1, x2, x3, x4)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)

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

(57) PiDPToQDPProof (SOUND transformation)

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

(58) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, split17_in_ggaa(T24, T25))
U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(59) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, split17_in_ggaa(T24, T25)) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

(60) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(61) 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.

(62) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(63) 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_ggaa(x0, x1)

(64) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U17_GA(x1, ..., x4) ) = max{0, 2x1 + x2 + 2x4 - 2}


POL( U2_ggaa(x1, ..., x3) ) = 2x3 + 2


POL( split21_in_gaa(x1) ) = 2x1


POL( [] ) = 0


POL( split21_out_gaa(x1, ..., x3) ) = x2 + x3


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


POL( U1_gaa(x1, ..., x3) ) = x1 + 2x3 + 1


POL( split17_out_ggaa(x1, ..., x4) ) = 2x4 + 2


POL( MERGESORT1_IN_GA(x1) ) = max{0, 2x1 - 1}



The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

(66) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(67) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(68) YES

(69) 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: (b,b,f,f)
split21_in: (b,f,f)
mergesort32_in: (b,f)
merge33_in: (f,f,f)
le68_in: (f,f)
gt85_in: (f,f)
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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(70) 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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)

(71) 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) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(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) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_AA(T151, T153)
GT85_IN_AA(s(T169), s(T170)) → U15_AA(T169, T170, gt85_in_aa(T169, T170))
GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_AAA(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4)  =  SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x5, x6)
U20_GA(x1, x2, x3, x4, x5)  =  U20_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)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x6)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA
U14_AA(x1, x2, x3)  =  U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6)  =  U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA
U15_AA(x1, x2, x3)  =  U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)

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

(72) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(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) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_AA(T151, T153)
GT85_IN_AA(s(T169), s(T170)) → U15_AA(T169, T170, gt85_in_aa(T169, T170))
GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_AAA(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4)  =  SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5)  =  U2_GGAA(x1, x5)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x5, x6)
U20_GA(x1, x2, x3, x4, x5)  =  U20_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)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x6)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA
U14_AA(x1, x2, x3)  =  U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6)  =  U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA
U15_AA(x1, x2, x3)  =  U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6)  =  U21_GA(x6)
U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x5)

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

(73) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes.

(74) Complex Obligation (AND)

(75) Obligation:

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

GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA

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

(76) UsableRulesProof (EQUIVALENT transformation)

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

(77) Obligation:

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

GT85_IN_AA(s(T169), s(T170)) → GT85_IN_AA(T169, T170)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT85_IN_AA(x1, x2)  =  GT85_IN_AA

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

(78) PiDPToQDPProof (SOUND transformation)

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

(79) Obligation:

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

GT85_IN_AAGT85_IN_AA

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

(80) 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 = GT85_IN_AA evaluates to t =GT85_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT85_IN_AA to GT85_IN_AA.



(81) NO

(82) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA

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

(83) UsableRulesProof (EQUIVALENT transformation)

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

(84) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE68_IN_AA(x1, x2)  =  LE68_IN_AA

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

(85) PiDPToQDPProof (SOUND transformation)

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

(86) Obligation:

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

LE68_IN_AALE68_IN_AA

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

(87) 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 = LE68_IN_AA evaluates to t =LE68_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE68_IN_AA to LE68_IN_AA.



(88) NO

(89) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)

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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)

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

(90) UsableRulesProof (EQUIVALENT transformation)

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

(91) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_AAA(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U12_AAA(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → MERGE33_IN_AAA(.(T151, T152), T154, T156)

The TRS R consists of the following rules:

le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
s(x1)  =  s(x1)
MERGE33_IN_AAA(x1, x2, x3)  =  MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6)  =  U12_AAA(x6)

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

(92) PiDPToQDPProof (SOUND transformation)

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

(93) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(le68_in_aa)
MERGE33_IN_AAAU12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAAU10_AAA(le68_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))

(95) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAAU12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(96) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAAU12_AAA(gt85_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAAU12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAAU12_AAA(gt85_out_aa(0))

(97) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
U12_AAA(gt85_out_aa(T153)) → MERGE33_IN_AAA
MERGE33_IN_AAAU10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAAU10_AAA(le68_out_aa(0))
MERGE33_IN_AAAU12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAAU12_AAA(gt85_out_aa(0))

The TRS R consists of the following rules:

le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(98) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

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

MERGE33_IN_AAAU10_AAA(le68_out_aa(0))
with rule MERGE33_IN_AAAU10_AAA(le68_out_aa(0)) at position [] and matcher [ ]

U10_AAA(le68_out_aa(0))MERGE33_IN_AAA
with rule U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA

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


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



(99) NO

(100) 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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(101) UsableRulesProof (EQUIVALENT transformation)

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

(102) 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)  =  .(x1, x2)
SPLIT21_IN_GAA(x1, x2, x3)  =  SPLIT21_IN_GAA(x1)

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

(103) PiDPToQDPProof (SOUND transformation)

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

(104) Obligation:

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

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

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

(105) 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(.(T42, T43)) → SPLIT21_IN_GAA(T43)
    The graph contains the following edges 1 > 1

(106) YES

(107) 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_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)

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

(108) UsableRulesProof (EQUIVALENT transformation)

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

(109) 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_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
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_ggaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
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_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))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
s(x1)  =  s(x1)
MERGESORT32_IN_GA(x1, x2)  =  MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x5, x6)

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

(110) PiDPToQDPProof (SOUND transformation)

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

(111) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(split17_out_ggaa(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_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
U9_ga(merge33_out_aaa) → mergesort32_out_ga
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(112) 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(.(T54, [])) → mergesort32_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(0) = 0   
POL(MERGESORT32_IN_GA(x1)) = x1   
POL(U10_aaa(x1)) = x1   
POL(U11_aaa(x1)) = x1   
POL(U12_aaa(x1)) = x1   
POL(U13_aaa(x1)) = x1   
POL(U14_aa(x1)) = x1   
POL(U15_aa(x1)) = x1   
POL(U1_gaa(x1, x2)) = 1 + x1 + x2   
POL(U2_ggaa(x1, x2)) = x1 + x2   
POL(U3_ga(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL(U4_ga(x1)) = x1   
POL(U5_ga(x1)) = x1   
POL(U6_GA(x1, x2)) = x1 + x2   
POL(U6_ga(x1, x2)) = x1 + x2   
POL(U7_ga(x1)) = x1   
POL(U8_ga(x1)) = x1   
POL(U9_ga(x1)) = x1   
POL([]) = 0   
POL(gt85_in_aa) = 0   
POL(gt85_out_aa(x1)) = x1   
POL(le68_in_aa) = 0   
POL(le68_out_aa(x1)) = x1   
POL(merge33_in_aaa) = 0   
POL(merge33_out_aaa) = 0   
POL(mergesort32_in_ga(x1)) = x1   
POL(mergesort32_out_ga) = 0   
POL(s(x1)) = x1   
POL(split17_in_ggaa(x1, x2)) = 1 + x1 + x2   
POL(split17_out_ggaa(x1, x2)) = x1 + x2   
POL(split21_in_gaa(x1)) = 1 + x1   
POL(split21_out_gaa(x1, x2)) = 1 + x1 + x2   

(113) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(split17_out_ggaa(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_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
U9_ga(merge33_out_aaa) → mergesort32_out_ga
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(114) 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_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

Strictly oriented rules of the TRS R:

mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U9_ga(merge33_out_aaa) → mergesort32_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(0) = 0   
POL(MERGESORT32_IN_GA(x1)) = 2 + 2·x1   
POL(U10_aaa(x1)) = 1 + 2·x1   
POL(U11_aaa(x1)) = x1   
POL(U12_aaa(x1)) = 1 + 2·x1   
POL(U13_aaa(x1)) = x1   
POL(U14_aa(x1)) = 2·x1   
POL(U15_aa(x1)) = 2·x1   
POL(U1_gaa(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U2_ggaa(x1, x2)) = 2 + 2·x1 + x2   
POL(U3_ga(x1)) = 2 + x1   
POL(U4_GA(x1)) = 1 + 2·x1   
POL(U4_ga(x1)) = 2 + 2·x1   
POL(U5_ga(x1)) = 1 + x1   
POL(U6_GA(x1, x2)) = 1 + 2·x1 + x2   
POL(U6_ga(x1, x2)) = 2 + 2·x1 + x2   
POL(U7_ga(x1)) = 1 + x1   
POL(U8_ga(x1)) = 1 + x1   
POL(U9_ga(x1)) = 1 + 2·x1   
POL([]) = 0   
POL(gt85_in_aa) = 0   
POL(gt85_out_aa(x1)) = x1   
POL(le68_in_aa) = 0   
POL(le68_out_aa(x1)) = 2·x1   
POL(merge33_in_aaa) = 1   
POL(merge33_out_aaa) = 1   
POL(mergesort32_in_ga(x1)) = 2 + 2·x1   
POL(mergesort32_out_ga) = 2   
POL(s(x1)) = x1   
POL(split17_in_ggaa(x1, x2)) = 2 + 2·x1 + x2   
POL(split17_out_ggaa(x1, x2)) = 1 + x1 + x2   
POL(split21_in_gaa(x1)) = x1   
POL(split21_out_gaa(x1, x2)) = x1 + 2·x2   

(115) Obligation:

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

U4_GA(split17_out_ggaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
merge33_in_aaamerge33_out_aaa
merge33_in_aaaU10_aaa(le68_in_aa)
merge33_in_aaaU12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T153)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T153)) → U13_aaa(merge33_in_aaa)
le68_in_aaU14_aa(le68_in_aa)
le68_in_aale68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aaU15_aa(gt85_in_aa)
gt85_in_aagt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T170)) → gt85_out_aa(s(T170))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(116) DependencyGraphProof (EQUIVALENT transformation)

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

(117) TRUE

(118) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(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) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(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_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_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_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(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_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
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_ggaa(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)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa(T82, [], T82) → merge33_out_aaa(T82, [], T82)
merge33_in_aaa([], T87, T87) → merge33_out_aaa([], T87, T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T151, T152), .(T153, T154), .(T153, T156)) → U12_aaa(T151, T152, T153, T154, T156, gt85_in_aa(T151, T153))
gt85_in_aa(s(T169), s(T170)) → U15_aa(T169, T170, gt85_in_aa(T169, T170))
gt85_in_aa(s(T175), 0) → gt85_out_aa(s(T175), 0)
U15_aa(T169, T170, gt85_out_aa(T169, T170)) → gt85_out_aa(s(T169), s(T170))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U12_aaa(T151, T152, T153, T154, T156, gt85_out_aa(T151, T153)) → U13_aaa(T151, T152, T153, T154, T156, merge33_in_aaa(.(T151, T152), T154, T156))
U13_aaa(T151, T152, T153, T154, T156, merge33_out_aaa(.(T151, T152), T154, T156)) → merge33_out_aaa(.(T151, T152), .(T153, T154), .(T153, T156))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_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)  =  .(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6)  =  U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5)  =  U20_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)
U6_ga(x1, x2, x3, x4, x5, x6)  =  U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6)  =  U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
merge33_in_aaa(x1, x2, x3)  =  merge33_in_aaa
merge33_out_aaa(x1, x2, x3)  =  merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
le68_in_aa(x1, x2)  =  le68_in_aa
U14_aa(x1, x2, x3)  =  U14_aa(x3)
le68_out_aa(x1, x2)  =  le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6)  =  U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6)  =  U12_aaa(x6)
gt85_in_aa(x1, x2)  =  gt85_in_aa
U15_aa(x1, x2, x3)  =  U15_aa(x3)
gt85_out_aa(x1, x2)  =  gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6)  =  U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5)  =  U22_ga(x5)
s(x1)  =  s(x1)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)

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

(119) UsableRulesProof (EQUIVALENT transformation)

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

(120) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(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)  =  .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4)  =  split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5)  =  U2_ggaa(x1, 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(x1, x5)
split17_out_ggaa(x1, x2, x3, x4)  =  split17_out_ggaa(x3, x4)
MERGESORT1_IN_GA(x1, x2)  =  MERGESORT1_IN_GA(x1)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x5)

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

(121) PiDPToQDPProof (SOUND transformation)

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

(122) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, split17_in_ggaa(T24, T25))
U17_GA(T23, split17_out_ggaa(T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(123) 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:

U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + 2·x2   
POL(MERGESORT1_IN_GA(x1)) = 2·x1   
POL(U17_GA(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(U1_gaa(x1, x2)) = x1 + 2·x2   
POL(U2_ggaa(x1, x2)) = x1 + 2·x2   
POL([]) = 0   
POL(split17_in_ggaa(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(split17_out_ggaa(x1, x2)) = x1 + 2·x2   
POL(split21_in_gaa(x1)) = 1 + x1   
POL(split21_out_gaa(x1, x2)) = 1 + x1 + 2·x2   

(124) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, split17_in_ggaa(T24, T25))
U17_GA(T23, split17_out_ggaa(T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(125) DependencyGraphProof (EQUIVALENT transformation)

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

(126) TRUE