(0) Obligation:

Clauses:

ackermann(0, N, s(N)).
ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val).
ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)).

Queries:

ackermann(g,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

ackermann21(T28, X73) :- ackermann27(T28, X73).
ackermann27(0, s(s(0))).
ackermann27(s(T32), X97) :- ackermann21(T32, X96).
ackermann27(s(T32), X97) :- ','(ackermann21(T32, T34), ackermann38(T32, T34, X97)).
ackermann38(0, T42, s(T42)).
ackermann38(s(T47), 0, X133) :- ackermann27(T47, X133).
ackermann38(s(T52), s(T53), X151) :- ackermann38(s(T52), T53, X150).
ackermann38(s(T52), s(T53), X151) :- ','(ackermann38(s(T52), T53, T55), ackermann38(T52, T55, X151)).
ackermann57(T79, 0, X211) :- ackermann27(T79, X211).
ackermann57(T84, s(T86), X229) :- p65(T84, T86, X228, X229).
p65(T84, T86, X228, X229) :- ackermann57(T84, T86, X228).
p65(T84, T86, T88, X229) :- ','(ackermann57(T84, T86, T88), ackermann68(T84, T88, X229)).
ackermann68(0, T96, s(T96)).
ackermann68(s(T101), 0, X265) :- ackermann27(T101, X265).
ackermann68(s(T106), s(T108), X283) :- p65(T106, T108, X282, X283).
ackermann1(0, T5, s(T5)).
ackermann1(s(0), 0, s(s(0))).
ackermann1(s(s(T19)), 0, T20) :- ackermann21(T19, X40).
ackermann1(s(s(T19)), 0, T20) :- ','(ackermann21(T19, T22), ackermann1(T19, T22, T20)).
ackermann1(s(T68), s(T71), T70) :- ackermann57(T68, T71, X182).
ackermann1(s(T68), s(T71), T70) :- ','(ackermann57(T68, T71, T73), ackermann1(T68, T73, T70)).
ackermann1(s(T123), s(0), T117) :- ackermann27(T123, X319).
ackermann1(s(T123), s(0), T117) :- ','(ackermann27(T123, T124), ackermann1(T123, T124, T117)).
ackermann1(s(T131), s(s(T133)), T117) :- ackermann57(T131, T133, X340).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermann57(T131, T133, T135), ackermann68(T131, T135, X341)).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermann57(T131, T133, T135), ','(ackermann68(T131, T135, T139), ackermann1(T131, T139, T117))).

Queries:

ackermann1(g,a,g).

(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:
ackermann1_in: (b,f,b)
ackermann21_in: (b,f)
ackermann27_in: (b,f)
ackermann38_in: (b,f,f)
ackermann57_in: (b,f,f)
p65_in: (b,f,f,f)
ackermann68_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)

(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:

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3)  =  U16_GAG(x1, x2, x3)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3)  =  U5_GAA(x1, x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3)  =  U18_GAG(x1, x2, x3)
U19_GAG(x1, x2, x3, x4)  =  U19_GAG(x1, x3, x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3)  =  U9_GAA(x1, x3)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5)  =  U11_GAAA(x1, x5)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5)  =  U13_GAAA(x1, x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3)  =  U14_GAA(x1, x3)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x1, x4)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4)  =  U21_GAG(x1, x2, x3, x4)
U22_GAG(x1, x2, x3)  =  U22_GAG(x1, x2, x3)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3)  =  U24_GAG(x1, x2, x3)
U25_GAG(x1, x2, x3, x4)  =  U25_GAG(x1, x3, x4)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4)  =  U27_GAG(x1, x2, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4)  =  U29_GAG(x1, x2, x3, x4)

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

(6) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3)  =  U16_GAG(x1, x2, x3)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3)  =  U5_GAA(x1, x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3)  =  U18_GAG(x1, x2, x3)
U19_GAG(x1, x2, x3, x4)  =  U19_GAG(x1, x3, x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3)  =  U9_GAA(x1, x3)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5)  =  U11_GAAA(x1, x5)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5)  =  U13_GAAA(x1, x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3)  =  U14_GAA(x1, x3)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x1, x4)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4)  =  U21_GAG(x1, x2, x3, x4)
U22_GAG(x1, x2, x3)  =  U22_GAG(x1, x2, x3)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3)  =  U24_GAG(x1, x2, x3)
U25_GAG(x1, x2, x3, x4)  =  U25_GAG(x1, x3, x4)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4)  =  U27_GAG(x1, x2, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4)  =  U29_GAG(x1, x2, x3, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 28 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)

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:

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)

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:

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermann21_in_ga(T32))
U3_GA(T32, ackermann21_out_ga(T32)) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa(s(T52))) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(14) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
U3_GA(T32, ackermann21_out_ga(T32)) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
U7_GAA(T52, ackermann38_out_gaa(s(T52))) → ACKERMANN38_IN_GAA(T52)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ACKERMANN21_IN_GA(x1)) = x1   
POL(ACKERMANN27_IN_GA(x1)) = x1   
POL(ACKERMANN38_IN_GAA(x1)) = x1   
POL(U1_ga(x1, x2)) = 0   
POL(U2_ga(x1, x2)) = 0   
POL(U3_GA(x1, x2)) = 1 + x1   
POL(U3_ga(x1, x2)) = 0   
POL(U4_ga(x1, x2)) = 0   
POL(U5_gaa(x1, x2)) = 0   
POL(U6_gaa(x1, x2)) = 0   
POL(U7_GAA(x1, x2)) = 1 + x1   
POL(U7_gaa(x1, x2)) = 0   
POL(U8_gaa(x1, x2)) = 0   
POL(ackermann21_in_ga(x1)) = 0   
POL(ackermann21_out_ga(x1)) = 0   
POL(ackermann27_in_ga(x1)) = 1 + x1   
POL(ackermann27_out_ga(x1)) = 1   
POL(ackermann38_in_gaa(x1)) = 1 + x1   
POL(ackermann38_out_gaa(x1)) = 1   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(15) Obligation:

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

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermann21_in_ga(T32))
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(16) DependencyGraphProof (EQUIVALENT transformation)

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

(17) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(18) 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.

(19) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(20) 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].

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

(21) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

(22) 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 = ACKERMANN38_IN_GAA(s(T52)) evaluates to t =ACKERMANN38_IN_GAA(s(T52))

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 ACKERMANN38_IN_GAA(s(T52)) to ACKERMANN38_IN_GAA(s(T52)).



(23) NO

(24) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)

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

(25) UsableRulesProof (EQUIVALENT transformation)

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

(26) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)

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

(27) PiDPToQDPProof (SOUND transformation)

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

(28) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
U12_GAAA(T84, ackermann57_out_gaa(T84, T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACKERMANN57_IN_GAA(x1)) = 1 + x1   
POL(ACKERMANN68_IN_GAA(x1)) = x1   
POL(P65_IN_GAAA(x1)) = 1 + x1   
POL(U10_gaa(x1, x2)) = 0   
POL(U11_gaaa(x1, x2)) = 0   
POL(U12_GAAA(x1, x2)) = x1   
POL(U12_gaaa(x1, x2)) = 0   
POL(U13_gaaa(x1, x2, x3)) = 0   
POL(U14_gaa(x1, x2)) = 1   
POL(U15_gaa(x1, x2)) = 0   
POL(U1_ga(x1, x2)) = 0   
POL(U2_ga(x1, x2)) = 0   
POL(U3_ga(x1, x2)) = 0   
POL(U4_ga(x1, x2)) = 0   
POL(U5_gaa(x1, x2)) = 0   
POL(U6_gaa(x1, x2)) = 0   
POL(U7_gaa(x1, x2)) = 0   
POL(U8_gaa(x1, x2)) = 0   
POL(U9_gaa(x1, x2)) = 0   
POL(ackermann21_in_ga(x1)) = 0   
POL(ackermann21_out_ga(x1)) = 0   
POL(ackermann27_in_ga(x1)) = 0   
POL(ackermann27_out_ga(x1)) = 0   
POL(ackermann38_in_gaa(x1)) = 0   
POL(ackermann38_out_gaa(x1)) = 0   
POL(ackermann57_in_gaa(x1)) = 0   
POL(ackermann57_out_gaa(x1, x2)) = 0   
POL(ackermann68_in_gaa(x1)) = x1   
POL(ackermann68_out_gaa(x1)) = 0   
POL(p65_in_gaaa(x1)) = 0   
POL(p65_out_gaaa(x1, x2)) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
U12_GAAA(T84, ackermann57_out_gaa(T84, T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(32) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(33) 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.

(34) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

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

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(35) 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].

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

(36) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

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

(37) 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 = ACKERMANN57_IN_GAA(T84') evaluates to t =ACKERMANN57_IN_GAA(T84')

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




Rewriting sequence

ACKERMANN57_IN_GAA(T84')P65_IN_GAAA(T84')
with rule ACKERMANN57_IN_GAA(T84'') → P65_IN_GAAA(T84'') at position [] and matcher [T84'' / T84']

P65_IN_GAAA(T84')ACKERMANN57_IN_GAA(T84')
with rule P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)

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.



(38) NO

(39) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x1, x2, x3)
U16_gag(x1, x2, x3)  =  U16_gag(x1, x2, x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x1, x2, x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x1, x3, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x2, x3, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x1, x2, x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x1, x2, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x1, x2, x3, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x1, x2, x3, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x1, x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x1, x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x1, x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3)  =  U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x1, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
U17_GAG(T19, T20, ackermann21_out_ga(T19)) → ACKERMANN1_IN_GAG(T19, T20)
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
U20_GAG(T68, T70, ackermann57_out_gaa(T68, T71)) → ACKERMANN1_IN_GAG(T68, T70)
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
U23_GAG(T123, T117, ackermann27_out_ga(T123)) → ACKERMANN1_IN_GAG(T123, T117)
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
U26_GAG(T131, T117, ackermann57_out_gaa(T131, T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131)) → ACKERMANN1_IN_GAG(T131, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann57_in_gaa(x0)
ackermann27_in_ga(x0)
ackermann68_in_gaa(x0)
U1_ga(x0, x1)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
p65_in_gaaa(x0)
U4_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann38_in_gaa(x0)
U13_gaaa(x0, x1, x2)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1)

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

(44) 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:

  • U17_GAG(T19, T20, ackermann21_out_ga(T19)) → ACKERMANN1_IN_GAG(T19, T20)
    The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
    The graph contains the following edges 1 > 1, 2 >= 2

  • U26_GAG(T131, T117, ackermann57_out_gaa(T131, T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
    The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3

  • U20_GAG(T68, T70, ackermann57_out_gaa(T68, T71)) → ACKERMANN1_IN_GAG(T68, T70)
    The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
    The graph contains the following edges 1 > 1, 2 >= 2

  • U23_GAG(T123, T117, ackermann27_out_ga(T123)) → ACKERMANN1_IN_GAG(T123, T117)
    The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2

  • U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131)) → ACKERMANN1_IN_GAG(T131, T117)
    The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

  • ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
    The graph contains the following edges 1 > 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
    The graph contains the following edges 1 > 1, 2 >= 2

(45) YES

(46) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ackermann1_in: (b,f,b)
ackermann21_in: (b,f)
ackermann27_in: (b,f)
ackermann38_in: (b,f,f)
ackermann57_in: (b,f,f)
p65_in: (b,f,f,f)
ackermann68_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(47) Obligation:

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

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)

(48) 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:

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3)  =  U16_GAG(x3)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3)  =  U5_GAA(x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3)  =  U18_GAG(x3)
U19_GAG(x1, x2, x3, x4)  =  U19_GAG(x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3)  =  U9_GAA(x3)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x4)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5)  =  U11_GAAA(x5)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5)  =  U13_GAAA(x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3)  =  U14_GAA(x3)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x4)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4)  =  U21_GAG(x2, x4)
U22_GAG(x1, x2, x3)  =  U22_GAG(x3)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3)  =  U24_GAG(x3)
U25_GAG(x1, x2, x3, x4)  =  U25_GAG(x4)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4)  =  U27_GAG(x2, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4)  =  U29_GAG(x2, x4)

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

(49) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3)  =  U16_GAG(x3)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3)  =  U5_GAA(x3)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3)  =  U18_GAG(x3)
U19_GAG(x1, x2, x3, x4)  =  U19_GAG(x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3)  =  U9_GAA(x3)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x4)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5)  =  U11_GAAA(x5)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5)  =  U13_GAAA(x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3)  =  U14_GAA(x3)
U15_GAA(x1, x2, x3, x4)  =  U15_GAA(x4)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4)  =  U21_GAG(x2, x4)
U22_GAG(x1, x2, x3)  =  U22_GAG(x3)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3)  =  U24_GAG(x3)
U25_GAG(x1, x2, x3, x4)  =  U25_GAG(x4)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4)  =  U27_GAG(x2, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4)  =  U29_GAG(x2, x4)

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

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 28 less nodes.

(51) Complex Obligation (AND)

(52) Obligation:

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

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)

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

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) Obligation:

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

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
ACKERMANN21_IN_GA(x1, x2)  =  ACKERMANN21_IN_GA(x1)
ACKERMANN27_IN_GA(x1, x2)  =  ACKERMANN27_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
ACKERMANN38_IN_GAA(x1, x2, x3)  =  ACKERMANN38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermann21_in_ga(T32))
U3_GA(T32, ackermann21_out_ga) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermann21_in_ga(T32))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(ACKERMANN21_IN_GA(x1)) = 0 +
[1,1]
·x1

POL(ACKERMANN27_IN_GA(x1)) = 0 +
[1,1]
·x1

POL(s(x1)) =
/1\
\0/
+
/10\
\11/
·x1

POL(U3_GA(x1, x2)) = 0 +
[0,1]
·x1 +
[0,0]
·x2

POL(ackermann21_in_ga(x1)) =
/0\
\0/
+
/01\
\00/
·x1

POL(ackermann21_out_ga) =
/1\
\1/

POL(ACKERMANN38_IN_GAA(x1)) = 0 +
[0,1]
·x1

POL(U7_GAA(x1, x2)) = 0 +
[0,1]
·x1 +
[0,0]
·x2

POL(ackermann38_in_gaa(x1)) =
/1\
\1/
+
/10\
\10/
·x1

POL(ackermann38_out_gaa) =
/1\
\1/

POL(U1_ga(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(ackermann27_in_ga(x1)) =
/1\
\0/
+
/10\
\00/
·x1

POL(U5_gaa(x1)) =
/1\
\0/
+
/10\
\00/
·x1

POL(U6_gaa(x1)) =
/1\
\0/
+
/11\
\10/
·x1

POL(U7_gaa(x1, x2)) =
/1\
\1/
+
/11\
\10/
·x1 +
/10\
\11/
·x2

POL(U2_ga(x1)) =
/0\
\0/
+
/00\
\00/
·x1

POL(ackermann27_out_ga) =
/1\
\0/

POL(0) =
/0\
\0/

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

POL(U4_ga(x1)) =
/0\
\0/
+
/00\
\11/
·x1

POL(U8_gaa(x1)) =
/0\
\0/
+
/00\
\00/
·x1

The following usable rules [FROCOS05] were oriented: none

(58) Obligation:

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

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
U3_GA(T32, ackermann21_out_ga) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ACKERMANN38_IN_GAA(x1)) = x1   
POL(U1_ga(x1)) = 0   
POL(U2_ga(x1)) = 0   
POL(U3_ga(x1, x2)) = 0   
POL(U4_ga(x1)) = 0   
POL(U5_gaa(x1)) = 0   
POL(U6_gaa(x1)) = 0   
POL(U7_GAA(x1, x2)) = 1 + x1   
POL(U7_gaa(x1, x2)) = 0   
POL(U8_gaa(x1)) = 0   
POL(ackermann21_in_ga(x1)) = 0   
POL(ackermann21_out_ga) = 0   
POL(ackermann27_in_ga(x1)) = 1 + x1   
POL(ackermann27_out_ga) = 1   
POL(ackermann38_in_gaa(x1)) = 1 + x1   
POL(ackermann38_out_gaa) = 1   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

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

(64) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(65) 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.

(66) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(67) 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].

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

(68) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

(69) 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 = ACKERMANN38_IN_GAA(s(T52)) evaluates to t =ACKERMANN38_IN_GAA(s(T52))

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 ACKERMANN38_IN_GAA(s(T52)) to ACKERMANN38_IN_GAA(s(T52)).



(70) NO

(71) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)

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

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
ACKERMANN57_IN_GAA(x1, x2, x3)  =  ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4)  =  P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5)  =  U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3)  =  ACKERMANN68_IN_GAA(x1)

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
U12_GAAA(T84, ackermann57_out_gaa(T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACKERMANN57_IN_GAA(x1)) = 1 + x1   
POL(ACKERMANN68_IN_GAA(x1)) = x1   
POL(P65_IN_GAAA(x1)) = 1 + x1   
POL(U10_gaa(x1)) = 0   
POL(U11_gaaa(x1)) = 0   
POL(U12_GAAA(x1, x2)) = x1   
POL(U12_gaaa(x1, x2)) = 0   
POL(U13_gaaa(x1, x2)) = 0   
POL(U14_gaa(x1)) = 0   
POL(U15_gaa(x1)) = 0   
POL(U1_ga(x1)) = 0   
POL(U2_ga(x1)) = 0   
POL(U3_ga(x1, x2)) = 0   
POL(U4_ga(x1)) = 0   
POL(U5_gaa(x1)) = 0   
POL(U6_gaa(x1)) = 0   
POL(U7_gaa(x1, x2)) = 0   
POL(U8_gaa(x1)) = 0   
POL(U9_gaa(x1)) = 0   
POL(ackermann21_in_ga(x1)) = 0   
POL(ackermann21_out_ga) = 0   
POL(ackermann27_in_ga(x1)) = 0   
POL(ackermann27_out_ga) = 0   
POL(ackermann38_in_gaa(x1)) = 0   
POL(ackermann38_out_gaa) = 0   
POL(ackermann57_in_gaa(x1)) = 0   
POL(ackermann57_out_gaa(x1)) = 0   
POL(ackermann68_in_gaa(x1)) = 1 + x1   
POL(ackermann68_out_gaa) = 0   
POL(p65_in_gaaa(x1)) = 0   
POL(p65_out_gaaa(x1)) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
U12_GAAA(T84, ackermann57_out_gaa(T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(78) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(79) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(80) 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.

(81) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

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

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(82) 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].

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

(83) Obligation:

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

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

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

(84) 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 = ACKERMANN57_IN_GAA(T84') evaluates to t =ACKERMANN57_IN_GAA(T84')

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




Rewriting sequence

ACKERMANN57_IN_GAA(T84')P65_IN_GAAA(T84')
with rule ACKERMANN57_IN_GAA(T84'') → P65_IN_GAAA(T84'') at position [] and matcher [T84'' / T84']

P65_IN_GAAA(T84')ACKERMANN57_IN_GAA(T84')
with rule P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)

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.



(85) NO

(86) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3)  =  ackermann1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
ackermann1_out_gag(x1, x2, x3)  =  ackermann1_out_gag(x2)
U16_gag(x1, x2, x3)  =  U16_gag(x3)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U17_gag(x1, x2, x3)  =  U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3)  =  U18_gag(x3)
U19_gag(x1, x2, x3, x4)  =  U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
U20_gag(x1, x2, x3, x4)  =  U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x2, x4)
U22_gag(x1, x2, x3)  =  U22_gag(x3)
U23_gag(x1, x2, x3)  =  U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3)  =  U24_gag(x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x4)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4)  =  U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4)  =  U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4)  =  U29_gag(x2, x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)

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

(87) UsableRulesProof (EQUIVALENT transformation)

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

(88) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
ackermann21_in_ga(x1, x2)  =  ackermann21_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
ackermann27_in_ga(x1, x2)  =  ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2)  =  ackermann27_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
ackermann21_out_ga(x1, x2)  =  ackermann21_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3)  =  ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3)  =  ackermann38_out_gaa
U5_gaa(x1, x2, x3)  =  U5_gaa(x3)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x4)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
ackermann57_in_gaa(x1, x2, x3)  =  ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3)  =  U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3)  =  ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4)  =  U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4)  =  p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5)  =  U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4)  =  p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5)  =  U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5)  =  U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3)  =  ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3)  =  ackermann68_out_gaa
U14_gaa(x1, x2, x3)  =  U14_gaa(x3)
U15_gaa(x1, x2, x3, x4)  =  U15_gaa(x4)
ACKERMANN1_IN_GAG(x1, x2, x3)  =  ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3)  =  U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4)  =  U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3)  =  U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4)  =  U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4)  =  U28_GAG(x1, x2, x3, x4)

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

(89) PiDPToQDPProof (SOUND transformation)

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

(90) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
U17_GAG(T19, T20, ackermann21_out_ga) → ACKERMANN1_IN_GAG(T19, T20)
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
U20_GAG(T68, T70, ackermann57_out_gaa(T71)) → ACKERMANN1_IN_GAG(T68, T70)
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
U23_GAG(T123, T117, ackermann27_out_ga) → ACKERMANN1_IN_GAG(T123, T117)
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
U26_GAG(T131, T117, ackermann57_out_gaa(T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
U28_GAG(T131, T133, T117, ackermann68_out_gaa) → ACKERMANN1_IN_GAG(T131, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann57_in_gaa(x0)
ackermann27_in_ga(x0)
ackermann68_in_gaa(x0)
U1_ga(x0)
U9_gaa(x0)
U10_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
p65_in_gaaa(x0)
U4_ga(x0)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann38_in_gaa(x0)
U13_gaaa(x0, x1)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U8_gaa(x0)

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

(91) 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:

  • U17_GAG(T19, T20, ackermann21_out_ga) → ACKERMANN1_IN_GAG(T19, T20)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
    The graph contains the following edges 1 > 1, 2 >= 2

  • U26_GAG(T131, T117, ackermann57_out_gaa(T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
    The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3

  • U20_GAG(T68, T70, ackermann57_out_gaa(T71)) → ACKERMANN1_IN_GAG(T68, T70)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
    The graph contains the following edges 1 > 1, 2 >= 2

  • U23_GAG(T123, T117, ackermann27_out_ga) → ACKERMANN1_IN_GAG(T123, T117)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U28_GAG(T131, T133, T117, ackermann68_out_gaa) → ACKERMANN1_IN_GAG(T131, T117)
    The graph contains the following edges 1 >= 1, 3 >= 2

  • ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
    The graph contains the following edges 1 > 1, 2 >= 2

  • ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
    The graph contains the following edges 1 > 1, 2 >= 2

(92) YES