(0) Obligation:

Clauses:

s2(plus(A, plus(B, C)), D) :- s2(plus(plus(A, B), C), D).
s2(plus(A, B), C) :- s2(plus(B, A), C).
s2(plus(X, 0), X).
s2(plus(X, Y), Z) :- ','(s2(X, A), ','(s2(Y, B), s2(plus(A, B), Z))).
s2(plus(A, B), C) :- ','(isNat(A), ','(isNat(B), add(A, B, C))).
isNat(s(X)) :- isNat(X).
isNat(0).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).

Queries:

s2(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

isNat33(s(T165)) :- isNat33(T165).
isNat33(0).
p22(T115, X104, T121, T117) :- s21(T115, X104).
p22(T115, T128, T121, T117) :- ','(s21(T115, T128), s21(plus(T121, T128), T117)).
add60(s(T282), T283, T284, s(T286)) :- add60(T282, T283, T284, T286).
add60(0, T295, T296, plus(T295, T296)).
add99(s(T457), T458, s(T460)) :- add99(T457, T458, T460).
add99(0, T466, T466).
s21(plus(T32, plus(T33, plus(T34, T35))), T37) :- s21(plus(plus(plus(T32, T33), T34), T35), T37).
s21(plus(T70, plus(T71, T72)), T74) :- s21(plus(T72, plus(T70, T71)), T74).
s21(plus(T87, plus(T88, 0)), plus(T87, T88)).
s21(plus(T113, plus(T114, T115)), T117) :- s21(plus(T113, T114), X103).
s21(plus(T113, plus(T114, T115)), T117) :- ','(s21(plus(T113, T114), T121), p22(T115, X104, T121, T117)).
s21(plus(T195, plus(T196, T197)), T199) :- s21(plus(plus(T196, T197), T195), T199).
s21(plus(T230, plus(T231, T232)), T234) :- s21(T230, X224).
s21(plus(T230, plus(T231, T232)), T234) :- ','(s21(T230, T238), p22(plus(T231, T232), X225, T238, T234)).
s21(plus(T255, plus(T256, T257)), T259) :- isNat33(T255).
s21(plus(plus(T321, T322), T320), T324) :- s21(plus(plus(T320, T321), T322), T324).
s21(plus(T353, T352), T355) :- s21(plus(T353, T352), T355).
s21(plus(0, T364), T364).
s21(plus(T386, T385), T388) :- s21(T385, X382).
s21(plus(T386, T385), T388) :- ','(s21(T385, T392), p22(T386, X383, T392, T388)).
s21(plus(T416, T415), T418) :- isNat33(T415).
s21(plus(T416, T415), T418) :- ','(isNat33(T415), isNat33(T416)).
s21(plus(T439, s(T438)), s(T441)) :- ','(isNat33(s(T438)), ','(isNat33(T439), add99(T438, T439, T441))).
s21(plus(T469, 0), T469) :- ','(isNat33(0), isNat33(T469)).
s21(plus(T474, 0), T474).
s21(plus(T495, T496), T498) :- s21(T495, X508).
s21(plus(T495, T496), T498) :- ','(s21(T495, T502), p22(T496, X509, T502, T498)).
s21(plus(T517, T518), T520) :- isNat33(T517).
s21(plus(T517, T518), T520) :- ','(isNat33(T517), isNat33(T518)).
s21(plus(T517, T518), T520) :- ','(isNat33(T517), ','(isNat33(T518), add99(T517, T518, T520))).

Queries:

s21(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
s21_in: (b,f) (f,f)
isNat33_in: (b) (f)
add99_in: (b,b,f)
p22_in: (b,f,f,f) (f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)

(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:

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_GA(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → U8_GA(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U9_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → U12_GA(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U13_GA(T230, T231, T232, T234, s21_in_ga(T230, X224))
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → U16_GA(T255, T256, T257, T259, isNat33_in_g(T255))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_G(T255)
ISNAT33_IN_G(s(T165)) → U1_G(T165, isNat33_in_g(T165))
ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → U17_GA(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → U18_GA(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → U19_GA(T386, T385, T388, s21_in_ga(T385, X382))
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
S21_IN_GA(plus(T416, T415), T418) → U22_GA(T416, T415, T418, isNat33_in_g(T415))
S21_IN_GA(plus(T416, T415), T418) → ISNAT33_IN_G(T415)
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → U23_GA(T416, T415, T418, isNat33_in_g(T416))
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U24_GA(T439, T438, T441, isNat33_in_g(s(T438)))
S21_IN_GA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_G(s(T438))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → U25_GA(T439, T438, T441, isNat33_in_g(T439))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → ISNAT33_IN_G(T439)
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → U26_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → ADD99_IN_GGA(T438, T439, T441)
ADD99_IN_GGA(s(T457), T458, s(T460)) → U6_GGA(T457, T458, T460, add99_in_gga(T457, T458, T460))
ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)
S21_IN_GA(plus(T469, 0), T469) → U27_GA(T469, isNat33_in_g(0))
S21_IN_GA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_GA(T469, isNat33_out_g(0)) → U28_GA(T469, isNat33_in_g(T469))
U27_GA(T469, isNat33_out_g(0)) → ISNAT33_IN_G(T469)
S21_IN_GA(plus(T495, T496), T498) → U29_GA(T495, T496, T498, s21_in_ga(T495, X508))
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → ISNAT33_IN_G(T518)
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → U34_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → U31_GA(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)
P22_IN_GAAA(T115, X104, T121, T117) → U2_GAAA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
P22_IN_GAAA(T115, T128, T121, T117) → U3_GAAA(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_GAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_AA(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → U8_AA(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U9_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → U12_AA(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U13_AA(T230, T231, T232, T234, s21_in_aa(T230, X224))
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → U16_AA(T255, T256, T257, T259, isNat33_in_a(T255))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_A(T255)
ISNAT33_IN_A(s(T165)) → U1_A(T165, isNat33_in_a(T165))
ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → U17_AA(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → U18_AA(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → U19_AA(T386, T385, T388, s21_in_aa(T385, X382))
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
S21_IN_AA(plus(T416, T415), T418) → U22_AA(T416, T415, T418, isNat33_in_a(T415))
S21_IN_AA(plus(T416, T415), T418) → ISNAT33_IN_A(T415)
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → U23_AA(T416, T415, T418, isNat33_in_a(T416))
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → ISNAT33_IN_A(T416)
S21_IN_AA(plus(T439, s(T438)), s(T441)) → U24_AA(T439, T438, T441, isNat33_in_a(s(T438)))
S21_IN_AA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_A(s(T438))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → U25_AA(T439, T438, T441, isNat33_in_a(T439))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → ISNAT33_IN_A(T439)
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → U26_AA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → ADD99_IN_GGA(T438, T439, T441)
S21_IN_AA(plus(T469, 0), T469) → U27_AA(T469, isNat33_in_g(0))
S21_IN_AA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_AA(T469, isNat33_out_g(0)) → U28_AA(T469, isNat33_in_a(T469))
U27_AA(T469, isNat33_out_g(0)) → ISNAT33_IN_A(T469)
S21_IN_AA(plus(T495, T496), T498) → U29_AA(T495, T496, T498, s21_in_aa(T495, X508))
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
S21_IN_AA(plus(T517, T518), T520) → U32_AA(T517, T518, T520, isNat33_in_a(T517))
S21_IN_AA(plus(T517, T518), T520) → ISNAT33_IN_A(T517)
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → U33_AA(T517, T518, T520, isNat33_in_a(T518))
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → ISNAT33_IN_A(T518)
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → U34_AA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → U31_AA(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)
P22_IN_AAAA(T115, X104, T121, T117) → U2_AAAA(T115, X104, T121, T117, s21_in_aa(T115, X104))
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_AAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → U21_AA(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_AA(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_AA(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x2, x3, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x2, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x4)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U27_GA(x1, x2)  =  U27_GA(x1, x2)
U28_GA(x1, x2)  =  U28_GA(x2)
U29_GA(x1, x2, x3, x4)  =  U29_GA(x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x2, x4)
U32_GA(x1, x2, x3, x4)  =  U32_GA(x1, x2, x4)
U33_GA(x1, x2, x3, x4)  =  U33_GA(x1, x2, x4)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x5)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x5)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U7_AA(x1, x2, x3, x4, x5, x6)  =  U7_AA(x6)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4, x5)  =  U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4)  =  U18_AA(x4)
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U25_AA(x1, x2, x3, x4)  =  U25_AA(x2, x4)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U27_AA(x1, x2)  =  U27_AA(x2)
U28_AA(x1, x2)  =  U28_AA(x2)
U29_AA(x1, x2, x3, x4)  =  U29_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
U32_AA(x1, x2, x3, x4)  =  U32_AA(x4)
U33_AA(x1, x2, x3, x4)  =  U33_AA(x1, x4)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x4)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5)  =  U2_AAAA(x5)
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)
U4_AAAA(x1, x2, x3, x4, x5)  =  U4_AAAA(x5)
U21_AA(x1, x2, x3, x4)  =  U21_AA(x4)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(6) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_GA(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → U8_GA(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U9_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → U12_GA(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U13_GA(T230, T231, T232, T234, s21_in_ga(T230, X224))
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → U16_GA(T255, T256, T257, T259, isNat33_in_g(T255))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_G(T255)
ISNAT33_IN_G(s(T165)) → U1_G(T165, isNat33_in_g(T165))
ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → U17_GA(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → U18_GA(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → U19_GA(T386, T385, T388, s21_in_ga(T385, X382))
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
S21_IN_GA(plus(T416, T415), T418) → U22_GA(T416, T415, T418, isNat33_in_g(T415))
S21_IN_GA(plus(T416, T415), T418) → ISNAT33_IN_G(T415)
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → U23_GA(T416, T415, T418, isNat33_in_g(T416))
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U24_GA(T439, T438, T441, isNat33_in_g(s(T438)))
S21_IN_GA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_G(s(T438))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → U25_GA(T439, T438, T441, isNat33_in_g(T439))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → ISNAT33_IN_G(T439)
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → U26_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → ADD99_IN_GGA(T438, T439, T441)
ADD99_IN_GGA(s(T457), T458, s(T460)) → U6_GGA(T457, T458, T460, add99_in_gga(T457, T458, T460))
ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)
S21_IN_GA(plus(T469, 0), T469) → U27_GA(T469, isNat33_in_g(0))
S21_IN_GA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_GA(T469, isNat33_out_g(0)) → U28_GA(T469, isNat33_in_g(T469))
U27_GA(T469, isNat33_out_g(0)) → ISNAT33_IN_G(T469)
S21_IN_GA(plus(T495, T496), T498) → U29_GA(T495, T496, T498, s21_in_ga(T495, X508))
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → ISNAT33_IN_G(T518)
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → U34_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → U31_GA(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)
P22_IN_GAAA(T115, X104, T121, T117) → U2_GAAA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
P22_IN_GAAA(T115, T128, T121, T117) → U3_GAAA(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_GAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_AA(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → U8_AA(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U9_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → U12_AA(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U13_AA(T230, T231, T232, T234, s21_in_aa(T230, X224))
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → U16_AA(T255, T256, T257, T259, isNat33_in_a(T255))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_A(T255)
ISNAT33_IN_A(s(T165)) → U1_A(T165, isNat33_in_a(T165))
ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → U17_AA(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → U18_AA(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → U19_AA(T386, T385, T388, s21_in_aa(T385, X382))
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
S21_IN_AA(plus(T416, T415), T418) → U22_AA(T416, T415, T418, isNat33_in_a(T415))
S21_IN_AA(plus(T416, T415), T418) → ISNAT33_IN_A(T415)
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → U23_AA(T416, T415, T418, isNat33_in_a(T416))
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → ISNAT33_IN_A(T416)
S21_IN_AA(plus(T439, s(T438)), s(T441)) → U24_AA(T439, T438, T441, isNat33_in_a(s(T438)))
S21_IN_AA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_A(s(T438))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → U25_AA(T439, T438, T441, isNat33_in_a(T439))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → ISNAT33_IN_A(T439)
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → U26_AA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → ADD99_IN_GGA(T438, T439, T441)
S21_IN_AA(plus(T469, 0), T469) → U27_AA(T469, isNat33_in_g(0))
S21_IN_AA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_AA(T469, isNat33_out_g(0)) → U28_AA(T469, isNat33_in_a(T469))
U27_AA(T469, isNat33_out_g(0)) → ISNAT33_IN_A(T469)
S21_IN_AA(plus(T495, T496), T498) → U29_AA(T495, T496, T498, s21_in_aa(T495, X508))
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
S21_IN_AA(plus(T517, T518), T520) → U32_AA(T517, T518, T520, isNat33_in_a(T517))
S21_IN_AA(plus(T517, T518), T520) → ISNAT33_IN_A(T517)
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → U33_AA(T517, T518, T520, isNat33_in_a(T518))
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → ISNAT33_IN_A(T518)
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → U34_AA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → U31_AA(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)
P22_IN_AAAA(T115, X104, T121, T117) → U2_AAAA(T115, X104, T121, T117, s21_in_aa(T115, X104))
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_AAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → U21_AA(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_AA(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_AA(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x2, x3, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x2, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x4)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U27_GA(x1, x2)  =  U27_GA(x1, x2)
U28_GA(x1, x2)  =  U28_GA(x2)
U29_GA(x1, x2, x3, x4)  =  U29_GA(x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x2, x4)
U32_GA(x1, x2, x3, x4)  =  U32_GA(x1, x2, x4)
U33_GA(x1, x2, x3, x4)  =  U33_GA(x1, x2, x4)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x5)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x5)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U7_AA(x1, x2, x3, x4, x5, x6)  =  U7_AA(x6)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4, x5)  =  U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4)  =  U18_AA(x4)
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U25_AA(x1, x2, x3, x4)  =  U25_AA(x2, x4)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U27_AA(x1, x2)  =  U27_AA(x2)
U28_AA(x1, x2)  =  U28_AA(x2)
U29_AA(x1, x2, x3, x4)  =  U29_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
U32_AA(x1, x2, x3, x4)  =  U32_AA(x4)
U33_AA(x1, x2, x3, x4)  =  U33_AA(x1, x4)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x4)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5)  =  U2_AAAA(x5)
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)
U4_AAAA(x1, x2, x3, x4, x5)  =  U4_AAAA(x5)
U21_AA(x1, x2, x3, x4)  =  U21_AA(x4)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 79 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A

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:

ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A

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:

ISNAT33_IN_AISNAT33_IN_A

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = ISNAT33_IN_A evaluates to t =ISNAT33_IN_A

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 ISNAT33_IN_A to ISNAT33_IN_A.



(15) NO

(16) Obligation:

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

ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

ADD99_IN_GGA(s(T457), T458) → ADD99_IN_GGA(T457, T458)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • ADD99_IN_GGA(s(T457), T458) → ADD99_IN_GGA(T457, T458)
    The graph contains the following edges 1 > 1, 2 >= 2

(22) YES

(23) Obligation:

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

S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The argument filtering Pi contains the following mapping:
plus(x1, x2)  =  plus(x1, x2)
0  =  0
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
isNat33_out_g(x1)  =  isNat33_out_g
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

S21_IN_AAS21_IN_AA
S21_IN_AAU10_AA(s21_in_aa)
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
S21_IN_AAU14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU10_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))

(29) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
S21_IN_AAU14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU14_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))

(31) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(32) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU10_AA(U27_aa(isNat33_out_g))

(33) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(34) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU14_AA(U27_aa(isNat33_out_g))

(35) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(36) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P22_IN_AAAAU3_AAAA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))

(37) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(38) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))

(39) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(40) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU20_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))

(41) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(42) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU20_AA(U27_aa(isNat33_out_g))

(43) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(44) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU30_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))

(45) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g))
S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(46) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU30_AA(U27_aa(isNat33_out_g))

(47) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g))
S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T441)) → s21_out_aa
U34_aa(add99_out_gga(T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0)

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

(48) 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 = S21_IN_AA evaluates to t =S21_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from S21_IN_AA to S21_IN_AA.



(49) NO

(50) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

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

(53) PiDPToQDPProof (EQUIVALENT transformation)

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

(54) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

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

(55) 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:

  • ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
    The graph contains the following edges 1 > 1

(56) YES

(57) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
isNat33_out_g(x1)  =  isNat33_out_g
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x2, x3, x5)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x2, x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)

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

(58) PiDPToQDPProof (SOUND transformation)

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

(59) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
S21_IN_GA(plus(T113, plus(T114, T115))) → U10_GA(T115, s21_in_ga(plus(T113, T114)))
U10_GA(T115, s21_out_ga) → P22_IN_GAAA(T115)
P22_IN_GAAA(T115) → S21_IN_GA(T115)
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(T230, plus(T231, T232))) → S21_IN_GA(T230)
S21_IN_GA(plus(T230, plus(T231, T232))) → U14_GA(T231, T232, s21_in_ga(T230))
U14_GA(T231, T232, s21_out_ga) → P22_IN_GAAA(plus(T231, T232))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))
S21_IN_GA(plus(T386, T385)) → S21_IN_GA(T385)
S21_IN_GA(plus(T386, T385)) → U20_GA(T386, s21_in_ga(T385))
U20_GA(T386, s21_out_ga) → P22_IN_GAAA(T386)
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
S21_IN_GA(plus(T495, T496)) → U30_GA(T496, s21_in_ga(T495))
U30_GA(T496, s21_out_ga) → P22_IN_GAAA(T496)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T87, plus(T88, 0))) → s21_out_ga
s21_in_ga(plus(T113, plus(T114, T115))) → U9_ga(s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(s21_in_ga(plus(plus(T196, T197), T195)))
s21_in_ga(plus(T230, plus(T231, T232))) → U13_ga(s21_in_ga(T230))
s21_in_ga(plus(T230, plus(T231, T232))) → U14_ga(T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T255, plus(T256, T257))) → U16_ga(isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g
U1_g(isNat33_out_g) → isNat33_out_g
U16_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(s21_in_ga(plus(T353, T352)))
s21_in_ga(plus(0, T364)) → s21_out_ga
s21_in_ga(plus(T386, T385)) → U19_ga(s21_in_ga(T385))
s21_in_ga(plus(T386, T385)) → U20_ga(T386, s21_in_ga(T385))
s21_in_ga(plus(T416, T415)) → U22_ga(T416, isNat33_in_g(T415))
U22_ga(T416, isNat33_out_g) → s21_out_ga
U22_ga(T416, isNat33_out_g) → U23_ga(isNat33_in_g(T416))
U23_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g) → U25_ga(T439, T438, isNat33_in_g(T439))
U25_ga(T439, T438, isNat33_out_g) → U26_ga(add99_in_gga(T438, T439))
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))
U26_ga(add99_out_gga(T441)) → s21_out_ga
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g) → U28_ga(isNat33_in_g(T469))
U28_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(T474, 0)) → s21_out_ga
s21_in_ga(plus(T495, T496)) → U29_ga(s21_in_ga(T495))
s21_in_ga(plus(T495, T496)) → U30_ga(T496, s21_in_ga(T495))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g) → s21_out_ga
U32_ga(T517, T518, isNat33_out_g) → U33_ga(T517, T518, isNat33_in_g(T518))
U33_ga(T517, T518, isNat33_out_g) → s21_out_ga
U33_ga(T517, T518, isNat33_out_g) → U34_ga(add99_in_gga(T517, T518))
U34_ga(add99_out_gga(T520)) → s21_out_ga
U30_ga(T496, s21_out_ga) → U31_ga(p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(s21_in_ga(T115))
U2_gaaa(s21_out_ga) → p22_out_gaaa
p22_in_gaaa(T115) → U3_gaaa(s21_in_ga(T115))
U3_gaaa(s21_out_ga) → U4_gaaa(s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(s21_out_aa) → p22_out_gaaa
U31_ga(p22_out_gaaa) → s21_out_ga
U29_ga(s21_out_ga) → s21_out_ga
U20_ga(T386, s21_out_ga) → U21_ga(p22_in_gaaa(T386))
U21_ga(p22_out_gaaa) → s21_out_ga
U19_ga(s21_out_ga) → s21_out_ga
U18_ga(s21_out_ga) → s21_out_ga
U17_ga(s21_out_ga) → s21_out_ga
U14_ga(T231, T232, s21_out_ga) → U15_ga(p22_in_gaaa(plus(T231, T232)))
U15_ga(p22_out_gaaa) → s21_out_ga
U13_ga(s21_out_ga) → s21_out_ga
U12_ga(s21_out_ga) → s21_out_ga
U10_ga(T115, s21_out_ga) → U11_ga(p22_in_gaaa(T115))
U11_ga(p22_out_gaaa) → s21_out_ga
U9_ga(s21_out_ga) → s21_out_ga
U8_ga(s21_out_ga) → s21_out_ga
U7_ga(s21_out_ga) → s21_out_ga

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0)
U16_ga(x0)
U22_ga(x0, x1)
U23_ga(x0)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0)
U26_ga(x0)
U27_ga(x0, x1)
U28_ga(x0)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0)
U30_ga(x0, x1)
p22_in_gaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0)
U31_ga(x0)
U29_ga(x0)
U20_ga(x0, x1)
U21_ga(x0)
U19_ga(x0)
U18_ga(x0)
U17_ga(x0)
U14_ga(x0, x1, x2)
U15_ga(x0)
U13_ga(x0)
U12_ga(x0)
U10_ga(x0, x1)
U11_ga(x0)
U9_ga(x0)
U8_ga(x0)
U7_ga(x0)

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

(60) MRRProof (EQUIVALENT transformation)

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

S21_IN_GA(plus(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
S21_IN_GA(plus(T113, plus(T114, T115))) → U10_GA(T115, s21_in_ga(plus(T113, T114)))
U10_GA(T115, s21_out_ga) → P22_IN_GAAA(T115)
S21_IN_GA(plus(T230, plus(T231, T232))) → S21_IN_GA(T230)
S21_IN_GA(plus(T230, plus(T231, T232))) → U14_GA(T231, T232, s21_in_ga(T230))
S21_IN_GA(plus(T386, T385)) → S21_IN_GA(T385)
S21_IN_GA(plus(T386, T385)) → U20_GA(T386, s21_in_ga(T385))
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
S21_IN_GA(plus(T495, T496)) → U30_GA(T496, s21_in_ga(T495))
U30_GA(T496, s21_out_ga) → P22_IN_GAAA(T496)

Strictly oriented rules of the TRS R:

s21_in_ga(plus(T87, plus(T88, 0))) → s21_out_ga
s21_in_ga(plus(T113, plus(T114, T115))) → U9_ga(s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T230, plus(T231, T232))) → U13_ga(s21_in_ga(T230))
s21_in_ga(plus(T230, plus(T231, T232))) → U14_ga(T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T255, plus(T256, T257))) → U16_ga(isNat33_in_g(T255))
U16_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(0, T364)) → s21_out_ga
s21_in_ga(plus(T386, T385)) → U19_ga(s21_in_ga(T385))
s21_in_ga(plus(T386, T385)) → U20_ga(T386, s21_in_ga(T385))
s21_in_ga(plus(T416, T415)) → U22_ga(T416, isNat33_in_g(T415))
U25_ga(T439, T438, isNat33_out_g) → U26_ga(add99_in_gga(T438, T439))
U27_ga(T469, isNat33_out_g) → U28_ga(isNat33_in_g(T469))
U28_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(T474, 0)) → s21_out_ga
s21_in_ga(plus(T495, T496)) → U29_ga(s21_in_ga(T495))
s21_in_ga(plus(T495, T496)) → U30_ga(T496, s21_in_ga(T495))
U32_ga(T517, T518, isNat33_out_g) → s21_out_ga
U33_ga(T517, T518, isNat33_out_g) → s21_out_ga
U33_ga(T517, T518, isNat33_out_g) → U34_ga(add99_in_gga(T517, T518))
U29_ga(s21_out_ga) → s21_out_ga
U13_ga(s21_out_ga) → s21_out_ga
U10_ga(T115, s21_out_ga) → U11_ga(p22_in_gaaa(T115))
U11_ga(p22_out_gaaa) → s21_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P22_IN_GAAA(x1)) = 2·x1   
POL(S21_IN_GA(x1)) = 2·x1   
POL(U10_GA(x1, x2)) = 1 + 2·x1 + x2   
POL(U10_aa(x1)) = 2·x1   
POL(U10_ga(x1, x2)) = 2 + 2·x1 + x2   
POL(U11_aa(x1)) = 2·x1   
POL(U11_ga(x1)) = 1 + x1   
POL(U12_aa(x1)) = 2·x1   
POL(U12_ga(x1)) = x1   
POL(U13_aa(x1)) = x1   
POL(U13_ga(x1)) = 1 + x1   
POL(U14_GA(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3   
POL(U14_aa(x1)) = 2·x1   
POL(U14_ga(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3   
POL(U15_aa(x1)) = x1   
POL(U15_ga(x1)) = x1   
POL(U16_aa(x1)) = 2·x1   
POL(U16_ga(x1)) = 2 + x1   
POL(U17_aa(x1)) = x1   
POL(U17_ga(x1)) = x1   
POL(U18_aa(x1)) = 2·x1   
POL(U18_ga(x1)) = x1   
POL(U19_aa(x1)) = 2·x1   
POL(U19_ga(x1)) = x1   
POL(U1_a(x1)) = 2·x1   
POL(U1_g(x1)) = x1   
POL(U20_GA(x1, x2)) = 2·x1 + x2   
POL(U20_aa(x1)) = x1   
POL(U20_ga(x1, x2)) = 2·x1 + x2   
POL(U21_aa(x1)) = x1   
POL(U21_ga(x1)) = x1   
POL(U22_aa(x1)) = x1   
POL(U22_ga(x1, x2)) = x1 + x2   
POL(U23_aa(x1)) = x1   
POL(U23_ga(x1)) = x1   
POL(U24_aa(x1)) = 2·x1   
POL(U24_ga(x1, x2, x3)) = 2 + 2·x1 + x2 + x3   
POL(U25_aa(x1, x2)) = 2·x1 + 2·x2   
POL(U25_ga(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(U26_aa(x1)) = x1   
POL(U26_ga(x1)) = x1   
POL(U27_aa(x1)) = 2·x1   
POL(U27_ga(x1, x2)) = 2 + x1 + x2   
POL(U28_aa(x1)) = x1   
POL(U28_ga(x1)) = 1 + x1   
POL(U29_aa(x1)) = x1   
POL(U29_ga(x1)) = 1 + x1   
POL(U2_aaaa(x1)) = 2·x1   
POL(U2_gaaa(x1)) = x1   
POL(U30_GA(x1, x2)) = 1 + 2·x1 + x2   
POL(U30_aa(x1)) = 2·x1   
POL(U30_ga(x1, x2)) = 2·x1 + x2   
POL(U31_aa(x1)) = x1   
POL(U31_ga(x1)) = x1   
POL(U32_aa(x1)) = x1   
POL(U32_ga(x1, x2, x3)) = 2 + x1 + 2·x2 + x3   
POL(U33_aa(x1, x2)) = 2·x1 + x2   
POL(U33_ga(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(U34_aa(x1)) = 2·x1   
POL(U34_ga(x1)) = x1   
POL(U3_aaaa(x1)) = 2·x1   
POL(U3_gaaa(x1)) = x1   
POL(U4_aaaa(x1)) = x1   
POL(U4_gaaa(x1)) = 2·x1   
POL(U6_gga(x1)) = x1   
POL(U7_aa(x1)) = 2·x1   
POL(U7_ga(x1)) = x1   
POL(U8_aa(x1)) = 2·x1   
POL(U8_ga(x1)) = x1   
POL(U9_aa(x1)) = 2·x1   
POL(U9_ga(x1)) = x1   
POL(add99_in_gga(x1, x2)) = x1 + x2   
POL(add99_out_gga(x1)) = x1   
POL(isNat33_in_a) = 0   
POL(isNat33_in_g(x1)) = x1   
POL(isNat33_out_a(x1)) = 2·x1   
POL(isNat33_out_g) = 0   
POL(p22_in_aaaa) = 0   
POL(p22_in_gaaa(x1)) = 2·x1   
POL(p22_out_aaaa) = 0   
POL(p22_out_gaaa) = 0   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = x1   
POL(s21_in_aa) = 0   
POL(s21_in_ga(x1)) = 2·x1   
POL(s21_out_aa) = 0   
POL(s21_out_ga) = 0   

(61) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
P22_IN_GAAA(T115) → S21_IN_GA(T115)
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
U14_GA(T231, T232, s21_out_ga) → P22_IN_GAAA(plus(T231, T232))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))
U20_GA(T386, s21_out_ga) → P22_IN_GAAA(T386)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(s21_in_ga(plus(plus(T196, T197), T195)))
isNat33_in_g(s(T165)) → U1_g(isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g
U1_g(isNat33_out_g) → isNat33_out_g
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(s21_in_ga(plus(T353, T352)))
U22_ga(T416, isNat33_out_g) → s21_out_ga
U22_ga(T416, isNat33_out_g) → U23_ga(isNat33_in_g(T416))
U23_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g) → U25_ga(T439, T438, isNat33_in_g(T439))
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))
U26_ga(add99_out_gga(T441)) → s21_out_ga
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g) → U33_ga(T517, T518, isNat33_in_g(T518))
U34_ga(add99_out_gga(T520)) → s21_out_ga
U30_ga(T496, s21_out_ga) → U31_ga(p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(s21_in_ga(T115))
U2_gaaa(s21_out_ga) → p22_out_gaaa
p22_in_gaaa(T115) → U3_gaaa(s21_in_ga(T115))
U3_gaaa(s21_out_ga) → U4_gaaa(s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(s21_out_aa) → p22_out_gaaa
U31_ga(p22_out_gaaa) → s21_out_ga
U20_ga(T386, s21_out_ga) → U21_ga(p22_in_gaaa(T386))
U21_ga(p22_out_gaaa) → s21_out_ga
U19_ga(s21_out_ga) → s21_out_ga
U18_ga(s21_out_ga) → s21_out_ga
U17_ga(s21_out_ga) → s21_out_ga
U14_ga(T231, T232, s21_out_ga) → U15_ga(p22_in_gaaa(plus(T231, T232)))
U15_ga(p22_out_gaaa) → s21_out_ga
U12_ga(s21_out_ga) → s21_out_ga
U9_ga(s21_out_ga) → s21_out_ga
U8_ga(s21_out_ga) → s21_out_ga
U7_ga(s21_out_ga) → s21_out_ga

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0)
U16_ga(x0)
U22_ga(x0, x1)
U23_ga(x0)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0)
U26_ga(x0)
U27_ga(x0, x1)
U28_ga(x0)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0)
U30_ga(x0, x1)
p22_in_gaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0)
U31_ga(x0)
U29_ga(x0)
U20_ga(x0, x1)
U21_ga(x0)
U19_ga(x0)
U18_ga(x0)
U17_ga(x0)
U14_ga(x0, x1, x2)
U15_ga(x0)
U13_ga(x0)
U12_ga(x0)
U10_ga(x0, x1)
U11_ga(x0)
U9_ga(x0)
U8_ga(x0)
U7_ga(x0)

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

(62) DependencyGraphProof (EQUIVALENT transformation)

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

(63) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(s21_in_ga(plus(plus(T196, T197), T195)))
isNat33_in_g(s(T165)) → U1_g(isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g
U1_g(isNat33_out_g) → isNat33_out_g
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(s21_in_ga(plus(T353, T352)))
U22_ga(T416, isNat33_out_g) → s21_out_ga
U22_ga(T416, isNat33_out_g) → U23_ga(isNat33_in_g(T416))
U23_ga(isNat33_out_g) → s21_out_ga
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g) → U25_ga(T439, T438, isNat33_in_g(T439))
add99_in_gga(s(T457), T458) → U6_gga(add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(T466)
U6_gga(add99_out_gga(T460)) → add99_out_gga(s(T460))
U26_ga(add99_out_gga(T441)) → s21_out_ga
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g) → U33_ga(T517, T518, isNat33_in_g(T518))
U34_ga(add99_out_gga(T520)) → s21_out_ga
U30_ga(T496, s21_out_ga) → U31_ga(p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(s21_in_ga(T115))
U2_gaaa(s21_out_ga) → p22_out_gaaa
p22_in_gaaa(T115) → U3_gaaa(s21_in_ga(T115))
U3_gaaa(s21_out_ga) → U4_gaaa(s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(s21_out_aa) → p22_out_gaaa
U31_ga(p22_out_gaaa) → s21_out_ga
U20_ga(T386, s21_out_ga) → U21_ga(p22_in_gaaa(T386))
U21_ga(p22_out_gaaa) → s21_out_ga
U19_ga(s21_out_ga) → s21_out_ga
U18_ga(s21_out_ga) → s21_out_ga
U17_ga(s21_out_ga) → s21_out_ga
U14_ga(T231, T232, s21_out_ga) → U15_ga(p22_in_gaaa(plus(T231, T232)))
U15_ga(p22_out_gaaa) → s21_out_ga
U12_ga(s21_out_ga) → s21_out_ga
U9_ga(s21_out_ga) → s21_out_ga
U8_ga(s21_out_ga) → s21_out_ga
U7_ga(s21_out_ga) → s21_out_ga

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0)
U16_ga(x0)
U22_ga(x0, x1)
U23_ga(x0)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0)
U26_ga(x0)
U27_ga(x0, x1)
U28_ga(x0)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0)
U30_ga(x0, x1)
p22_in_gaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0)
U31_ga(x0)
U29_ga(x0)
U20_ga(x0, x1)
U21_ga(x0)
U19_ga(x0)
U18_ga(x0)
U17_ga(x0)
U14_ga(x0, x1, x2)
U15_ga(x0)
U13_ga(x0)
U12_ga(x0)
U10_ga(x0, x1)
U11_ga(x0)
U9_ga(x0)
U8_ga(x0)
U7_ga(x0)

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

(64) 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.

(65) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

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

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0)
U16_ga(x0)
U22_ga(x0, x1)
U23_ga(x0)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0)
U26_ga(x0)
U27_ga(x0, x1)
U28_ga(x0)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0)
U30_ga(x0, x1)
p22_in_gaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0)
U31_ga(x0)
U29_ga(x0)
U20_ga(x0, x1)
U21_ga(x0)
U19_ga(x0)
U18_ga(x0)
U17_ga(x0)
U14_ga(x0, x1, x2)
U15_ga(x0)
U13_ga(x0)
U12_ga(x0)
U10_ga(x0, x1)
U11_ga(x0)
U9_ga(x0)
U8_ga(x0)
U7_ga(x0)

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

(66) 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].

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0)
U16_ga(x0)
U22_ga(x0, x1)
U23_ga(x0)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0)
U26_ga(x0)
U27_ga(x0, x1)
U28_ga(x0)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0)
U30_ga(x0, x1)
p22_in_gaaa(x0)
U2_gaaa(x0)
U3_gaaa(x0)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0)
U31_ga(x0)
U29_ga(x0)
U20_ga(x0, x1)
U21_ga(x0)
U19_ga(x0)
U18_ga(x0)
U17_ga(x0)
U14_ga(x0, x1, x2)
U15_ga(x0)
U13_ga(x0)
U12_ga(x0)
U10_ga(x0, x1)
U11_ga(x0)
U9_ga(x0)
U8_ga(x0)
U7_ga(x0)

(67) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

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

(68) 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 = S21_IN_GA(plus(T70, plus(T71, T72))) evaluates to t =S21_IN_GA(plus(T72, plus(T70, T71)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [T70 / T72, T71 / T70, T72 / T71]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from S21_IN_GA(plus(T70, plus(T71, T72))) to S21_IN_GA(plus(T72, plus(T70, T71))).



(69) NO

(70) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
s21_in: (b,f) (f,f)
isNat33_in: (b) (f)
add99_in: (b,b,f)
p22_in: (b,f,f,f) (f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(71) Obligation:

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

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)

(72) 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:

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_GA(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → U8_GA(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U9_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → U12_GA(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U13_GA(T230, T231, T232, T234, s21_in_ga(T230, X224))
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → U16_GA(T255, T256, T257, T259, isNat33_in_g(T255))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_G(T255)
ISNAT33_IN_G(s(T165)) → U1_G(T165, isNat33_in_g(T165))
ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → U17_GA(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → U18_GA(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → U19_GA(T386, T385, T388, s21_in_ga(T385, X382))
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
S21_IN_GA(plus(T416, T415), T418) → U22_GA(T416, T415, T418, isNat33_in_g(T415))
S21_IN_GA(plus(T416, T415), T418) → ISNAT33_IN_G(T415)
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → U23_GA(T416, T415, T418, isNat33_in_g(T416))
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U24_GA(T439, T438, T441, isNat33_in_g(s(T438)))
S21_IN_GA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_G(s(T438))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → U25_GA(T439, T438, T441, isNat33_in_g(T439))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → ISNAT33_IN_G(T439)
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → U26_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → ADD99_IN_GGA(T438, T439, T441)
ADD99_IN_GGA(s(T457), T458, s(T460)) → U6_GGA(T457, T458, T460, add99_in_gga(T457, T458, T460))
ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)
S21_IN_GA(plus(T469, 0), T469) → U27_GA(T469, isNat33_in_g(0))
S21_IN_GA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_GA(T469, isNat33_out_g(0)) → U28_GA(T469, isNat33_in_g(T469))
U27_GA(T469, isNat33_out_g(0)) → ISNAT33_IN_G(T469)
S21_IN_GA(plus(T495, T496), T498) → U29_GA(T495, T496, T498, s21_in_ga(T495, X508))
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → ISNAT33_IN_G(T518)
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → U34_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → U31_GA(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)
P22_IN_GAAA(T115, X104, T121, T117) → U2_GAAA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
P22_IN_GAAA(T115, T128, T121, T117) → U3_GAAA(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_GAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_AA(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → U8_AA(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U9_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → U12_AA(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U13_AA(T230, T231, T232, T234, s21_in_aa(T230, X224))
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → U16_AA(T255, T256, T257, T259, isNat33_in_a(T255))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_A(T255)
ISNAT33_IN_A(s(T165)) → U1_A(T165, isNat33_in_a(T165))
ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → U17_AA(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → U18_AA(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → U19_AA(T386, T385, T388, s21_in_aa(T385, X382))
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
S21_IN_AA(plus(T416, T415), T418) → U22_AA(T416, T415, T418, isNat33_in_a(T415))
S21_IN_AA(plus(T416, T415), T418) → ISNAT33_IN_A(T415)
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → U23_AA(T416, T415, T418, isNat33_in_a(T416))
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → ISNAT33_IN_A(T416)
S21_IN_AA(plus(T439, s(T438)), s(T441)) → U24_AA(T439, T438, T441, isNat33_in_a(s(T438)))
S21_IN_AA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_A(s(T438))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → U25_AA(T439, T438, T441, isNat33_in_a(T439))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → ISNAT33_IN_A(T439)
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → U26_AA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → ADD99_IN_GGA(T438, T439, T441)
S21_IN_AA(plus(T469, 0), T469) → U27_AA(T469, isNat33_in_g(0))
S21_IN_AA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_AA(T469, isNat33_out_g(0)) → U28_AA(T469, isNat33_in_a(T469))
U27_AA(T469, isNat33_out_g(0)) → ISNAT33_IN_A(T469)
S21_IN_AA(plus(T495, T496), T498) → U29_AA(T495, T496, T498, s21_in_aa(T495, X508))
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
S21_IN_AA(plus(T517, T518), T520) → U32_AA(T517, T518, T520, isNat33_in_a(T517))
S21_IN_AA(plus(T517, T518), T520) → ISNAT33_IN_A(T517)
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → U33_AA(T517, T518, T520, isNat33_in_a(T518))
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → ISNAT33_IN_A(T518)
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → U34_AA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → U31_AA(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)
P22_IN_AAAA(T115, X104, T121, T117) → U2_AAAA(T115, X104, T121, T117, s21_in_aa(T115, X104))
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_AAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → U21_AA(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_AA(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_AA(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x1, x2, x3, x4, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x3, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x3, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x3, x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x2, x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x2, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x2, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x2, x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x2, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x2, x4)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U27_GA(x1, x2)  =  U27_GA(x1, x2)
U28_GA(x1, x2)  =  U28_GA(x1, x2)
U29_GA(x1, x2, x3, x4)  =  U29_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x1, x2, x4)
U32_GA(x1, x2, x3, x4)  =  U32_GA(x1, x2, x4)
U33_GA(x1, x2, x3, x4)  =  U33_GA(x1, x2, x4)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x2, x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x2, x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x1, x5)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x1, x5)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U7_AA(x1, x2, x3, x4, x5, x6)  =  U7_AA(x6)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4, x5)  =  U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4)  =  U18_AA(x4)
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U25_AA(x1, x2, x3, x4)  =  U25_AA(x2, x4)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U27_AA(x1, x2)  =  U27_AA(x2)
U28_AA(x1, x2)  =  U28_AA(x2)
U29_AA(x1, x2, x3, x4)  =  U29_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
U32_AA(x1, x2, x3, x4)  =  U32_AA(x4)
U33_AA(x1, x2, x3, x4)  =  U33_AA(x1, x4)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x4)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5)  =  U2_AAAA(x5)
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)
U4_AAAA(x1, x2, x3, x4, x5)  =  U4_AAAA(x5)
U21_AA(x1, x2, x3, x4)  =  U21_AA(x4)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)

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

(73) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_GA(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → U8_GA(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U9_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → U12_GA(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U13_GA(T230, T231, T232, T234, s21_in_ga(T230, X224))
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → U16_GA(T255, T256, T257, T259, isNat33_in_g(T255))
S21_IN_GA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_G(T255)
ISNAT33_IN_G(s(T165)) → U1_G(T165, isNat33_in_g(T165))
ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → U17_GA(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → U18_GA(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → U19_GA(T386, T385, T388, s21_in_ga(T385, X382))
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
S21_IN_GA(plus(T416, T415), T418) → U22_GA(T416, T415, T418, isNat33_in_g(T415))
S21_IN_GA(plus(T416, T415), T418) → ISNAT33_IN_G(T415)
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → U23_GA(T416, T415, T418, isNat33_in_g(T416))
U22_GA(T416, T415, T418, isNat33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U24_GA(T439, T438, T441, isNat33_in_g(s(T438)))
S21_IN_GA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_G(s(T438))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → U25_GA(T439, T438, T441, isNat33_in_g(T439))
U24_GA(T439, T438, T441, isNat33_out_g(s(T438))) → ISNAT33_IN_G(T439)
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → U26_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_GA(T439, T438, T441, isNat33_out_g(T439)) → ADD99_IN_GGA(T438, T439, T441)
ADD99_IN_GGA(s(T457), T458, s(T460)) → U6_GGA(T457, T458, T460, add99_in_gga(T457, T458, T460))
ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)
S21_IN_GA(plus(T469, 0), T469) → U27_GA(T469, isNat33_in_g(0))
S21_IN_GA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_GA(T469, isNat33_out_g(0)) → U28_GA(T469, isNat33_in_g(T469))
U27_GA(T469, isNat33_out_g(0)) → ISNAT33_IN_G(T469)
S21_IN_GA(plus(T495, T496), T498) → U29_GA(T495, T496, T498, s21_in_ga(T495, X508))
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNat33_out_g(T517)) → ISNAT33_IN_G(T518)
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → U34_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_GA(T517, T518, T520, isNat33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → U31_GA(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)
P22_IN_GAAA(T115, X104, T121, T117) → U2_GAAA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
P22_IN_GAAA(T115, T128, T121, T117) → U3_GAAA(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_GAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_GAAA(T115, T128, T121, T117, s21_out_ga(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_AA(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → U8_AA(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U9_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → U12_AA(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U13_AA(T230, T231, T232, T234, s21_in_aa(T230, X224))
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → U16_AA(T255, T256, T257, T259, isNat33_in_a(T255))
S21_IN_AA(plus(T255, plus(T256, T257)), T259) → ISNAT33_IN_A(T255)
ISNAT33_IN_A(s(T165)) → U1_A(T165, isNat33_in_a(T165))
ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → U17_AA(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → U18_AA(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → U19_AA(T386, T385, T388, s21_in_aa(T385, X382))
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
S21_IN_AA(plus(T416, T415), T418) → U22_AA(T416, T415, T418, isNat33_in_a(T415))
S21_IN_AA(plus(T416, T415), T418) → ISNAT33_IN_A(T415)
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → U23_AA(T416, T415, T418, isNat33_in_a(T416))
U22_AA(T416, T415, T418, isNat33_out_a(T415)) → ISNAT33_IN_A(T416)
S21_IN_AA(plus(T439, s(T438)), s(T441)) → U24_AA(T439, T438, T441, isNat33_in_a(s(T438)))
S21_IN_AA(plus(T439, s(T438)), s(T441)) → ISNAT33_IN_A(s(T438))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → U25_AA(T439, T438, T441, isNat33_in_a(T439))
U24_AA(T439, T438, T441, isNat33_out_a(s(T438))) → ISNAT33_IN_A(T439)
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → U26_AA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U25_AA(T439, T438, T441, isNat33_out_a(T439)) → ADD99_IN_GGA(T438, T439, T441)
S21_IN_AA(plus(T469, 0), T469) → U27_AA(T469, isNat33_in_g(0))
S21_IN_AA(plus(T469, 0), T469) → ISNAT33_IN_G(0)
U27_AA(T469, isNat33_out_g(0)) → U28_AA(T469, isNat33_in_a(T469))
U27_AA(T469, isNat33_out_g(0)) → ISNAT33_IN_A(T469)
S21_IN_AA(plus(T495, T496), T498) → U29_AA(T495, T496, T498, s21_in_aa(T495, X508))
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
S21_IN_AA(plus(T517, T518), T520) → U32_AA(T517, T518, T520, isNat33_in_a(T517))
S21_IN_AA(plus(T517, T518), T520) → ISNAT33_IN_A(T517)
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → U33_AA(T517, T518, T520, isNat33_in_a(T518))
U32_AA(T517, T518, T520, isNat33_out_a(T517)) → ISNAT33_IN_A(T518)
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → U34_AA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U33_AA(T517, T518, T520, isNat33_out_a(T518)) → ADD99_IN_GGA(T517, T518, T520)
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → U31_AA(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)
P22_IN_AAAA(T115, X104, T121, T117) → U2_AAAA(T115, X104, T121, T117, s21_in_aa(T115, X104))
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_AAAA(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → U21_AA(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_AA(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_AA(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x1, x2, x3, x4, x6)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x3, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x3, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x3, x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x3, x5)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x2, x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x2, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x2, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x2, x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x2, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x2, x4)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U27_GA(x1, x2)  =  U27_GA(x1, x2)
U28_GA(x1, x2)  =  U28_GA(x1, x2)
U29_GA(x1, x2, x3, x4)  =  U29_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x1, x2, x4)
U32_GA(x1, x2, x3, x4)  =  U32_GA(x1, x2, x4)
U33_GA(x1, x2, x3, x4)  =  U33_GA(x1, x2, x4)
U34_GA(x1, x2, x3, x4)  =  U34_GA(x1, x2, x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x1, x2, x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x1, x5)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x1, x5)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U7_AA(x1, x2, x3, x4, x5, x6)  =  U7_AA(x6)
U8_AA(x1, x2, x3, x4, x5)  =  U8_AA(x5)
U9_AA(x1, x2, x3, x4, x5)  =  U9_AA(x5)
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U12_AA(x1, x2, x3, x4, x5)  =  U12_AA(x5)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4)  =  U18_AA(x4)
U19_AA(x1, x2, x3, x4)  =  U19_AA(x4)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x4)
U23_AA(x1, x2, x3, x4)  =  U23_AA(x4)
U24_AA(x1, x2, x3, x4)  =  U24_AA(x4)
U25_AA(x1, x2, x3, x4)  =  U25_AA(x2, x4)
U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)
U27_AA(x1, x2)  =  U27_AA(x2)
U28_AA(x1, x2)  =  U28_AA(x2)
U29_AA(x1, x2, x3, x4)  =  U29_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
U32_AA(x1, x2, x3, x4)  =  U32_AA(x4)
U33_AA(x1, x2, x3, x4)  =  U33_AA(x1, x4)
U34_AA(x1, x2, x3, x4)  =  U34_AA(x4)
U31_AA(x1, x2, x3, x4)  =  U31_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U2_AAAA(x1, x2, x3, x4, x5)  =  U2_AAAA(x5)
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)
U4_AAAA(x1, x2, x3, x4, x5)  =  U4_AAAA(x5)
U21_AA(x1, x2, x3, x4)  =  U21_AA(x4)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)

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

(74) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 79 less nodes.

(75) Complex Obligation (AND)

(76) Obligation:

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

ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

ISNAT33_IN_A(s(T165)) → ISNAT33_IN_A(T165)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ISNAT33_IN_A(x1)  =  ISNAT33_IN_A

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

ISNAT33_IN_AISNAT33_IN_A

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

(81) 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 = ISNAT33_IN_A evaluates to t =ISNAT33_IN_A

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 ISNAT33_IN_A to ISNAT33_IN_A.



(82) NO

(83) Obligation:

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

ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

ADD99_IN_GGA(s(T457), T458, s(T460)) → ADD99_IN_GGA(T457, T458, T460)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD99_IN_GGA(x1, x2, x3)  =  ADD99_IN_GGA(x1, x2)

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

ADD99_IN_GGA(s(T457), T458) → ADD99_IN_GGA(T457, T458)

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

(88) 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:

  • ADD99_IN_GGA(s(T457), T458) → ADD99_IN_GGA(T457, T458)
    The graph contains the following edges 1 > 1, 2 >= 2

(89) YES

(90) Obligation:

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

S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)

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

(91) UsableRulesProof (EQUIVALENT transformation)

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

(92) Obligation:

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

S21_IN_AA(plus(T70, plus(T71, T72)), T74) → S21_IN_AA(plus(T72, plus(T70, T71)), T74)
S21_IN_AA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_AA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → S21_IN_AA(plus(T113, T114), X103)
S21_IN_AA(plus(T113, plus(T114, T115)), T117) → U10_AA(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
U10_AA(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → P22_IN_AAAA(T115, X104, T121, T117)
P22_IN_AAAA(T115, X104, T121, T117) → S21_IN_AA(T115, X104)
S21_IN_AA(plus(T195, plus(T196, T197)), T199) → S21_IN_AA(plus(plus(T196, T197), T195), T199)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → S21_IN_AA(T230, X224)
S21_IN_AA(plus(T230, plus(T231, T232)), T234) → U14_AA(T230, T231, T232, T234, s21_in_aa(T230, T238))
U14_AA(T230, T231, T232, T234, s21_out_aa(T230, T238)) → P22_IN_AAAA(plus(T231, T232), X225, T238, T234)
P22_IN_AAAA(T115, T128, T121, T117) → U3_AAAA(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_AAAA(T115, T128, T121, T117, s21_out_aa(T115, T128)) → S21_IN_AA(plus(T121, T128), T117)
S21_IN_AA(plus(plus(T321, T322), T320), T324) → S21_IN_AA(plus(plus(T320, T321), T322), T324)
S21_IN_AA(plus(T353, T352), T355) → S21_IN_AA(plus(T353, T352), T355)
S21_IN_AA(plus(T386, T385), T388) → S21_IN_AA(T385, X382)
S21_IN_AA(plus(T386, T385), T388) → U20_AA(T386, T385, T388, s21_in_aa(T385, T392))
U20_AA(T386, T385, T388, s21_out_aa(T385, T392)) → P22_IN_AAAA(T386, X383, T392, T388)
S21_IN_AA(plus(T495, T496), T498) → S21_IN_AA(T495, X508)
S21_IN_AA(plus(T495, T496), T498) → U30_AA(T495, T496, T498, s21_in_aa(T495, T502))
U30_AA(T495, T496, T498, s21_out_aa(T495, T502)) → P22_IN_AAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The argument filtering Pi contains the following mapping:
plus(x1, x2)  =  plus(x1, x2)
0  =  0
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
S21_IN_AA(x1, x2)  =  S21_IN_AA
U10_AA(x1, x2, x3, x4, x5)  =  U10_AA(x5)
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
U20_AA(x1, x2, x3, x4)  =  U20_AA(x4)
U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)
P22_IN_AAAA(x1, x2, x3, x4)  =  P22_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5)  =  U3_AAAA(x5)

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

(93) PiDPToQDPProof (SOUND transformation)

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

(94) Obligation:

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

S21_IN_AAS21_IN_AA
S21_IN_AAU10_AA(s21_in_aa)
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
S21_IN_AAU14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(95) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU10_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))

(96) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
S21_IN_AAU14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(97) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU10_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))

(98) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
S21_IN_AAU14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(99) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU14_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))

(100) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(101) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU14_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))

(102) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAU3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(103) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P22_IN_AAAAU3_AAAA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))

(104) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(105) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule P22_IN_AAAAU3_AAAA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))

(106) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AAU20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU20_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))

(108) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(109) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU20_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU20_AA(U27_aa(isNat33_out_g(0)))

(110) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(111) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S21_IN_AAU30_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))

(112) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(113) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule S21_IN_AAU30_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AAU30_AA(U27_aa(isNat33_out_g(0)))

(114) Obligation:

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

S21_IN_AAS21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAAS21_IN_AA
U14_AA(s21_out_aa) → P22_IN_AAAA
U3_AAAA(s21_out_aa) → S21_IN_AA
U20_AA(s21_out_aa) → P22_IN_AAAA
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AAU10_AA(U7_aa(s21_in_aa))
S21_IN_AAU10_AA(U8_aa(s21_in_aa))
S21_IN_AAU10_AA(s21_out_aa)
S21_IN_AAU10_AA(U9_aa(s21_in_aa))
S21_IN_AAU10_AA(U10_aa(s21_in_aa))
S21_IN_AAU10_AA(U12_aa(s21_in_aa))
S21_IN_AAU10_AA(U13_aa(s21_in_aa))
S21_IN_AAU10_AA(U14_aa(s21_in_aa))
S21_IN_AAU10_AA(U16_aa(isNat33_in_a))
S21_IN_AAU10_AA(U17_aa(s21_in_aa))
S21_IN_AAU10_AA(U18_aa(s21_in_aa))
S21_IN_AAU10_AA(U19_aa(s21_in_aa))
S21_IN_AAU10_AA(U20_aa(s21_in_aa))
S21_IN_AAU10_AA(U22_aa(isNat33_in_a))
S21_IN_AAU10_AA(U24_aa(isNat33_in_a))
S21_IN_AAU10_AA(U29_aa(s21_in_aa))
S21_IN_AAU10_AA(U30_aa(s21_in_aa))
S21_IN_AAU10_AA(U32_aa(isNat33_in_a))
S21_IN_AAU10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU14_AA(U7_aa(s21_in_aa))
S21_IN_AAU14_AA(U8_aa(s21_in_aa))
S21_IN_AAU14_AA(s21_out_aa)
S21_IN_AAU14_AA(U9_aa(s21_in_aa))
S21_IN_AAU14_AA(U10_aa(s21_in_aa))
S21_IN_AAU14_AA(U12_aa(s21_in_aa))
S21_IN_AAU14_AA(U13_aa(s21_in_aa))
S21_IN_AAU14_AA(U14_aa(s21_in_aa))
S21_IN_AAU14_AA(U16_aa(isNat33_in_a))
S21_IN_AAU14_AA(U17_aa(s21_in_aa))
S21_IN_AAU14_AA(U18_aa(s21_in_aa))
S21_IN_AAU14_AA(U19_aa(s21_in_aa))
S21_IN_AAU14_AA(U20_aa(s21_in_aa))
S21_IN_AAU14_AA(U22_aa(isNat33_in_a))
S21_IN_AAU14_AA(U24_aa(isNat33_in_a))
S21_IN_AAU14_AA(U29_aa(s21_in_aa))
S21_IN_AAU14_AA(U30_aa(s21_in_aa))
S21_IN_AAU14_AA(U32_aa(isNat33_in_a))
S21_IN_AAU14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAAU3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(s21_out_aa)
P22_IN_AAAAU3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAAU3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAAU3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU20_AA(U7_aa(s21_in_aa))
S21_IN_AAU20_AA(U8_aa(s21_in_aa))
S21_IN_AAU20_AA(s21_out_aa)
S21_IN_AAU20_AA(U9_aa(s21_in_aa))
S21_IN_AAU20_AA(U10_aa(s21_in_aa))
S21_IN_AAU20_AA(U12_aa(s21_in_aa))
S21_IN_AAU20_AA(U13_aa(s21_in_aa))
S21_IN_AAU20_AA(U14_aa(s21_in_aa))
S21_IN_AAU20_AA(U16_aa(isNat33_in_a))
S21_IN_AAU20_AA(U17_aa(s21_in_aa))
S21_IN_AAU20_AA(U18_aa(s21_in_aa))
S21_IN_AAU20_AA(U19_aa(s21_in_aa))
S21_IN_AAU20_AA(U20_aa(s21_in_aa))
S21_IN_AAU20_AA(U22_aa(isNat33_in_a))
S21_IN_AAU20_AA(U24_aa(isNat33_in_a))
S21_IN_AAU20_AA(U29_aa(s21_in_aa))
S21_IN_AAU20_AA(U30_aa(s21_in_aa))
S21_IN_AAU20_AA(U32_aa(isNat33_in_a))
S21_IN_AAU20_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AAU30_AA(U7_aa(s21_in_aa))
S21_IN_AAU30_AA(U8_aa(s21_in_aa))
S21_IN_AAU30_AA(s21_out_aa)
S21_IN_AAU30_AA(U9_aa(s21_in_aa))
S21_IN_AAU30_AA(U10_aa(s21_in_aa))
S21_IN_AAU30_AA(U12_aa(s21_in_aa))
S21_IN_AAU30_AA(U13_aa(s21_in_aa))
S21_IN_AAU30_AA(U14_aa(s21_in_aa))
S21_IN_AAU30_AA(U16_aa(isNat33_in_a))
S21_IN_AAU30_AA(U17_aa(s21_in_aa))
S21_IN_AAU30_AA(U18_aa(s21_in_aa))
S21_IN_AAU30_AA(U19_aa(s21_in_aa))
S21_IN_AAU30_AA(U20_aa(s21_in_aa))
S21_IN_AAU30_AA(U22_aa(isNat33_in_a))
S21_IN_AAU30_AA(U24_aa(isNat33_in_a))
S21_IN_AAU30_AA(U29_aa(s21_in_aa))
S21_IN_AAU30_AA(U30_aa(s21_in_aa))
S21_IN_AAU30_AA(U32_aa(isNat33_in_a))
S21_IN_AAU30_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
s21_in_aaU24_aa(isNat33_in_a)
s21_in_aaU27_aa(isNat33_in_g(0))
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U7_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U12_aa(s21_out_aa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U16_aa(isNat33_out_a(T255)) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U29_aa(s21_out_aa) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U11_aa(p22_out_aaaa) → s21_out_aa
U15_aa(p22_out_aaaa) → s21_out_aa
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U21_aa(p22_out_aaaa) → s21_out_aa
U23_aa(isNat33_out_a(T416)) → s21_out_aa
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
isNat33_in_g(0) → isNat33_out_g(0)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
U31_aa(p22_out_aaaa) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
p22_in_aaaaU2_aaaa(s21_in_aa)
p22_in_aaaaU3_aaaa(s21_in_aa)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U2_aaaa(s21_out_aa) → p22_out_aaaa
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))

The set Q consists of the following terms:

s21_in_aa
U7_aa(x0)
U8_aa(x0)
U9_aa(x0)
U10_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
U16_aa(x0)
U17_aa(x0)
U18_aa(x0)
U19_aa(x0)
U20_aa(x0)
U22_aa(x0)
U24_aa(x0)
U27_aa(x0)
U29_aa(x0)
U30_aa(x0)
U32_aa(x0)
U11_aa(x0)
U15_aa(x0)
isNat33_in_a
U21_aa(x0)
U23_aa(x0)
U25_aa(x0, x1)
isNat33_in_g(x0)
U28_aa(x0)
U31_aa(x0)
U33_aa(x0, x1)
p22_in_aaaa
U1_a(x0)
U26_aa(x0)
U34_aa(x0)
U2_aaaa(x0)
U3_aaaa(x0)
add99_in_gga(x0, x1)
U4_aaaa(x0)
U6_gga(x0, x1, x2)

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

(115) 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 = S21_IN_AA evaluates to t =S21_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from S21_IN_AA to S21_IN_AA.



(116) NO

(117) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
ISNAT33_IN_G(x1)  =  ISNAT33_IN_G(x1)

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

(118) UsableRulesProof (EQUIVALENT transformation)

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

(119) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

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

(120) PiDPToQDPProof (EQUIVALENT transformation)

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

(121) Obligation:

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

ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)

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

(122) 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:

  • ISNAT33_IN_G(s(T165)) → ISNAT33_IN_G(T165)
    The graph contains the following edges 1 > 1

(123) YES

(124) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72)), T74) → S21_IN_GA(plus(T72, plus(T70, T71)), T74)
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35))), T37) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35), T37)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → S21_IN_GA(plus(T113, T114), X103)
S21_IN_GA(plus(T113, plus(T114, T115)), T117) → U10_GA(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
U10_GA(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → P22_IN_GAAA(T115, X104, T121, T117)
P22_IN_GAAA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
S21_IN_GA(plus(T195, plus(T196, T197)), T199) → S21_IN_GA(plus(plus(T196, T197), T195), T199)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → S21_IN_GA(T230, X224)
S21_IN_GA(plus(T230, plus(T231, T232)), T234) → U14_GA(T230, T231, T232, T234, s21_in_ga(T230, T238))
U14_GA(T230, T231, T232, T234, s21_out_ga(T230, T238)) → P22_IN_GAAA(plus(T231, T232), X225, T238, T234)
S21_IN_GA(plus(plus(T321, T322), T320), T324) → S21_IN_GA(plus(plus(T320, T321), T322), T324)
S21_IN_GA(plus(T353, T352), T355) → S21_IN_GA(plus(T353, T352), T355)
S21_IN_GA(plus(T386, T385), T388) → S21_IN_GA(T385, X382)
S21_IN_GA(plus(T386, T385), T388) → U20_GA(T386, T385, T388, s21_in_ga(T385, T392))
U20_GA(T386, T385, T388, s21_out_ga(T385, T392)) → P22_IN_GAAA(T386, X383, T392, T388)
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U30_GA(T495, T496, T498, s21_in_ga(T495, T502))
U30_GA(T495, T496, T498, s21_out_ga(T495, T502)) → P22_IN_GAAA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_ga(T32, T33, T34, T35, T37, s21_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_ga(plus(T70, plus(T71, T72)), T74) → U8_ga(T70, T71, T72, T74, s21_in_ga(plus(T72, plus(T70, T71)), T74))
s21_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U9_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), X103))
s21_in_ga(plus(T113, plus(T114, T115)), T117) → U10_ga(T113, T114, T115, T117, s21_in_ga(plus(T113, T114), T121))
s21_in_ga(plus(T195, plus(T196, T197)), T199) → U12_ga(T195, T196, T197, T199, s21_in_ga(plus(plus(T196, T197), T195), T199))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U13_ga(T230, T231, T232, T234, s21_in_ga(T230, X224))
s21_in_ga(plus(T230, plus(T231, T232)), T234) → U14_ga(T230, T231, T232, T234, s21_in_ga(T230, T238))
s21_in_ga(plus(T255, plus(T256, T257)), T259) → U16_ga(T255, T256, T257, T259, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, T259, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)), T259)
s21_in_ga(plus(plus(T321, T322), T320), T324) → U17_ga(T321, T322, T320, T324, s21_in_ga(plus(plus(T320, T321), T322), T324))
s21_in_ga(plus(T353, T352), T355) → U18_ga(T353, T352, T355, s21_in_ga(plus(T353, T352), T355))
s21_in_ga(plus(0, T364), T364) → s21_out_ga(plus(0, T364), T364)
s21_in_ga(plus(T386, T385), T388) → U19_ga(T386, T385, T388, s21_in_ga(T385, X382))
s21_in_ga(plus(T386, T385), T388) → U20_ga(T386, T385, T388, s21_in_ga(T385, T392))
s21_in_ga(plus(T416, T415), T418) → U22_ga(T416, T415, T418, isNat33_in_g(T415))
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415), T418)
U22_ga(T416, T415, T418, isNat33_out_g(T415)) → U23_ga(T416, T415, T418, isNat33_in_g(T416))
U23_ga(T416, T415, T418, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415), T418)
s21_in_ga(plus(T439, s(T438)), s(T441)) → U24_ga(T439, T438, T441, isNat33_in_g(s(T438)))
U24_ga(T439, T438, T441, isNat33_out_g(s(T438))) → U25_ga(T439, T438, T441, isNat33_in_g(T439))
U25_ga(T439, T438, T441, isNat33_out_g(T439)) → U26_ga(T439, T438, T441, add99_in_gga(T438, T439, T441))
add99_in_gga(s(T457), T458, s(T460)) → U6_gga(T457, T458, T460, add99_in_gga(T457, T458, T460))
add99_in_gga(0, T466, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, T460, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)), s(T441))
s21_in_ga(plus(T469, 0), T469) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0), T469)
s21_in_ga(plus(T474, 0), T474) → s21_out_ga(plus(T474, 0), T474)
s21_in_ga(plus(T495, T496), T498) → U29_ga(T495, T496, T498, s21_in_ga(T495, X508))
s21_in_ga(plus(T495, T496), T498) → U30_ga(T495, T496, T498, s21_in_ga(T495, T502))
s21_in_ga(plus(T517, T518), T520) → U32_ga(T517, T518, T520, isNat33_in_g(T517))
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518), T520)
U32_ga(T517, T518, T520, isNat33_out_g(T517)) → U33_ga(T517, T518, T520, isNat33_in_g(T518))
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518), T520)
U33_ga(T517, T518, T520, isNat33_out_g(T518)) → U34_ga(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_ga(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518), T520)
U30_ga(T495, T496, T498, s21_out_ga(T495, T502)) → U31_ga(T495, T496, T498, p22_in_gaaa(T496, X509, T502, T498))
p22_in_gaaa(T115, X104, T121, T117) → U2_gaaa(T115, X104, T121, T117, s21_in_ga(T115, X104))
U2_gaaa(T115, X104, T121, T117, s21_out_ga(T115, X104)) → p22_out_gaaa(T115, X104, T121, T117)
p22_in_gaaa(T115, T128, T121, T117) → U3_gaaa(T115, T128, T121, T117, s21_in_ga(T115, T128))
U3_gaaa(T115, T128, T121, T117, s21_out_ga(T115, T128)) → U4_gaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
s21_in_aa(plus(T32, plus(T33, plus(T34, T35))), T37) → U7_aa(T32, T33, T34, T35, T37, s21_in_aa(plus(plus(plus(T32, T33), T34), T35), T37))
s21_in_aa(plus(T70, plus(T71, T72)), T74) → U8_aa(T70, T71, T72, T74, s21_in_aa(plus(T72, plus(T70, T71)), T74))
s21_in_aa(plus(T87, plus(T88, 0)), plus(T87, T88)) → s21_out_aa(plus(T87, plus(T88, 0)), plus(T87, T88))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U9_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), X103))
s21_in_aa(plus(T113, plus(T114, T115)), T117) → U10_aa(T113, T114, T115, T117, s21_in_aa(plus(T113, T114), T121))
s21_in_aa(plus(T195, plus(T196, T197)), T199) → U12_aa(T195, T196, T197, T199, s21_in_aa(plus(plus(T196, T197), T195), T199))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U13_aa(T230, T231, T232, T234, s21_in_aa(T230, X224))
s21_in_aa(plus(T230, plus(T231, T232)), T234) → U14_aa(T230, T231, T232, T234, s21_in_aa(T230, T238))
s21_in_aa(plus(T255, plus(T256, T257)), T259) → U16_aa(T255, T256, T257, T259, isNat33_in_a(T255))
isNat33_in_a(s(T165)) → U1_a(T165, isNat33_in_a(T165))
isNat33_in_a(0) → isNat33_out_a(0)
U1_a(T165, isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(T255, T256, T257, T259, isNat33_out_a(T255)) → s21_out_aa(plus(T255, plus(T256, T257)), T259)
s21_in_aa(plus(plus(T321, T322), T320), T324) → U17_aa(T321, T322, T320, T324, s21_in_aa(plus(plus(T320, T321), T322), T324))
s21_in_aa(plus(T353, T352), T355) → U18_aa(T353, T352, T355, s21_in_aa(plus(T353, T352), T355))
s21_in_aa(plus(0, T364), T364) → s21_out_aa(plus(0, T364), T364)
s21_in_aa(plus(T386, T385), T388) → U19_aa(T386, T385, T388, s21_in_aa(T385, X382))
s21_in_aa(plus(T386, T385), T388) → U20_aa(T386, T385, T388, s21_in_aa(T385, T392))
s21_in_aa(plus(T416, T415), T418) → U22_aa(T416, T415, T418, isNat33_in_a(T415))
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → s21_out_aa(plus(T416, T415), T418)
U22_aa(T416, T415, T418, isNat33_out_a(T415)) → U23_aa(T416, T415, T418, isNat33_in_a(T416))
U23_aa(T416, T415, T418, isNat33_out_a(T416)) → s21_out_aa(plus(T416, T415), T418)
s21_in_aa(plus(T439, s(T438)), s(T441)) → U24_aa(T439, T438, T441, isNat33_in_a(s(T438)))
U24_aa(T439, T438, T441, isNat33_out_a(s(T438))) → U25_aa(T439, T438, T441, isNat33_in_a(T439))
U25_aa(T439, T438, T441, isNat33_out_a(T439)) → U26_aa(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_aa(T439, T438, T441, add99_out_gga(T438, T439, T441)) → s21_out_aa(plus(T439, s(T438)), s(T441))
s21_in_aa(plus(T469, 0), T469) → U27_aa(T469, isNat33_in_g(0))
U27_aa(T469, isNat33_out_g(0)) → U28_aa(T469, isNat33_in_a(T469))
U28_aa(T469, isNat33_out_a(T469)) → s21_out_aa(plus(T469, 0), T469)
s21_in_aa(plus(T474, 0), T474) → s21_out_aa(plus(T474, 0), T474)
s21_in_aa(plus(T495, T496), T498) → U29_aa(T495, T496, T498, s21_in_aa(T495, X508))
s21_in_aa(plus(T495, T496), T498) → U30_aa(T495, T496, T498, s21_in_aa(T495, T502))
s21_in_aa(plus(T517, T518), T520) → U32_aa(T517, T518, T520, isNat33_in_a(T517))
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → s21_out_aa(plus(T517, T518), T520)
U32_aa(T517, T518, T520, isNat33_out_a(T517)) → U33_aa(T517, T518, T520, isNat33_in_a(T518))
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → s21_out_aa(plus(T517, T518), T520)
U33_aa(T517, T518, T520, isNat33_out_a(T518)) → U34_aa(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_aa(T517, T518, T520, add99_out_gga(T517, T518, T520)) → s21_out_aa(plus(T517, T518), T520)
U30_aa(T495, T496, T498, s21_out_aa(T495, T502)) → U31_aa(T495, T496, T498, p22_in_aaaa(T496, X509, T502, T498))
p22_in_aaaa(T115, X104, T121, T117) → U2_aaaa(T115, X104, T121, T117, s21_in_aa(T115, X104))
U2_aaaa(T115, X104, T121, T117, s21_out_aa(T115, X104)) → p22_out_aaaa(T115, X104, T121, T117)
p22_in_aaaa(T115, T128, T121, T117) → U3_aaaa(T115, T128, T121, T117, s21_in_aa(T115, T128))
U3_aaaa(T115, T128, T121, T117, s21_out_aa(T115, T128)) → U4_aaaa(T115, T128, T121, T117, s21_in_aa(plus(T121, T128), T117))
U4_aaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_aaaa(T115, T128, T121, T117)
U31_aa(T495, T496, T498, p22_out_aaaa(T496, X509, T502, T498)) → s21_out_aa(plus(T495, T496), T498)
U29_aa(T495, T496, T498, s21_out_aa(T495, X508)) → s21_out_aa(plus(T495, T496), T498)
U20_aa(T386, T385, T388, s21_out_aa(T385, T392)) → U21_aa(T386, T385, T388, p22_in_aaaa(T386, X383, T392, T388))
U21_aa(T386, T385, T388, p22_out_aaaa(T386, X383, T392, T388)) → s21_out_aa(plus(T386, T385), T388)
U19_aa(T386, T385, T388, s21_out_aa(T385, X382)) → s21_out_aa(plus(T386, T385), T388)
U18_aa(T353, T352, T355, s21_out_aa(plus(T353, T352), T355)) → s21_out_aa(plus(T353, T352), T355)
U17_aa(T321, T322, T320, T324, s21_out_aa(plus(plus(T320, T321), T322), T324)) → s21_out_aa(plus(plus(T321, T322), T320), T324)
U14_aa(T230, T231, T232, T234, s21_out_aa(T230, T238)) → U15_aa(T230, T231, T232, T234, p22_in_aaaa(plus(T231, T232), X225, T238, T234))
U15_aa(T230, T231, T232, T234, p22_out_aaaa(plus(T231, T232), X225, T238, T234)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U13_aa(T230, T231, T232, T234, s21_out_aa(T230, X224)) → s21_out_aa(plus(T230, plus(T231, T232)), T234)
U12_aa(T195, T196, T197, T199, s21_out_aa(plus(plus(T196, T197), T195), T199)) → s21_out_aa(plus(T195, plus(T196, T197)), T199)
U10_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), T121)) → U11_aa(T113, T114, T115, T117, p22_in_aaaa(T115, X104, T121, T117))
U11_aa(T113, T114, T115, T117, p22_out_aaaa(T115, X104, T121, T117)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U9_aa(T113, T114, T115, T117, s21_out_aa(plus(T113, T114), X103)) → s21_out_aa(plus(T113, plus(T114, T115)), T117)
U8_aa(T70, T71, T72, T74, s21_out_aa(plus(T72, plus(T70, T71)), T74)) → s21_out_aa(plus(T70, plus(T71, T72)), T74)
U7_aa(T32, T33, T34, T35, T37, s21_out_aa(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_aa(plus(T32, plus(T33, plus(T34, T35))), T37)
U4_gaaa(T115, T128, T121, T117, s21_out_aa(plus(T121, T128), T117)) → p22_out_gaaa(T115, T128, T121, T117)
U31_ga(T495, T496, T498, p22_out_gaaa(T496, X509, T502, T498)) → s21_out_ga(plus(T495, T496), T498)
U29_ga(T495, T496, T498, s21_out_ga(T495, X508)) → s21_out_ga(plus(T495, T496), T498)
U20_ga(T386, T385, T388, s21_out_ga(T385, T392)) → U21_ga(T386, T385, T388, p22_in_gaaa(T386, X383, T392, T388))
U21_ga(T386, T385, T388, p22_out_gaaa(T386, X383, T392, T388)) → s21_out_ga(plus(T386, T385), T388)
U19_ga(T386, T385, T388, s21_out_ga(T385, X382)) → s21_out_ga(plus(T386, T385), T388)
U18_ga(T353, T352, T355, s21_out_ga(plus(T353, T352), T355)) → s21_out_ga(plus(T353, T352), T355)
U17_ga(T321, T322, T320, T324, s21_out_ga(plus(plus(T320, T321), T322), T324)) → s21_out_ga(plus(plus(T321, T322), T320), T324)
U14_ga(T230, T231, T232, T234, s21_out_ga(T230, T238)) → U15_ga(T230, T231, T232, T234, p22_in_gaaa(plus(T231, T232), X225, T238, T234))
U15_ga(T230, T231, T232, T234, p22_out_gaaa(plus(T231, T232), X225, T238, T234)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U13_ga(T230, T231, T232, T234, s21_out_ga(T230, X224)) → s21_out_ga(plus(T230, plus(T231, T232)), T234)
U12_ga(T195, T196, T197, T199, s21_out_ga(plus(plus(T196, T197), T195), T199)) → s21_out_ga(plus(T195, plus(T196, T197)), T199)
U10_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), T121)) → U11_ga(T113, T114, T115, T117, p22_in_gaaa(T115, X104, T121, T117))
U11_ga(T113, T114, T115, T117, p22_out_gaaa(T115, X104, T121, T117)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U9_ga(T113, T114, T115, T117, s21_out_ga(plus(T113, T114), X103)) → s21_out_ga(plus(T113, plus(T114, T115)), T117)
U8_ga(T70, T71, T72, T74, s21_out_ga(plus(T72, plus(T70, T71)), T74)) → s21_out_ga(plus(T70, plus(T71, T72)), T74)
U7_ga(T32, T33, T34, T35, T37, s21_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
s21_in_ga(x1, x2)  =  s21_in_ga(x1)
plus(x1, x2)  =  plus(x1, x2)
U7_ga(x1, x2, x3, x4, x5, x6)  =  U7_ga(x1, x2, x3, x4, x6)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x1, x2, x3, x5)
0  =  0
s21_out_ga(x1, x2)  =  s21_out_ga(x1)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x1, x2, x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x1, x2, x3, x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x3, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x3, x5)
isNat33_in_g(x1)  =  isNat33_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
isNat33_out_g(x1)  =  isNat33_out_g(x1)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4)  =  U18_ga(x1, x2, x4)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x2, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x2, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x2, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
add99_in_gga(x1, x2, x3)  =  add99_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
add99_out_gga(x1, x2, x3)  =  add99_out_gga(x1, x2, x3)
U27_ga(x1, x2)  =  U27_ga(x1, x2)
U28_ga(x1, x2)  =  U28_ga(x1, x2)
U29_ga(x1, x2, x3, x4)  =  U29_ga(x1, x2, x4)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
U32_ga(x1, x2, x3, x4)  =  U32_ga(x1, x2, x4)
U33_ga(x1, x2, x3, x4)  =  U33_ga(x1, x2, x4)
U34_ga(x1, x2, x3, x4)  =  U34_ga(x1, x2, x4)
U31_ga(x1, x2, x3, x4)  =  U31_ga(x1, x2, x4)
p22_in_gaaa(x1, x2, x3, x4)  =  p22_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5)  =  U2_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4)  =  p22_out_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
s21_in_aa(x1, x2)  =  s21_in_aa
U7_aa(x1, x2, x3, x4, x5, x6)  =  U7_aa(x6)
U8_aa(x1, x2, x3, x4, x5)  =  U8_aa(x5)
s21_out_aa(x1, x2)  =  s21_out_aa
U9_aa(x1, x2, x3, x4, x5)  =  U9_aa(x5)
U10_aa(x1, x2, x3, x4, x5)  =  U10_aa(x5)
U12_aa(x1, x2, x3, x4, x5)  =  U12_aa(x5)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
isNat33_in_a(x1)  =  isNat33_in_a
U1_a(x1, x2)  =  U1_a(x2)
isNat33_out_a(x1)  =  isNat33_out_a(x1)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4)  =  U18_aa(x4)
U19_aa(x1, x2, x3, x4)  =  U19_aa(x4)
U20_aa(x1, x2, x3, x4)  =  U20_aa(x4)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x4)
U23_aa(x1, x2, x3, x4)  =  U23_aa(x4)
U24_aa(x1, x2, x3, x4)  =  U24_aa(x4)
U25_aa(x1, x2, x3, x4)  =  U25_aa(x2, x4)
U26_aa(x1, x2, x3, x4)  =  U26_aa(x4)
U27_aa(x1, x2)  =  U27_aa(x2)
U28_aa(x1, x2)  =  U28_aa(x2)
U29_aa(x1, x2, x3, x4)  =  U29_aa(x4)
U30_aa(x1, x2, x3, x4)  =  U30_aa(x4)
U32_aa(x1, x2, x3, x4)  =  U32_aa(x4)
U33_aa(x1, x2, x3, x4)  =  U33_aa(x1, x4)
U34_aa(x1, x2, x3, x4)  =  U34_aa(x4)
U31_aa(x1, x2, x3, x4)  =  U31_aa(x4)
p22_in_aaaa(x1, x2, x3, x4)  =  p22_in_aaaa
U2_aaaa(x1, x2, x3, x4, x5)  =  U2_aaaa(x5)
p22_out_aaaa(x1, x2, x3, x4)  =  p22_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5)  =  U3_aaaa(x5)
U4_aaaa(x1, x2, x3, x4, x5)  =  U4_aaaa(x5)
U21_aa(x1, x2, x3, x4)  =  U21_aa(x4)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x2, x4)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x1, x2, x3, x5)
S21_IN_GA(x1, x2)  =  S21_IN_GA(x1)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x3, x5)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x1, x2, x4)
P22_IN_GAAA(x1, x2, x3, x4)  =  P22_IN_GAAA(x1)

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

(125) PiDPToQDPProof (SOUND transformation)

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

(126) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
S21_IN_GA(plus(T113, plus(T114, T115))) → U10_GA(T113, T114, T115, s21_in_ga(plus(T113, T114)))
U10_GA(T113, T114, T115, s21_out_ga(plus(T113, T114))) → P22_IN_GAAA(T115)
P22_IN_GAAA(T115) → S21_IN_GA(T115)
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(T230, plus(T231, T232))) → S21_IN_GA(T230)
S21_IN_GA(plus(T230, plus(T231, T232))) → U14_GA(T230, T231, T232, s21_in_ga(T230))
U14_GA(T230, T231, T232, s21_out_ga(T230)) → P22_IN_GAAA(plus(T231, T232))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))
S21_IN_GA(plus(T386, T385)) → S21_IN_GA(T385)
S21_IN_GA(plus(T386, T385)) → U20_GA(T386, T385, s21_in_ga(T385))
U20_GA(T386, T385, s21_out_ga(T385)) → P22_IN_GAAA(T386)
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
S21_IN_GA(plus(T495, T496)) → U30_GA(T495, T496, s21_in_ga(T495))
U30_GA(T495, T496, s21_out_ga(T495)) → P22_IN_GAAA(T496)

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(T32, T33, T34, T35, s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(T70, T71, T72, s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T87, plus(T88, 0))) → s21_out_ga(plus(T87, plus(T88, 0)))
s21_in_ga(plus(T113, plus(T114, T115))) → U9_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(T195, T196, T197, s21_in_ga(plus(plus(T196, T197), T195)))
s21_in_ga(plus(T230, plus(T231, T232))) → U13_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T230, plus(T231, T232))) → U14_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T255, plus(T256, T257))) → U16_ga(T255, T256, T257, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)))
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(T321, T322, T320, s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(T353, T352, s21_in_ga(plus(T353, T352)))
s21_in_ga(plus(0, T364)) → s21_out_ga(plus(0, T364))
s21_in_ga(plus(T386, T385)) → U19_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T386, T385)) → U20_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T416, T415)) → U22_ga(T416, T415, isNat33_in_g(T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → U23_ga(T416, T415, isNat33_in_g(T416))
U23_ga(T416, T415, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415))
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g(s(T438))) → U25_ga(T439, T438, isNat33_in_g(T439))
U25_ga(T439, T438, isNat33_out_g(T439)) → U26_ga(T439, T438, add99_in_gga(T438, T439))
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)))
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0))
s21_in_ga(plus(T474, 0)) → s21_out_ga(plus(T474, 0))
s21_in_ga(plus(T495, T496)) → U29_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T495, T496)) → U30_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518))
U32_ga(T517, T518, isNat33_out_g(T517)) → U33_ga(T517, T518, isNat33_in_g(T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → U34_ga(T517, T518, add99_in_gga(T517, T518))
U34_ga(T517, T518, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518))
U30_ga(T495, T496, s21_out_ga(T495)) → U31_ga(T495, T496, p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(T115, s21_in_ga(T115))
U2_gaaa(T115, s21_out_ga(T115)) → p22_out_gaaa(T115)
p22_in_gaaa(T115) → U3_gaaa(T115, s21_in_ga(T115))
U3_gaaa(T115, s21_out_ga(T115)) → U4_gaaa(T115, s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(T115, s21_out_aa) → p22_out_gaaa(T115)
U31_ga(T495, T496, p22_out_gaaa(T496)) → s21_out_ga(plus(T495, T496))
U29_ga(T495, T496, s21_out_ga(T495)) → s21_out_ga(plus(T495, T496))
U20_ga(T386, T385, s21_out_ga(T385)) → U21_ga(T386, T385, p22_in_gaaa(T386))
U21_ga(T386, T385, p22_out_gaaa(T386)) → s21_out_ga(plus(T386, T385))
U19_ga(T386, T385, s21_out_ga(T385)) → s21_out_ga(plus(T386, T385))
U18_ga(T353, T352, s21_out_ga(plus(T353, T352))) → s21_out_ga(plus(T353, T352))
U17_ga(T321, T322, T320, s21_out_ga(plus(plus(T320, T321), T322))) → s21_out_ga(plus(plus(T321, T322), T320))
U14_ga(T230, T231, T232, s21_out_ga(T230)) → U15_ga(T230, T231, T232, p22_in_gaaa(plus(T231, T232)))
U15_ga(T230, T231, T232, p22_out_gaaa(plus(T231, T232))) → s21_out_ga(plus(T230, plus(T231, T232)))
U13_ga(T230, T231, T232, s21_out_ga(T230)) → s21_out_ga(plus(T230, plus(T231, T232)))
U12_ga(T195, T196, T197, s21_out_ga(plus(plus(T196, T197), T195))) → s21_out_ga(plus(T195, plus(T196, T197)))
U10_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → U11_ga(T113, T114, T115, p22_in_gaaa(T115))
U11_ga(T113, T114, T115, p22_out_gaaa(T115)) → s21_out_ga(plus(T113, plus(T114, T115)))
U9_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → s21_out_ga(plus(T113, plus(T114, T115)))
U8_ga(T70, T71, T72, s21_out_ga(plus(T72, plus(T70, T71)))) → s21_out_ga(plus(T70, plus(T71, T72)))
U7_ga(T32, T33, T34, T35, s21_out_ga(plus(plus(plus(T32, T33), T34), T35))) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))))

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0, x1)
U16_ga(x0, x1, x2, x3)
U22_ga(x0, x1, x2)
U23_ga(x0, x1, x2)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0, x1, x2)
U26_ga(x0, x1, x2)
U27_ga(x0, x1)
U28_ga(x0, x1)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U30_ga(x0, x1, x2)
p22_in_gaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0, x1)
U31_ga(x0, x1, x2)
U29_ga(x0, x1, x2)
U20_ga(x0, x1, x2)
U21_ga(x0, x1, x2)
U19_ga(x0, x1, x2)
U18_ga(x0, x1, x2)
U17_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
U15_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U12_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U7_ga(x0, x1, x2, x3, x4)

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

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S21_IN_GA(plus(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
S21_IN_GA(plus(T113, plus(T114, T115))) → U10_GA(T113, T114, T115, s21_in_ga(plus(T113, T114)))
S21_IN_GA(plus(T230, plus(T231, T232))) → S21_IN_GA(T230)
S21_IN_GA(plus(T230, plus(T231, T232))) → U14_GA(T230, T231, T232, s21_in_ga(T230))
S21_IN_GA(plus(T386, T385)) → S21_IN_GA(T385)
S21_IN_GA(plus(T386, T385)) → U20_GA(T386, T385, s21_in_ga(T385))
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
U30_GA(T495, T496, s21_out_ga(T495)) → P22_IN_GAAA(T496)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P22_IN_GAAA(x1)) = x1   
POL(S21_IN_GA(x1)) = x1   
POL(U10_GA(x1, x2, x3, x4)) = x1 + x3   
POL(U10_aa(x1)) = 0   
POL(U10_ga(x1, x2, x3, x4)) = 0   
POL(U11_aa(x1)) = 0   
POL(U11_ga(x1, x2, x3, x4)) = 0   
POL(U12_aa(x1)) = 0   
POL(U12_ga(x1, x2, x3, x4)) = 0   
POL(U13_aa(x1)) = 0   
POL(U13_ga(x1, x2, x3, x4)) = 0   
POL(U14_GA(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(U14_aa(x1)) = 0   
POL(U14_ga(x1, x2, x3, x4)) = 0   
POL(U15_aa(x1)) = 0   
POL(U15_ga(x1, x2, x3, x4)) = 0   
POL(U16_aa(x1)) = 0   
POL(U16_ga(x1, x2, x3, x4)) = 0   
POL(U17_aa(x1)) = 0   
POL(U17_ga(x1, x2, x3, x4)) = 0   
POL(U18_aa(x1)) = 0   
POL(U18_ga(x1, x2, x3)) = 0   
POL(U19_aa(x1)) = 0   
POL(U19_ga(x1, x2, x3)) = 0   
POL(U1_a(x1)) = 0   
POL(U1_g(x1, x2)) = 0   
POL(U20_GA(x1, x2, x3)) = x1 + x2   
POL(U20_aa(x1)) = 0   
POL(U20_ga(x1, x2, x3)) = 0   
POL(U21_aa(x1)) = 0   
POL(U21_ga(x1, x2, x3)) = 0   
POL(U22_aa(x1)) = 0   
POL(U22_ga(x1, x2, x3)) = 0   
POL(U23_aa(x1)) = 0   
POL(U23_ga(x1, x2, x3)) = 0   
POL(U24_aa(x1)) = 0   
POL(U24_ga(x1, x2, x3)) = 0   
POL(U25_aa(x1, x2)) = 0   
POL(U25_ga(x1, x2, x3)) = 0   
POL(U26_aa(x1)) = 0   
POL(U26_ga(x1, x2, x3)) = 0   
POL(U27_aa(x1)) = 0   
POL(U27_ga(x1, x2)) = 0   
POL(U28_aa(x1)) = 0   
POL(U28_ga(x1, x2)) = 0   
POL(U29_aa(x1)) = 0   
POL(U29_ga(x1, x2, x3)) = 0   
POL(U2_aaaa(x1)) = 0   
POL(U2_gaaa(x1, x2)) = 0   
POL(U30_GA(x1, x2, x3)) = 1 + x2   
POL(U30_aa(x1)) = 0   
POL(U30_ga(x1, x2, x3)) = 0   
POL(U31_aa(x1)) = 0   
POL(U31_ga(x1, x2, x3)) = 0   
POL(U32_aa(x1)) = 0   
POL(U32_ga(x1, x2, x3)) = 0   
POL(U33_aa(x1, x2)) = 0   
POL(U33_ga(x1, x2, x3)) = 0   
POL(U34_aa(x1)) = 0   
POL(U34_ga(x1, x2, x3)) = 0   
POL(U3_aaaa(x1)) = 0   
POL(U3_gaaa(x1, x2)) = 0   
POL(U4_aaaa(x1)) = 0   
POL(U4_gaaa(x1, x2)) = 0   
POL(U6_gga(x1, x2, x3)) = 0   
POL(U7_aa(x1)) = 0   
POL(U7_ga(x1, x2, x3, x4, x5)) = 0   
POL(U8_aa(x1)) = 0   
POL(U8_ga(x1, x2, x3, x4)) = 0   
POL(U9_aa(x1)) = 0   
POL(U9_ga(x1, x2, x3, x4)) = 0   
POL(add99_in_gga(x1, x2)) = x2   
POL(add99_out_gga(x1, x2, x3)) = x3   
POL(isNat33_in_a) = 0   
POL(isNat33_in_g(x1)) = 1   
POL(isNat33_out_a(x1)) = 0   
POL(isNat33_out_g(x1)) = 0   
POL(p22_in_aaaa) = 0   
POL(p22_in_gaaa(x1)) = 0   
POL(p22_out_aaaa) = 0   
POL(p22_out_gaaa(x1)) = 0   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = 0   
POL(s21_in_aa) = 0   
POL(s21_in_ga(x1)) = 0   
POL(s21_out_aa) = 0   
POL(s21_out_ga(x1)) = 0   

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
U10_GA(T113, T114, T115, s21_out_ga(plus(T113, T114))) → P22_IN_GAAA(T115)
P22_IN_GAAA(T115) → S21_IN_GA(T115)
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
U14_GA(T230, T231, T232, s21_out_ga(T230)) → P22_IN_GAAA(plus(T231, T232))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))
U20_GA(T386, T385, s21_out_ga(T385)) → P22_IN_GAAA(T386)
S21_IN_GA(plus(T495, T496)) → U30_GA(T495, T496, s21_in_ga(T495))

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(T32, T33, T34, T35, s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(T70, T71, T72, s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T87, plus(T88, 0))) → s21_out_ga(plus(T87, plus(T88, 0)))
s21_in_ga(plus(T113, plus(T114, T115))) → U9_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(T195, T196, T197, s21_in_ga(plus(plus(T196, T197), T195)))
s21_in_ga(plus(T230, plus(T231, T232))) → U13_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T230, plus(T231, T232))) → U14_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T255, plus(T256, T257))) → U16_ga(T255, T256, T257, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)))
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(T321, T322, T320, s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(T353, T352, s21_in_ga(plus(T353, T352)))
s21_in_ga(plus(0, T364)) → s21_out_ga(plus(0, T364))
s21_in_ga(plus(T386, T385)) → U19_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T386, T385)) → U20_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T416, T415)) → U22_ga(T416, T415, isNat33_in_g(T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → U23_ga(T416, T415, isNat33_in_g(T416))
U23_ga(T416, T415, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415))
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g(s(T438))) → U25_ga(T439, T438, isNat33_in_g(T439))
U25_ga(T439, T438, isNat33_out_g(T439)) → U26_ga(T439, T438, add99_in_gga(T438, T439))
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)))
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0))
s21_in_ga(plus(T474, 0)) → s21_out_ga(plus(T474, 0))
s21_in_ga(plus(T495, T496)) → U29_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T495, T496)) → U30_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518))
U32_ga(T517, T518, isNat33_out_g(T517)) → U33_ga(T517, T518, isNat33_in_g(T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → U34_ga(T517, T518, add99_in_gga(T517, T518))
U34_ga(T517, T518, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518))
U30_ga(T495, T496, s21_out_ga(T495)) → U31_ga(T495, T496, p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(T115, s21_in_ga(T115))
U2_gaaa(T115, s21_out_ga(T115)) → p22_out_gaaa(T115)
p22_in_gaaa(T115) → U3_gaaa(T115, s21_in_ga(T115))
U3_gaaa(T115, s21_out_ga(T115)) → U4_gaaa(T115, s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(T115, s21_out_aa) → p22_out_gaaa(T115)
U31_ga(T495, T496, p22_out_gaaa(T496)) → s21_out_ga(plus(T495, T496))
U29_ga(T495, T496, s21_out_ga(T495)) → s21_out_ga(plus(T495, T496))
U20_ga(T386, T385, s21_out_ga(T385)) → U21_ga(T386, T385, p22_in_gaaa(T386))
U21_ga(T386, T385, p22_out_gaaa(T386)) → s21_out_ga(plus(T386, T385))
U19_ga(T386, T385, s21_out_ga(T385)) → s21_out_ga(plus(T386, T385))
U18_ga(T353, T352, s21_out_ga(plus(T353, T352))) → s21_out_ga(plus(T353, T352))
U17_ga(T321, T322, T320, s21_out_ga(plus(plus(T320, T321), T322))) → s21_out_ga(plus(plus(T321, T322), T320))
U14_ga(T230, T231, T232, s21_out_ga(T230)) → U15_ga(T230, T231, T232, p22_in_gaaa(plus(T231, T232)))
U15_ga(T230, T231, T232, p22_out_gaaa(plus(T231, T232))) → s21_out_ga(plus(T230, plus(T231, T232)))
U13_ga(T230, T231, T232, s21_out_ga(T230)) → s21_out_ga(plus(T230, plus(T231, T232)))
U12_ga(T195, T196, T197, s21_out_ga(plus(plus(T196, T197), T195))) → s21_out_ga(plus(T195, plus(T196, T197)))
U10_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → U11_ga(T113, T114, T115, p22_in_gaaa(T115))
U11_ga(T113, T114, T115, p22_out_gaaa(T115)) → s21_out_ga(plus(T113, plus(T114, T115)))
U9_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → s21_out_ga(plus(T113, plus(T114, T115)))
U8_ga(T70, T71, T72, s21_out_ga(plus(T72, plus(T70, T71)))) → s21_out_ga(plus(T70, plus(T71, T72)))
U7_ga(T32, T33, T34, T35, s21_out_ga(plus(plus(plus(T32, T33), T34), T35))) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))))

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0, x1)
U16_ga(x0, x1, x2, x3)
U22_ga(x0, x1, x2)
U23_ga(x0, x1, x2)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0, x1, x2)
U26_ga(x0, x1, x2)
U27_ga(x0, x1)
U28_ga(x0, x1)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U30_ga(x0, x1, x2)
p22_in_gaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0, x1)
U31_ga(x0, x1, x2)
U29_ga(x0, x1, x2)
U20_ga(x0, x1, x2)
U21_ga(x0, x1, x2)
U19_ga(x0, x1, x2)
U18_ga(x0, x1, x2)
U17_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
U15_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U12_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U7_ga(x0, x1, x2, x3, x4)

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

(129) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(130) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

The TRS R consists of the following rules:

s21_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U7_ga(T32, T33, T34, T35, s21_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s21_in_ga(plus(T70, plus(T71, T72))) → U8_ga(T70, T71, T72, s21_in_ga(plus(T72, plus(T70, T71))))
s21_in_ga(plus(T87, plus(T88, 0))) → s21_out_ga(plus(T87, plus(T88, 0)))
s21_in_ga(plus(T113, plus(T114, T115))) → U9_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T113, plus(T114, T115))) → U10_ga(T113, T114, T115, s21_in_ga(plus(T113, T114)))
s21_in_ga(plus(T195, plus(T196, T197))) → U12_ga(T195, T196, T197, s21_in_ga(plus(plus(T196, T197), T195)))
s21_in_ga(plus(T230, plus(T231, T232))) → U13_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T230, plus(T231, T232))) → U14_ga(T230, T231, T232, s21_in_ga(T230))
s21_in_ga(plus(T255, plus(T256, T257))) → U16_ga(T255, T256, T257, isNat33_in_g(T255))
isNat33_in_g(s(T165)) → U1_g(T165, isNat33_in_g(T165))
isNat33_in_g(0) → isNat33_out_g(0)
U1_g(T165, isNat33_out_g(T165)) → isNat33_out_g(s(T165))
U16_ga(T255, T256, T257, isNat33_out_g(T255)) → s21_out_ga(plus(T255, plus(T256, T257)))
s21_in_ga(plus(plus(T321, T322), T320)) → U17_ga(T321, T322, T320, s21_in_ga(plus(plus(T320, T321), T322)))
s21_in_ga(plus(T353, T352)) → U18_ga(T353, T352, s21_in_ga(plus(T353, T352)))
s21_in_ga(plus(0, T364)) → s21_out_ga(plus(0, T364))
s21_in_ga(plus(T386, T385)) → U19_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T386, T385)) → U20_ga(T386, T385, s21_in_ga(T385))
s21_in_ga(plus(T416, T415)) → U22_ga(T416, T415, isNat33_in_g(T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → s21_out_ga(plus(T416, T415))
U22_ga(T416, T415, isNat33_out_g(T415)) → U23_ga(T416, T415, isNat33_in_g(T416))
U23_ga(T416, T415, isNat33_out_g(T416)) → s21_out_ga(plus(T416, T415))
s21_in_ga(plus(T439, s(T438))) → U24_ga(T439, T438, isNat33_in_g(s(T438)))
U24_ga(T439, T438, isNat33_out_g(s(T438))) → U25_ga(T439, T438, isNat33_in_g(T439))
U25_ga(T439, T438, isNat33_out_g(T439)) → U26_ga(T439, T438, add99_in_gga(T438, T439))
add99_in_gga(s(T457), T458) → U6_gga(T457, T458, add99_in_gga(T457, T458))
add99_in_gga(0, T466) → add99_out_gga(0, T466, T466)
U6_gga(T457, T458, add99_out_gga(T457, T458, T460)) → add99_out_gga(s(T457), T458, s(T460))
U26_ga(T439, T438, add99_out_gga(T438, T439, T441)) → s21_out_ga(plus(T439, s(T438)))
s21_in_ga(plus(T469, 0)) → U27_ga(T469, isNat33_in_g(0))
U27_ga(T469, isNat33_out_g(0)) → U28_ga(T469, isNat33_in_g(T469))
U28_ga(T469, isNat33_out_g(T469)) → s21_out_ga(plus(T469, 0))
s21_in_ga(plus(T474, 0)) → s21_out_ga(plus(T474, 0))
s21_in_ga(plus(T495, T496)) → U29_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T495, T496)) → U30_ga(T495, T496, s21_in_ga(T495))
s21_in_ga(plus(T517, T518)) → U32_ga(T517, T518, isNat33_in_g(T517))
U32_ga(T517, T518, isNat33_out_g(T517)) → s21_out_ga(plus(T517, T518))
U32_ga(T517, T518, isNat33_out_g(T517)) → U33_ga(T517, T518, isNat33_in_g(T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → s21_out_ga(plus(T517, T518))
U33_ga(T517, T518, isNat33_out_g(T518)) → U34_ga(T517, T518, add99_in_gga(T517, T518))
U34_ga(T517, T518, add99_out_gga(T517, T518, T520)) → s21_out_ga(plus(T517, T518))
U30_ga(T495, T496, s21_out_ga(T495)) → U31_ga(T495, T496, p22_in_gaaa(T496))
p22_in_gaaa(T115) → U2_gaaa(T115, s21_in_ga(T115))
U2_gaaa(T115, s21_out_ga(T115)) → p22_out_gaaa(T115)
p22_in_gaaa(T115) → U3_gaaa(T115, s21_in_ga(T115))
U3_gaaa(T115, s21_out_ga(T115)) → U4_gaaa(T115, s21_in_aa)
s21_in_aaU7_aa(s21_in_aa)
s21_in_aaU8_aa(s21_in_aa)
s21_in_aas21_out_aa
s21_in_aaU9_aa(s21_in_aa)
s21_in_aaU10_aa(s21_in_aa)
s21_in_aaU12_aa(s21_in_aa)
s21_in_aaU13_aa(s21_in_aa)
s21_in_aaU14_aa(s21_in_aa)
s21_in_aaU16_aa(isNat33_in_a)
isNat33_in_aU1_a(isNat33_in_a)
isNat33_in_aisNat33_out_a(0)
U1_a(isNat33_out_a(T165)) → isNat33_out_a(s(T165))
U16_aa(isNat33_out_a(T255)) → s21_out_aa
s21_in_aaU17_aa(s21_in_aa)
s21_in_aaU18_aa(s21_in_aa)
s21_in_aaU19_aa(s21_in_aa)
s21_in_aaU20_aa(s21_in_aa)
s21_in_aaU22_aa(isNat33_in_a)
U22_aa(isNat33_out_a(T415)) → s21_out_aa
U22_aa(isNat33_out_a(T415)) → U23_aa(isNat33_in_a)
U23_aa(isNat33_out_a(T416)) → s21_out_aa
s21_in_aaU24_aa(isNat33_in_a)
U24_aa(isNat33_out_a(s(T438))) → U25_aa(T438, isNat33_in_a)
U25_aa(T438, isNat33_out_a(T439)) → U26_aa(add99_in_gga(T438, T439))
U26_aa(add99_out_gga(T438, T439, T441)) → s21_out_aa
s21_in_aaU27_aa(isNat33_in_g(0))
U27_aa(isNat33_out_g(0)) → U28_aa(isNat33_in_a)
U28_aa(isNat33_out_a(T469)) → s21_out_aa
s21_in_aaU29_aa(s21_in_aa)
s21_in_aaU30_aa(s21_in_aa)
s21_in_aaU32_aa(isNat33_in_a)
U32_aa(isNat33_out_a(T517)) → s21_out_aa
U32_aa(isNat33_out_a(T517)) → U33_aa(T517, isNat33_in_a)
U33_aa(T517, isNat33_out_a(T518)) → s21_out_aa
U33_aa(T517, isNat33_out_a(T518)) → U34_aa(add99_in_gga(T517, T518))
U34_aa(add99_out_gga(T517, T518, T520)) → s21_out_aa
U30_aa(s21_out_aa) → U31_aa(p22_in_aaaa)
p22_in_aaaaU2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaaU3_aaaa(s21_in_aa)
U3_aaaa(s21_out_aa) → U4_aaaa(s21_in_aa)
U4_aaaa(s21_out_aa) → p22_out_aaaa
U31_aa(p22_out_aaaa) → s21_out_aa
U29_aa(s21_out_aa) → s21_out_aa
U20_aa(s21_out_aa) → U21_aa(p22_in_aaaa)
U21_aa(p22_out_aaaa) → s21_out_aa
U19_aa(s21_out_aa) → s21_out_aa
U18_aa(s21_out_aa) → s21_out_aa
U17_aa(s21_out_aa) → s21_out_aa
U14_aa(s21_out_aa) → U15_aa(p22_in_aaaa)
U15_aa(p22_out_aaaa) → s21_out_aa
U13_aa(s21_out_aa) → s21_out_aa
U12_aa(s21_out_aa) → s21_out_aa
U10_aa(s21_out_aa) → U11_aa(p22_in_aaaa)
U11_aa(p22_out_aaaa) → s21_out_aa
U9_aa(s21_out_aa) → s21_out_aa
U8_aa(s21_out_aa) → s21_out_aa
U7_aa(s21_out_aa) → s21_out_aa
U4_gaaa(T115, s21_out_aa) → p22_out_gaaa(T115)
U31_ga(T495, T496, p22_out_gaaa(T496)) → s21_out_ga(plus(T495, T496))
U29_ga(T495, T496, s21_out_ga(T495)) → s21_out_ga(plus(T495, T496))
U20_ga(T386, T385, s21_out_ga(T385)) → U21_ga(T386, T385, p22_in_gaaa(T386))
U21_ga(T386, T385, p22_out_gaaa(T386)) → s21_out_ga(plus(T386, T385))
U19_ga(T386, T385, s21_out_ga(T385)) → s21_out_ga(plus(T386, T385))
U18_ga(T353, T352, s21_out_ga(plus(T353, T352))) → s21_out_ga(plus(T353, T352))
U17_ga(T321, T322, T320, s21_out_ga(plus(plus(T320, T321), T322))) → s21_out_ga(plus(plus(T321, T322), T320))
U14_ga(T230, T231, T232, s21_out_ga(T230)) → U15_ga(T230, T231, T232, p22_in_gaaa(plus(T231, T232)))
U15_ga(T230, T231, T232, p22_out_gaaa(plus(T231, T232))) → s21_out_ga(plus(T230, plus(T231, T232)))
U13_ga(T230, T231, T232, s21_out_ga(T230)) → s21_out_ga(plus(T230, plus(T231, T232)))
U12_ga(T195, T196, T197, s21_out_ga(plus(plus(T196, T197), T195))) → s21_out_ga(plus(T195, plus(T196, T197)))
U10_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → U11_ga(T113, T114, T115, p22_in_gaaa(T115))
U11_ga(T113, T114, T115, p22_out_gaaa(T115)) → s21_out_ga(plus(T113, plus(T114, T115)))
U9_ga(T113, T114, T115, s21_out_ga(plus(T113, T114))) → s21_out_ga(plus(T113, plus(T114, T115)))
U8_ga(T70, T71, T72, s21_out_ga(plus(T72, plus(T70, T71)))) → s21_out_ga(plus(T70, plus(T71, T72)))
U7_ga(T32, T33, T34, T35, s21_out_ga(plus(plus(plus(T32, T33), T34), T35))) → s21_out_ga(plus(T32, plus(T33, plus(T34, T35))))

The set Q consists of the following terms:

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0, x1)
U16_ga(x0, x1, x2, x3)
U22_ga(x0, x1, x2)
U23_ga(x0, x1, x2)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0, x1, x2)
U26_ga(x0, x1, x2)
U27_ga(x0, x1)
U28_ga(x0, x1)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U30_ga(x0, x1, x2)
p22_in_gaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0, x1)
U31_ga(x0, x1, x2)
U29_ga(x0, x1, x2)
U20_ga(x0, x1, x2)
U21_ga(x0, x1, x2)
U19_ga(x0, x1, x2)
U18_ga(x0, x1, x2)
U17_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
U15_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U12_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U7_ga(x0, x1, x2, x3, x4)

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

(131) 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.

(132) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

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

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0, x1)
U16_ga(x0, x1, x2, x3)
U22_ga(x0, x1, x2)
U23_ga(x0, x1, x2)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0, x1, x2)
U26_ga(x0, x1, x2)
U27_ga(x0, x1)
U28_ga(x0, x1)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U30_ga(x0, x1, x2)
p22_in_gaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0, x1)
U31_ga(x0, x1, x2)
U29_ga(x0, x1, x2)
U20_ga(x0, x1, x2)
U21_ga(x0, x1, x2)
U19_ga(x0, x1, x2)
U18_ga(x0, x1, x2)
U17_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
U15_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U12_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U7_ga(x0, x1, x2, x3, x4)

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

(133) 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].

s21_in_ga(x0)
isNat33_in_g(x0)
U1_g(x0, x1)
U16_ga(x0, x1, x2, x3)
U22_ga(x0, x1, x2)
U23_ga(x0, x1, x2)
U24_ga(x0, x1, x2)
U25_ga(x0, x1, x2)
add99_in_gga(x0, x1)
U6_gga(x0, x1, x2)
U26_ga(x0, x1, x2)
U27_ga(x0, x1)
U28_ga(x0, x1)
U32_ga(x0, x1, x2)
U33_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U30_ga(x0, x1, x2)
p22_in_gaaa(x0)
U2_gaaa(x0, x1)
U3_gaaa(x0, x1)
s21_in_aa
isNat33_in_a
U1_a(x0)
U16_aa(x0)
U22_aa(x0)
U23_aa(x0)
U24_aa(x0)
U25_aa(x0, x1)
U26_aa(x0)
U27_aa(x0)
U28_aa(x0)
U32_aa(x0)
U33_aa(x0, x1)
U34_aa(x0)
U30_aa(x0)
p22_in_aaaa
U2_aaaa(x0)
U3_aaaa(x0)
U4_aaaa(x0)
U31_aa(x0)
U29_aa(x0)
U20_aa(x0)
U21_aa(x0)
U19_aa(x0)
U18_aa(x0)
U17_aa(x0)
U14_aa(x0)
U15_aa(x0)
U13_aa(x0)
U12_aa(x0)
U10_aa(x0)
U11_aa(x0)
U9_aa(x0)
U8_aa(x0)
U7_aa(x0)
U4_gaaa(x0, x1)
U31_ga(x0, x1, x2)
U29_ga(x0, x1, x2)
U20_ga(x0, x1, x2)
U21_ga(x0, x1, x2)
U19_ga(x0, x1, x2)
U18_ga(x0, x1, x2)
U17_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
U15_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U12_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U7_ga(x0, x1, x2, x3, x4)

(134) Obligation:

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

S21_IN_GA(plus(T32, plus(T33, plus(T34, T35)))) → S21_IN_GA(plus(plus(plus(T32, T33), T34), T35))
S21_IN_GA(plus(T70, plus(T71, T72))) → S21_IN_GA(plus(T72, plus(T70, T71)))
S21_IN_GA(plus(T195, plus(T196, T197))) → S21_IN_GA(plus(plus(T196, T197), T195))
S21_IN_GA(plus(plus(T321, T322), T320)) → S21_IN_GA(plus(plus(T320, T321), T322))
S21_IN_GA(plus(T353, T352)) → S21_IN_GA(plus(T353, T352))

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

(135) 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 = S21_IN_GA(plus(T70, plus(T71, T72))) evaluates to t =S21_IN_GA(plus(T72, plus(T70, T71)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [T70 / T72, T71 / T70, T72 / T71]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from S21_IN_GA(plus(T70, plus(T71, T72))) to S21_IN_GA(plus(T72, plus(T70, T71))).



(136) NO