Left Termination of the query pattern s2_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(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.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: s2(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: s2(T1, T2), SCOPE: 1, CLAUSE: 0 | s2(T1, T2), SCOPE: 1, CLAUSE: 1 | s2(T1, T2), SCOPE: 1, CLAUSE: 2 | s2(T1, T2), SCOPE: 1, CLAUSE: 3 | s2(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: s2(plus(plus(T7, T8), T9), T11) | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2(T1, T2), SCOPE: 1, CLAUSE: 1 | s2(T1, T2), SCOPE: 1, CLAUSE: 2 | s2(T1, T2), SCOPE: 1, CLAUSE: 3 | s2(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\ns2(T1, T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 0 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 1 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 2 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 3 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4 | ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 1 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 2 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 3 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4 | ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2(plus(plus(plus(T32, T33), T34), T35), T37)\n\nT32 is ground\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 2 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 3 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4 | ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: s2(plus(T72, plus(T70, T71)), T74)\n\nT70 is ground\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 3 | s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4 | ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4 | ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(s2(plus(T113, T114), X103), ','(s2(T115, X104), s2(plus(X103, X104), T117)))\n\nT113 is ground\nT114 is ground\nT115 is ground\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2(plus(T113, T114), X103)\n\nT113 is ground\nT114 is ground\nX103 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(s2(T115, X104), s2(plus(T121, X104), T117))\n\nT115 is ground\nT121 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: s2(T115, X104)\n\nT115 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: s2(plus(T121, T128), T117)\n\nT121 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2(plus(plus(T7, T8), T9), T11), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(isNat(plus(T154, T155)), ','(isNat(T156), add(plus(T154, T155), T156, T158)))\n\nT154 is ground\nT155 is ground\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: isNat(plus(T154, T155))\n\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(isNat(T156), add(plus(T154, T155), T156, T158))\n\nT154 is ground\nT155 is ground\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: isNat(plus(T154, T155)), SCOPE: 3, CLAUSE: 5 | isNat(plus(T154, T155)), SCOPE: 3, CLAUSE: 6\n\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: isNat(plus(T154, T155)), SCOPE: 3, CLAUSE: 6\n\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: isNat(T156)\n\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: add(plus(T154, T155), T156, T158)\n\nT154 is ground\nT155 is ground\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: isNat(T156), SCOPE: 4, CLAUSE: 5 | isNat(T156), SCOPE: 4, CLAUSE: 6\n\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: isNat(T156), SCOPE: 4, CLAUSE: 5\n\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: isNat(T156), SCOPE: 4, CLAUSE: 6\n\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: isNat(T165)\n\nT165 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: add(plus(T154, T155), T156, T158), SCOPE: 5, CLAUSE: 7 | add(plus(T154, T155), T156, T158), SCOPE: 5, CLAUSE: 8\n\nT154 is ground\nT155 is ground\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: add(plus(T154, T155), T156, T158), SCOPE: 5, CLAUSE: 8\n\nT154 is ground\nT155 is ground\nT156 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: s2(plus(plus(T196, T197), T195), T199)\n\nT195 is ground\nT196 is ground\nT197 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: s2(plus(T7, plus(T8, T9)), T2), SCOPE: 1, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(s2(T230, X224), ','(s2(plus(T231, T232), X225), s2(plus(X224, X225), T234)))\n\nT230 is ground\nT231 is ground\nT232 is ground\nX224 is free; \nX225 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: s2(T230, X224)\n\nT230 is ground\nX224 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(s2(plus(T231, T232), X225), s2(plus(T238, X225), T234))\n\nT231 is ground\nT232 is ground\nT238 is ground\nX225 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(isNat(T255), ','(isNat(plus(T256, T257)), add(T255, plus(T256, T257), T259)))\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: isNat(T255)\n\nT255 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(isNat(plus(T256, T257)), add(T255, plus(T256, T257), T259))\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: isNat(plus(T256, T257))\n\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: add(T255, plus(T256, T257), T259)\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: add(T255, plus(T256, T257), T259), SCOPE: 6, CLAUSE: 7 | add(T255, plus(T256, T257), T259), SCOPE: 6, CLAUSE: 8\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: add(T255, plus(T256, T257), T259), SCOPE: 6, CLAUSE: 7\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: add(T255, plus(T256, T257), T259), SCOPE: 6, CLAUSE: 8\n\nT255 is ground\nT256 is ground\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: add(T282, plus(T283, T284), T286)\n\nT282 is ground\nT283 is ground\nT284 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: s2(plus(T301, T300), T303) | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: s2(T1, T2), SCOPE: 1, CLAUSE: 2 | s2(T1, T2), SCOPE: 1, CLAUSE: 3 | s2(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\ns2(T1, T2) !=? s2(plus(X5, plus(X6, X7)), X8)\ns2(T1, T2) !=? s2(plus(X286, X287), X288)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 0 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 1 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 2 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 3 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4 | ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 0\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 1 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 2 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 3 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4 | ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: s2(plus(plus(T320, T321), T322), T324)\n\nT320 is ground\nT321 is ground\nT322 is ground\ns2(plus(plus(T321, T322), T320), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 1\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 2 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 3 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4 | ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: s2(plus(T353, T352), T355)\n\nT352 is ground\nT353 is ground\ns2(plus(T353, T352), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 2\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 3 | s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4 | ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 3\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4 | ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(s2(T385, X382), ','(s2(T386, X383), s2(plus(X382, X383), T388)))\n\nT385 is ground\nT386 is ground\nX382 is free; \nX383 is free; \ns2(plus(T386, T385), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: s2(T385, X382)\n\nT385 is ground\nX382 is free; \ns2(plus(T386, T385), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(s2(T386, X383), s2(plus(T392, X383), T388))\n\nT386 is ground\nT392 is ground\nX383 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: s2(plus(T301, T300), T303), SCOPE: 7, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ?_7 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(isNat(T415), ','(isNat(T416), add(T415, T416, T418)))\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: isNat(T415)\n\nT415 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(isNat(T416), add(T415, T416, T418))\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: isNat(T416)\n\nT416 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: add(T415, T416, T418)\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: add(T415, T416, T418), SCOPE: 8, CLAUSE: 7 | add(T415, T416, T418), SCOPE: 8, CLAUSE: 8\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: add(T415, T416, T418), SCOPE: 8, CLAUSE: 7\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: add(T415, T416, T418), SCOPE: 8, CLAUSE: 8\n\nT415 is ground\nT416 is ground\ns2(plus(T416, T415), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: add(T438, T439, T441)\n\nT438 is ground\nT439 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: add(T438, T439, T441), SCOPE: 9, CLAUSE: 7 | add(T438, T439, T441), SCOPE: 9, CLAUSE: 8\n\nT438 is ground\nT439 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: add(T438, T439, T441), SCOPE: 9, CLAUSE: 7\n\nT438 is ground\nT439 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: add(T438, T439, T441), SCOPE: 9, CLAUSE: 8\n\nT438 is ground\nT439 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: add(T457, T458, T460)\n\nT457 is ground\nT458 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 2\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3 | s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 3\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: s2(plus(T300, T301), T2), SCOPE: 1, CLAUSE: 4\n\nT300 is ground\nT301 is ground\ns2(plus(T300, T301), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(s2(T495, X508), ','(s2(T496, X509), s2(plus(X508, X509), T498)))\n\nT495 is ground\nT496 is ground\nX508 is free; \nX509 is free; \ns2(plus(T495, T496), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: s2(T495, X508)\n\nT495 is ground\nX508 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(s2(T496, X509), s2(plus(T502, X509), T498))\n\nT496 is ground\nT502 is ground\nX509 is free; \ns2(plus(T495, T496), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(isNat(T517), ','(isNat(T518), add(T517, T518, T520)))\n\nT517 is ground\nT518 is ground\ns2(plus(T517, T518), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: isNat(T517)\n\nT517 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(isNat(T518), add(T517, T518, T520))\n\nT517 is ground\nT518 is ground\ns2(plus(T517, T518), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: isNat(T518)\n\nT518 is ground\ns2(plus(T517, T518), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: add(T517, T518, T520)\n\nT517 is ground\nT518 is ground\ns2(plus(T517, T518), T2) !=? s2(plus(X5, plus(X6, X7)), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: s2(T1, T2), SCOPE: 1, CLAUSE: 3 | s2(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\ns2(T1, T2) !=? s2(plus(X5, plus(X6, X7)), X8)\ns2(T1, T2) !=? s2(plus(X286, X287), X288)", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: s2(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\ns2(T1, T2) !=? s2(plus(X5, plus(X6, X7)), X8)\ns2(T1, T2) !=? s2(plus(X286, X287), X288)", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 131 [arrowhead = none ];
131 -> 2;

132 [label="EVAL with clause\ns2(plus(X5, plus(X6, X7)), X8) :- s2(plus(plus(X5, X6), X7), X8).\nand substitution\nX5 -> T7\nX6 -> T8\nX7 -> T9\nT1 -> plus(T7, plus(T8, T9))\nT2 -> T11\nX8 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 132 [arrowhead = none ];
132 -> 3;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 133 [arrowhead = none ];
133 -> 4;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 134 [arrowhead = none ];
134 -> 5;

135 [label="EVAL with clause\ns2(plus(X286, X287), X288) :- s2(plus(X287, X286), X288).\nand substitution\nX286 -> T300\nX287 -> T301\nT1 -> plus(T300, T301)\nT2 -> T303\nX288 -> T303\nT302 -> T303", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 135 [arrowhead = none ];
135 -> 69;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 136 [arrowhead = none ];
136 -> 70;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 137 [arrowhead = none ];
137 -> 6;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 138 [arrowhead = none ];
138 -> 7;

139 [label="EVAL with clause\ns2(plus(X29, plus(X30, X31)), X32) :- s2(plus(plus(X29, X30), X31), X32).\nand substitution\nT7 -> T32\nT8 -> T33\nX29 -> plus(T32, T33)\nX30 -> T34\nX31 -> T35\nT9 -> plus(T34, T35)\nT11 -> T37\nX32 -> T37\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 139 [arrowhead = none ];
139 -> 8;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 140 [arrowhead = none ];
140 -> 9;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 141 [arrowhead = none ];
141 -> 10;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 142 [arrowhead = none ];
142 -> 11;

143 [label="INSTANCE with matching:\nT1 -> plus(plus(plus(T32, T33), T34), T35)\nT2 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
8 -> 143 [arrowhead = none , style=dashed];
143 -> 1[style=dashed];

144 [label="ONLY EVAL with clause\ns2(plus(X61, X62), X63) :- s2(plus(X62, X61), X63).\nand substitution\nT7 -> T70\nT8 -> T71\nX61 -> plus(T70, T71)\nT9 -> T72\nX62 -> T72\nT11 -> T74\nX63 -> T74\nT73 -> T74", fontsize=14, style = filled, fillcolor = white];
10 -> 144 [arrowhead = none ];
144 -> 12;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 145 [arrowhead = none ];
145 -> 13;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 146 [arrowhead = none ];
146 -> 14;

147 [label="INSTANCE with matching:\nT1 -> plus(T72, plus(T70, T71))\nT2 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 147 [arrowhead = none , style=dashed];
147 -> 1[style=dashed];

148 [label="EVAL with clause\ns2(plus(X72, 0), X72).\nand substitution\nT7 -> T87\nT8 -> T88\nX72 -> plus(T87, T88)\nT9 -> 0\nT11 -> plus(T87, T88)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 148 [arrowhead = none ];
148 -> 15;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 149 [arrowhead = none ];
149 -> 16;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 150 [arrowhead = none ];
150 -> 18;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 151 [arrowhead = none ];
151 -> 19;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 152 [arrowhead = none ];
152 -> 17;

153 [label="ONLY EVAL with clause\ns2(plus(X100, X101), X102) :- ','(s2(X100, X103), ','(s2(X101, X104), s2(plus(X103, X104), X102))).\nand substitution\nT7 -> T113\nT8 -> T114\nX100 -> plus(T113, T114)\nT9 -> T115\nX101 -> T115\nT11 -> T117\nX102 -> T117\nT116 -> T117", fontsize=14, style = filled, fillcolor = white];
18 -> 153 [arrowhead = none ];
153 -> 20;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 154 [arrowhead = none ];
154 -> 25;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 155 [arrowhead = none ];
155 -> 26;

156 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 156 [arrowhead = none ];
156 -> 21;

157 [label="SPLIT 2\nnew knowledge:\nT121 is ground\nreplacements:\nX103 -> T121", fontsize=14, style = filled, fillcolor = violet];
20 -> 157 [arrowhead = none ];
157 -> 22;

158 [label="INSTANCE with matching:\nT1 -> plus(T113, T114)\nT2 -> X103", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 158 [arrowhead = none , style=dashed];
158 -> 1[style=dashed];

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 159 [arrowhead = none ];
159 -> 23;

160 [label="SPLIT 2\nnew knowledge:\nT128 is ground\nreplacements:\nX104 -> T128", fontsize=14, style = filled, fillcolor = violet];
22 -> 160 [arrowhead = none ];
160 -> 24;

161 [label="INSTANCE with matching:\nT1 -> T115\nT2 -> X104", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 161 [arrowhead = none , style=dashed];
161 -> 1[style=dashed];

162 [label="INSTANCE with matching:\nT1 -> plus(T121, T128)\nT2 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 162 [arrowhead = none , style=dashed];
162 -> 1[style=dashed];

163 [label="ONLY EVAL with clause\ns2(plus(X143, X144), X145) :- ','(isNat(X143), ','(isNat(X144), add(X143, X144, X145))).\nand substitution\nT7 -> T154\nT8 -> T155\nX143 -> plus(T154, T155)\nT9 -> T156\nX144 -> T156\nT11 -> T158\nX145 -> T158\nT157 -> T158", fontsize=14, style = filled, fillcolor = white];
25 -> 163 [arrowhead = none ];
163 -> 27;

164 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 164 [arrowhead = none ];
164 -> 46;

165 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 165 [arrowhead = none ];
165 -> 28;

166 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
27 -> 166 [arrowhead = none ];
166 -> 29;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 167 [arrowhead = none ];
167 -> 30;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 168 [arrowhead = none ];
168 -> 33;

169 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 169 [arrowhead = none ];
169 -> 34;

170 [label="BACKTRACK\nfor clause: isNat(s(X)) :- isNat(X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 170 [arrowhead = none ];
170 -> 31;

171 [label="BACKTRACK\nfor clause: isNat(0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 171 [arrowhead = none ];
171 -> 32;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 172 [arrowhead = none ];
172 -> 35;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 173 [arrowhead = none ];
173 -> 43;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 174 [arrowhead = none ];
174 -> 36;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 175 [arrowhead = none ];
175 -> 37;

176 [label="EVAL with clause\nisNat(s(X155)) :- isNat(X155).\nand substitution\nX155 -> T165\nT156 -> s(T165)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 176 [arrowhead = none ];
176 -> 38;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 177 [arrowhead = none ];
177 -> 39;

178 [label="EVAL with clause\nisNat(0).\nand substitution\nT156 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 178 [arrowhead = none ];
178 -> 40;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 179 [arrowhead = none ];
179 -> 41;

180 [label="INSTANCE with matching:\nT156 -> T165", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 180 [arrowhead = none , style=dashed];
180 -> 33[style=dashed];

181 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 181 [arrowhead = none ];
181 -> 42;

182 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 182 [arrowhead = none ];
182 -> 44;

183 [label="BACKTRACK\nfor clause: add(0, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 183 [arrowhead = none ];
183 -> 45;

184 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 184 [arrowhead = none ];
184 -> 47;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 185 [arrowhead = none ];
185 -> 48;

186 [label="ONLY EVAL with clause\ns2(plus(X187, X188), X189) :- s2(plus(X188, X187), X189).\nand substitution\nT7 -> T195\nX187 -> T195\nT8 -> T196\nT9 -> T197\nX188 -> plus(T196, T197)\nT2 -> T199\nX189 -> T199\nT198 -> T199", fontsize=14, style = filled, fillcolor = white];
47 -> 186 [arrowhead = none ];
186 -> 49;

187 [label="BACKTRACK\nfor clause: s2(plus(X, 0), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 187 [arrowhead = none ];
187 -> 50;

188 [label="INSTANCE with matching:\nT1 -> plus(plus(T196, T197), T195)\nT2 -> T199", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 188 [arrowhead = none , style=dashed];
188 -> 1[style=dashed];

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 189 [arrowhead = none ];
189 -> 51;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 190 [arrowhead = none ];
190 -> 52;

191 [label="ONLY EVAL with clause\ns2(plus(X221, X222), X223) :- ','(s2(X221, X224), ','(s2(X222, X225), s2(plus(X224, X225), X223))).\nand substitution\nT7 -> T230\nX221 -> T230\nT8 -> T231\nT9 -> T232\nX222 -> plus(T231, T232)\nT2 -> T234\nX223 -> T234\nT233 -> T234", fontsize=14, style = filled, fillcolor = white];
51 -> 191 [arrowhead = none ];
191 -> 53;

192 [label="ONLY EVAL with clause\ns2(plus(X248, X249), X250) :- ','(isNat(X248), ','(isNat(X249), add(X248, X249, X250))).\nand substitution\nT7 -> T255\nX248 -> T255\nT8 -> T256\nT9 -> T257\nX249 -> plus(T256, T257)\nT2 -> T259\nX250 -> T259\nT258 -> T259", fontsize=14, style = filled, fillcolor = white];
52 -> 192 [arrowhead = none ];
192 -> 56;

193 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 193 [arrowhead = none ];
193 -> 54;

194 [label="SPLIT 2\nnew knowledge:\nT238 is ground\nreplacements:\nX224 -> T238", fontsize=14, style = filled, fillcolor = violet];
53 -> 194 [arrowhead = none ];
194 -> 55;

195 [label="INSTANCE with matching:\nT1 -> T230\nT2 -> X224", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 195 [arrowhead = none , style=dashed];
195 -> 1[style=dashed];

196 [label="INSTANCE with matching:\nT115 -> plus(T231, T232)\nX104 -> X225\nT121 -> T238\nT117 -> T234", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 196 [arrowhead = none , style=dashed];
196 -> 22[style=dashed];

197 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 197 [arrowhead = none ];
197 -> 57;

198 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
56 -> 198 [arrowhead = none ];
198 -> 58;

199 [label="INSTANCE with matching:\nT156 -> T255", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 199 [arrowhead = none , style=dashed];
199 -> 33[style=dashed];

200 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 200 [arrowhead = none ];
200 -> 59;

201 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
58 -> 201 [arrowhead = none ];
201 -> 60;

202 [label="INSTANCE with matching:\nT154 -> T256\nT155 -> T257", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 202 [arrowhead = none , style=dashed];
202 -> 28[style=dashed];

203 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 203 [arrowhead = none ];
203 -> 61;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 204 [arrowhead = none ];
204 -> 62;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 205 [arrowhead = none ];
205 -> 63;

206 [label="EVAL with clause\nadd(s(X270), X271, s(X272)) :- add(X270, X271, X272).\nand substitution\nX270 -> T282\nT255 -> s(T282)\nT256 -> T283\nT257 -> T284\nX271 -> plus(T283, T284)\nX272 -> T286\nT259 -> s(T286)\nT285 -> T286", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 206 [arrowhead = none ];
206 -> 64;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 207 [arrowhead = none ];
207 -> 65;

208 [label="EVAL with clause\nadd(0, X281, X281).\nand substitution\nT255 -> 0\nT256 -> T295\nT257 -> T296\nX281 -> plus(T295, T296)\nT259 -> plus(T295, T296)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 208 [arrowhead = none ];
208 -> 66;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 209 [arrowhead = none ];
209 -> 67;

210 [label="INSTANCE with matching:\nT255 -> T282\nT256 -> T283\nT257 -> T284\nT259 -> T286", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 210 [arrowhead = none , style=dashed];
210 -> 60[style=dashed];

211 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 211 [arrowhead = none ];
211 -> 68;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 212 [arrowhead = none ];
212 -> 71;

213 [label="BACKTRACK\nfor clause: s2(plus(X, 0), X)\nwith clash: (s2(T1, T2), s2(plus(X286, X287), X288))", fontsize=14, style = filled, fillcolor = white];
70 -> 213 [arrowhead = none ];
213 -> 128;

214 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 214 [arrowhead = none ];
214 -> 72;

215 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 215 [arrowhead = none ];
215 -> 73;

216 [label="EVAL with clause\ns2(plus(X308, plus(X309, X310)), X311) :- s2(plus(plus(X308, X309), X310), X311).\nand substitution\nT301 -> T320\nX308 -> T320\nX309 -> T321\nX310 -> T322\nT300 -> plus(T321, T322)\nT303 -> T324\nX311 -> T324\nT323 -> T324", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 216 [arrowhead = none ];
216 -> 74;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 217 [arrowhead = none ];
217 -> 75;

218 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 218 [arrowhead = none ];
218 -> 76;

219 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 219 [arrowhead = none ];
219 -> 77;

220 [label="INSTANCE with matching:\nT1 -> plus(plus(T320, T321), T322)\nT2 -> T324", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 220 [arrowhead = none , style=dashed];
220 -> 1[style=dashed];

221 [label="ONLY EVAL with clause\ns2(plus(X340, X341), X342) :- s2(plus(X341, X340), X342).\nand substitution\nT301 -> T352\nX340 -> T352\nT300 -> T353\nX341 -> T353\nT303 -> T355\nX342 -> T355\nT354 -> T355", fontsize=14, style = filled, fillcolor = white];
76 -> 221 [arrowhead = none ];
221 -> 78;

222 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 222 [arrowhead = none ];
222 -> 79;

223 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 223 [arrowhead = none ];
223 -> 80;

224 [label="INSTANCE with matching:\nT1 -> plus(T353, T352)\nT2 -> T355", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 224 [arrowhead = none , style=dashed];
224 -> 1[style=dashed];

225 [label="EVAL with clause\ns2(plus(X351, 0), X351).\nand substitution\nT301 -> T364\nX351 -> T364\nT300 -> 0\nT303 -> T364", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 225 [arrowhead = none ];
225 -> 81;

226 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 226 [arrowhead = none ];
226 -> 82;

227 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 227 [arrowhead = none ];
227 -> 84;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 228 [arrowhead = none ];
228 -> 85;

229 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 229 [arrowhead = none ];
229 -> 83;

230 [label="ONLY EVAL with clause\ns2(plus(X379, X380), X381) :- ','(s2(X379, X382), ','(s2(X380, X383), s2(plus(X382, X383), X381))).\nand substitution\nT301 -> T385\nX379 -> T385\nT300 -> T386\nX380 -> T386\nT303 -> T388\nX381 -> T388\nT387 -> T388", fontsize=14, style = filled, fillcolor = white];
84 -> 230 [arrowhead = none ];
230 -> 86;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 231 [arrowhead = none ];
231 -> 89;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 232 [arrowhead = none ];
232 -> 90;

233 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
86 -> 233 [arrowhead = none ];
233 -> 87;

234 [label="SPLIT 2\nnew knowledge:\nT392 is ground\nreplacements:\nX382 -> T392", fontsize=14, style = filled, fillcolor = violet];
86 -> 234 [arrowhead = none ];
234 -> 88;

235 [label="INSTANCE with matching:\nT1 -> T385\nT2 -> X382", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 235 [arrowhead = none , style=dashed];
235 -> 1[style=dashed];

236 [label="INSTANCE with matching:\nT115 -> T386\nX104 -> X383\nT121 -> T392\nT117 -> T388", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 236 [arrowhead = none , style=dashed];
236 -> 22[style=dashed];

237 [label="ONLY EVAL with clause\ns2(plus(X413, X414), X415) :- ','(isNat(X413), ','(isNat(X414), add(X413, X414, X415))).\nand substitution\nT301 -> T415\nX413 -> T415\nT300 -> T416\nX414 -> T416\nT303 -> T418\nX415 -> T418\nT417 -> T418", fontsize=14, style = filled, fillcolor = white];
89 -> 237 [arrowhead = none ];
237 -> 91;

238 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 238 [arrowhead = none ];
238 -> 112;

239 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
91 -> 239 [arrowhead = none ];
239 -> 92;

240 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
91 -> 240 [arrowhead = none ];
240 -> 93;

241 [label="INSTANCE with matching:\nT156 -> T415", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 241 [arrowhead = none , style=dashed];
241 -> 33[style=dashed];

242 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
93 -> 242 [arrowhead = none ];
242 -> 94;

243 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
93 -> 243 [arrowhead = none ];
243 -> 95;

244 [label="INSTANCE with matching:\nT156 -> T416", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 244 [arrowhead = none , style=dashed];
244 -> 33[style=dashed];

245 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
95 -> 245 [arrowhead = none ];
245 -> 96;

246 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 246 [arrowhead = none ];
246 -> 97;

247 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 247 [arrowhead = none ];
247 -> 98;

248 [label="EVAL with clause\nadd(s(X435), X436, s(X437)) :- add(X435, X436, X437).\nand substitution\nX435 -> T438\nT415 -> s(T438)\nT416 -> T439\nX436 -> T439\nX437 -> T441\nT418 -> s(T441)\nT440 -> T441", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 248 [arrowhead = none ];
248 -> 99;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 249 [arrowhead = none ];
249 -> 100;

250 [label="EVAL with clause\nadd(0, X471, X471).\nand substitution\nT415 -> 0\nT416 -> T469\nX471 -> T469\nT418 -> T469", fontsize=14, style = filled, fillcolor = lightblue];
98 -> 250 [arrowhead = none ];
250 -> 109;

251 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
98 -> 251 [arrowhead = none ];
251 -> 110;

252 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 252 [arrowhead = none ];
252 -> 101;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 253 [arrowhead = none ];
253 -> 102;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 254 [arrowhead = none ];
254 -> 103;

255 [label="EVAL with clause\nadd(s(X456), X457, s(X458)) :- add(X456, X457, X458).\nand substitution\nX456 -> T457\nT438 -> s(T457)\nT439 -> T458\nX457 -> T458\nX458 -> T460\nT441 -> s(T460)\nT459 -> T460", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 255 [arrowhead = none ];
255 -> 104;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 256 [arrowhead = none ];
256 -> 105;

257 [label="EVAL with clause\nadd(0, X467, X467).\nand substitution\nT438 -> 0\nT439 -> T466\nX467 -> T466\nT441 -> T466", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 257 [arrowhead = none ];
257 -> 106;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 258 [arrowhead = none ];
258 -> 107;

259 [label="INSTANCE with matching:\nT438 -> T457\nT439 -> T458\nT441 -> T460", fontsize=14, style = filled, fillcolor = lightgrey];
104 -> 259 [arrowhead = none , style=dashed];
259 -> 99[style=dashed];

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 260 [arrowhead = none ];
260 -> 108;

261 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 261 [arrowhead = none ];
261 -> 111;

262 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 262 [arrowhead = none ];
262 -> 113;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
112 -> 263 [arrowhead = none ];
263 -> 114;

264 [label="EVAL with clause\ns2(plus(X477, 0), X477).\nand substitution\nT300 -> T474\nX477 -> T474\nT301 -> 0\nT2 -> T474", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 264 [arrowhead = none ];
264 -> 115;

265 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 265 [arrowhead = none ];
265 -> 116;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 266 [arrowhead = none ];
266 -> 118;

267 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 267 [arrowhead = none ];
267 -> 119;

268 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 268 [arrowhead = none ];
268 -> 117;

269 [label="ONLY EVAL with clause\ns2(plus(X505, X506), X507) :- ','(s2(X505, X508), ','(s2(X506, X509), s2(plus(X508, X509), X507))).\nand substitution\nT300 -> T495\nX505 -> T495\nT301 -> T496\nX506 -> T496\nT2 -> T498\nX507 -> T498\nT497 -> T498", fontsize=14, style = filled, fillcolor = white];
118 -> 269 [arrowhead = none ];
269 -> 120;

270 [label="ONLY EVAL with clause\ns2(plus(X532, X533), X534) :- ','(isNat(X532), ','(isNat(X533), add(X532, X533, X534))).\nand substitution\nT300 -> T517\nX532 -> T517\nT301 -> T518\nX533 -> T518\nT2 -> T520\nX534 -> T520\nT519 -> T520", fontsize=14, style = filled, fillcolor = white];
119 -> 270 [arrowhead = none ];
270 -> 123;

271 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
120 -> 271 [arrowhead = none ];
271 -> 121;

272 [label="SPLIT 2\nnew knowledge:\nT502 is ground\nreplacements:\nX508 -> T502", fontsize=14, style = filled, fillcolor = violet];
120 -> 272 [arrowhead = none ];
272 -> 122;

273 [label="INSTANCE with matching:\nT1 -> T495\nT2 -> X508", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 273 [arrowhead = none , style=dashed];
273 -> 1[style=dashed];

274 [label="INSTANCE with matching:\nT115 -> T496\nX104 -> X509\nT121 -> T502\nT117 -> T498", fontsize=14, style = filled, fillcolor = lightgrey];
122 -> 274 [arrowhead = none , style=dashed];
274 -> 22[style=dashed];

275 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
123 -> 275 [arrowhead = none ];
275 -> 124;

276 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
123 -> 276 [arrowhead = none ];
276 -> 125;

277 [label="INSTANCE with matching:\nT156 -> T517", fontsize=14, style = filled, fillcolor = lightgrey];
124 -> 277 [arrowhead = none , style=dashed];
277 -> 33[style=dashed];

278 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
125 -> 278 [arrowhead = none ];
278 -> 126;

279 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
125 -> 279 [arrowhead = none ];
279 -> 127;

280 [label="INSTANCE with matching:\nT156 -> T518", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 280 [arrowhead = none , style=dashed];
280 -> 33[style=dashed];

281 [label="INSTANCE with matching:\nT438 -> T517\nT439 -> T518\nT441 -> T520", fontsize=14, style = filled, fillcolor = lightgrey];
127 -> 281 [arrowhead = none , style=dashed];
281 -> 99[style=dashed];

282 [label="BACKTRACK\nfor clause: s2(plus(X, Y), Z) :- ','(s2(X, A), ','(s2(Y, B), s2(plus(A, B), Z)))\nwith clash: (s2(T1, T2), s2(plus(X286, X287), X288))", fontsize=14, style = filled, fillcolor = white];
128 -> 282 [arrowhead = none ];
282 -> 129;

283 [label="BACKTRACK\nfor clause: s2(plus(A, B), C) :- ','(isNat(A), ','(isNat(B), add(A, B, C)))\nwith clash: (s2(T1, T2), s2(plus(X286, X287), X288))", fontsize=14, style = filled, fillcolor = white];
129 -> 283 [arrowhead = none ];
283 -> 130;

}

(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_A → ISNAT33_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_AA → S21_IN_AA
S21_IN_AA → U10_AA(s21_in_aa)
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_IN_AA
S21_IN_AA → U14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → U3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AA → U20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U10_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))

(29) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U14_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))

(31) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U10_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U10_AA(U27_aa(isNat33_out_g))

(33) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U14_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U14_AA(U27_aa(isNat33_out_g))

(35) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AAAA → U3_AAAA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))

(37) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AAAA → U3_AAAA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g))

(39) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U20_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))

(41) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U20_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U20_AA(U27_aa(isNat33_out_g))

(43) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U30_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))

(45) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U30_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U30_AA(U27_aa(isNat33_out_g))

(47) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g))
S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_out_g))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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_A → ISNAT33_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_AA → S21_IN_AA
S21_IN_AA → U10_AA(s21_in_aa)
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_IN_AA
S21_IN_AA → U14_AA(s21_in_aa)
U14_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → U3_AAAA(s21_in_aa)
U3_AAAA(s21_out_aa) → S21_IN_AA
S21_IN_AA → U20_AA(s21_in_aa)
U20_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U10_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))

(96) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U10_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U10_AA(U27_aa(isNat33_out_g(0)))

(98) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U14_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))

(100) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U14_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U14_AA(U27_aa(isNat33_out_g(0)))

(102) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AAAA → U3_AAAA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_in_g(0)))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))

(104) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AAAA → U3_AAAA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g(0)))

(106) Obligation:

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

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

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U20_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))

(108) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U20_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U20_AA(U27_aa(isNat33_out_g(0)))

(110) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U30_AA(s21_in_aa)
U30_AA(s21_out_aa) → P22_IN_AAAA
S21_IN_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U30_AA(s21_in_aa) at position [0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))

(112) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_in_g(0)))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_AA → U30_AA(U27_aa(isNat33_in_g(0))) at position [0,0] we obtained the following new rules [LPAR04]:

S21_IN_AA → U30_AA(U27_aa(isNat33_out_g(0)))

(114) Obligation:

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

S21_IN_AA → S21_IN_AA
U10_AA(s21_out_aa) → P22_IN_AAAA
P22_IN_AAAA → S21_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_AA → U10_AA(U7_aa(s21_in_aa))
S21_IN_AA → U10_AA(U8_aa(s21_in_aa))
S21_IN_AA → U10_AA(s21_out_aa)
S21_IN_AA → U10_AA(U9_aa(s21_in_aa))
S21_IN_AA → U10_AA(U10_aa(s21_in_aa))
S21_IN_AA → U10_AA(U12_aa(s21_in_aa))
S21_IN_AA → U10_AA(U13_aa(s21_in_aa))
S21_IN_AA → U10_AA(U14_aa(s21_in_aa))
S21_IN_AA → U10_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U17_aa(s21_in_aa))
S21_IN_AA → U10_AA(U18_aa(s21_in_aa))
S21_IN_AA → U10_AA(U19_aa(s21_in_aa))
S21_IN_AA → U10_AA(U20_aa(s21_in_aa))
S21_IN_AA → U10_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U29_aa(s21_in_aa))
S21_IN_AA → U10_AA(U30_aa(s21_in_aa))
S21_IN_AA → U10_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U10_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U14_AA(U7_aa(s21_in_aa))
S21_IN_AA → U14_AA(U8_aa(s21_in_aa))
S21_IN_AA → U14_AA(s21_out_aa)
S21_IN_AA → U14_AA(U9_aa(s21_in_aa))
S21_IN_AA → U14_AA(U10_aa(s21_in_aa))
S21_IN_AA → U14_AA(U12_aa(s21_in_aa))
S21_IN_AA → U14_AA(U13_aa(s21_in_aa))
S21_IN_AA → U14_AA(U14_aa(s21_in_aa))
S21_IN_AA → U14_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U17_aa(s21_in_aa))
S21_IN_AA → U14_AA(U18_aa(s21_in_aa))
S21_IN_AA → U14_AA(U19_aa(s21_in_aa))
S21_IN_AA → U14_AA(U20_aa(s21_in_aa))
S21_IN_AA → U14_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U29_aa(s21_in_aa))
S21_IN_AA → U14_AA(U30_aa(s21_in_aa))
S21_IN_AA → U14_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U14_AA(U27_aa(isNat33_out_g(0)))
P22_IN_AAAA → U3_AAAA(U7_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U8_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(s21_out_aa)
P22_IN_AAAA → U3_AAAA(U9_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U10_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U12_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U13_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U14_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U16_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U17_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U18_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U19_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U20_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U22_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U24_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U29_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U30_aa(s21_in_aa))
P22_IN_AAAA → U3_AAAA(U32_aa(isNat33_in_a))
P22_IN_AAAA → U3_AAAA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U20_AA(U7_aa(s21_in_aa))
S21_IN_AA → U20_AA(U8_aa(s21_in_aa))
S21_IN_AA → U20_AA(s21_out_aa)
S21_IN_AA → U20_AA(U9_aa(s21_in_aa))
S21_IN_AA → U20_AA(U10_aa(s21_in_aa))
S21_IN_AA → U20_AA(U12_aa(s21_in_aa))
S21_IN_AA → U20_AA(U13_aa(s21_in_aa))
S21_IN_AA → U20_AA(U14_aa(s21_in_aa))
S21_IN_AA → U20_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U17_aa(s21_in_aa))
S21_IN_AA → U20_AA(U18_aa(s21_in_aa))
S21_IN_AA → U20_AA(U19_aa(s21_in_aa))
S21_IN_AA → U20_AA(U20_aa(s21_in_aa))
S21_IN_AA → U20_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U29_aa(s21_in_aa))
S21_IN_AA → U20_AA(U30_aa(s21_in_aa))
S21_IN_AA → U20_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U20_AA(U27_aa(isNat33_out_g(0)))
S21_IN_AA → U30_AA(U7_aa(s21_in_aa))
S21_IN_AA → U30_AA(U8_aa(s21_in_aa))
S21_IN_AA → U30_AA(s21_out_aa)
S21_IN_AA → U30_AA(U9_aa(s21_in_aa))
S21_IN_AA → U30_AA(U10_aa(s21_in_aa))
S21_IN_AA → U30_AA(U12_aa(s21_in_aa))
S21_IN_AA → U30_AA(U13_aa(s21_in_aa))
S21_IN_AA → U30_AA(U14_aa(s21_in_aa))
S21_IN_AA → U30_AA(U16_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U17_aa(s21_in_aa))
S21_IN_AA → U30_AA(U18_aa(s21_in_aa))
S21_IN_AA → U30_AA(U19_aa(s21_in_aa))
S21_IN_AA → U30_AA(U20_aa(s21_in_aa))
S21_IN_AA → U30_AA(U22_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U24_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U29_aa(s21_in_aa))
S21_IN_AA → U30_AA(U30_aa(s21_in_aa))
S21_IN_AA → U30_AA(U32_aa(isNat33_in_a))
S21_IN_AA → U30_AA(U27_aa(isNat33_out_g(0)))

The TRS R consists of the following rules:

s21_in_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
s21_in_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_aa(isNat33_in_a)
s21_in_aa → U24_aa(isNat33_in_a)
s21_in_aa → U27_aa(isNat33_in_g(0))
s21_in_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aaaa → U2_aaaa(s21_in_aa)
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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_aa → U7_aa(s21_in_aa)
s21_in_aa → U8_aa(s21_in_aa)
s21_in_aa → s21_out_aa
s21_in_aa → U9_aa(s21_in_aa)
s21_in_aa → U10_aa(s21_in_aa)
s21_in_aa → U12_aa(s21_in_aa)
s21_in_aa → U13_aa(s21_in_aa)
s21_in_aa → U14_aa(s21_in_aa)
s21_in_aa → U16_aa(isNat33_in_a)
isNat33_in_a → U1_a(isNat33_in_a)
isNat33_in_a → isNat33_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_aa → U17_aa(s21_in_aa)
s21_in_aa → U18_aa(s21_in_aa)
s21_in_aa → U19_aa(s21_in_aa)
s21_in_aa → U20_aa(s21_in_aa)
s21_in_aa → U22_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_aa → U24_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_aa → U27_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_aa → U29_aa(s21_in_aa)
s21_in_aa → U30_aa(s21_in_aa)
s21_in_aa → U32_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_aaaa → U2_aaaa(s21_in_aa)
U2_aaaa(s21_out_aa) → p22_out_aaaa
p22_in_aaaa → U3_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