Left Termination proof of /tmp/tmpxWrILn/pl4.5.2.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇐)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇐)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇔)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔)

        ↳29 QDP

        ↳30 UsableRulesProof (⇔)

        ↳31 QDP

        ↳32 QReductionProof (⇔)

        ↳33 QDP

        ↳34 MRRProof (⇔)

        ↳35 QDP

        ↳36 NonTerminationProof (⇔)

        ↳37 NO

(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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:

Triples:

isNat33(s(T165)) :- isNat33(T165).
p22(T115, X104, T121, T117) :- s21(T115, X104).
p22(T115, T128, T121, T117) :- ','(s2c1(T115, T128), s21(plus(T121, T128), T117)).
add60(s(T282), T283, T284, s(T286)) :- add60(T282, T283, T284, T286).
add99(s(T457), T458, s(T460)) :- add99(T457, T458, T460).
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(T113, plus(T114, T115)), T117) :- s21(plus(T113, T114), X103).
s21(plus(T113, plus(T114, T115)), T117) :- ','(s2c1(plus(T113, T114), T121), p22(T115, X104, T121, T117)).
s21(plus(T154, plus(T155, T156)), T158) :- isNat28(T154, T155).
s21(plus(T154, plus(T155, T156)), T158) :- ','(isNatc28(T154, T155), isNat33(T156)).
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) :- ','(s2c1(T230, T238), p22(plus(T231, T232), X225, T238, T234)).
s21(plus(T255, plus(T256, T257)), T259) :- isNat33(T255).
s21(plus(T255, plus(T256, T257)), T259) :- ','(isNatc33(T255), isNat28(T256, T257)).
s21(plus(T255, plus(T256, T257)), T259) :- ','(isNatc33(T255), ','(isNatc28(T256, T257), add60(T255, T256, T257, T259))).
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(T386, T385), T388) :- s21(T385, X382).
s21(plus(T386, T385), T388) :- ','(s2c1(T385, T392), p22(T386, X383, T392, T388)).
s21(plus(T416, T415), T418) :- isNat33(T415).
s21(plus(T416, T415), T418) :- ','(isNatc33(T415), isNat33(T416)).
s21(plus(T439, s(T438)), s(T441)) :- ','(isNatc33(s(T438)), ','(isNatc33(T439), add99(T438, T439, T441))).
s21(plus(T495, T496), T498) :- s21(T495, X508).
s21(plus(T495, T496), T498) :- ','(s2c1(T495, T502), p22(T496, X509, T502, T498)).
s21(plus(T517, T518), T520) :- isNat33(T517).
s21(plus(T517, T518), T520) :- ','(isNatc33(T517), isNat33(T518)).
s21(plus(T517, T518), T520) :- ','(isNatc33(T517), ','(isNatc33(T518), add99(T517, T518, T520))).

Clauses:

s2c1(plus(T32, plus(T33, plus(T34, T35))), T37) :- s2c1(plus(plus(plus(T32, T33), T34), T35), T37).
s2c1(plus(T70, plus(T71, T72)), T74) :- s2c1(plus(T72, plus(T70, T71)), T74).
s2c1(plus(T87, plus(T88, 0)), plus(T87, T88)).
s2c1(plus(T113, plus(T114, T115)), T117) :- ','(s2c1(plus(T113, T114), T121), qc22(T115, X104, T121, T117)).
s2c1(plus(T195, plus(T196, T197)), T199) :- s2c1(plus(plus(T196, T197), T195), T199).
s2c1(plus(T230, plus(T231, T232)), T234) :- ','(s2c1(T230, T238), qc22(plus(T231, T232), X225, T238, T234)).
s2c1(plus(T255, plus(T256, T257)), T259) :- ','(isNatc33(T255), ','(isNatc28(T256, T257), addc60(T255, T256, T257, T259))).
s2c1(plus(plus(T321, T322), T320), T324) :- s2c1(plus(plus(T320, T321), T322), T324).
s2c1(plus(T353, T352), T355) :- s2c1(plus(T353, T352), T355).
s2c1(plus(0, T364), T364).
s2c1(plus(T386, T385), T388) :- ','(s2c1(T385, T392), qc22(T386, X383, T392, T388)).
s2c1(plus(T439, s(T438)), s(T441)) :- ','(isNatc33(s(T438)), ','(isNatc33(T439), addc99(T438, T439, T441))).
s2c1(plus(T469, 0), T469) :- ','(isNatc33(0), isNatc33(T469)).
s2c1(plus(T474, 0), T474).
s2c1(plus(T495, T496), T498) :- ','(s2c1(T495, T502), qc22(T496, X509, T502, T498)).
s2c1(plus(T517, T518), T520) :- ','(isNatc33(T517), ','(isNatc33(T518), addc99(T517, T518, T520))).
isNatc33(s(T165)) :- isNatc33(T165).
isNatc33(0).
qc22(T115, T128, T121, T117) :- ','(s2c1(T115, T128), s2c1(plus(T121, T128), T117)).
addc60(s(T282), T283, T284, s(T286)) :- addc60(T282, T283, T284, T286).
addc60(0, T295, T296, plus(T295, T296)).
addc99(s(T457), T458, s(T460)) :- addc99(T457, T458, T460).
addc99(0, T466, T466).

Afs:

s21(x1, x2) = s21(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

isNat33(s(T165)) :- isNat33(T165).
p22(T115, X104, T121, T117) :- s21(T115, X104).
p22(T115, T128, T121, T117) :- ','(s2c1(T115, T128), s21(plus(T121, T128), T117)).
add60(s(T282), T283, T284, s(T286)) :- add60(T282, T283, T284, T286).
add99(s(T457), T458, s(T460)) :- add99(T457, T458, T460).
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(T113, plus(T114, T115)), T117) :- s21(plus(T113, T114), X103).
s21(plus(T113, plus(T114, T115)), T117) :- ','(s2c1(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) :- ','(s2c1(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(T386, T385), T388) :- s21(T385, X382).
s21(plus(T386, T385), T388) :- ','(s2c1(T385, T392), p22(T386, X383, T392, T388)).
s21(plus(T416, T415), T418) :- isNat33(T415).
s21(plus(T416, T415), T418) :- ','(isNatc33(T415), isNat33(T416)).
s21(plus(T439, s(T438)), s(T441)) :- ','(isNatc33(s(T438)), ','(isNatc33(T439), add99(T438, T439, T441))).
s21(plus(T495, T496), T498) :- s21(T495, X508).
s21(plus(T495, T496), T498) :- ','(s2c1(T495, T502), p22(T496, X509, T502, T498)).
s21(plus(T517, T518), T520) :- isNat33(T517).
s21(plus(T517, T518), T520) :- ','(isNatc33(T517), isNat33(T518)).
s21(plus(T517, T518), T520) :- ','(isNatc33(T517), ','(isNatc33(T518), add99(T517, T518, T520))).

Clauses:

s2c1(plus(T32, plus(T33, plus(T34, T35))), T37) :- s2c1(plus(plus(plus(T32, T33), T34), T35), T37).
s2c1(plus(T70, plus(T71, T72)), T74) :- s2c1(plus(T72, plus(T70, T71)), T74).
s2c1(plus(T87, plus(T88, 0)), plus(T87, T88)).
s2c1(plus(T113, plus(T114, T115)), T117) :- ','(s2c1(plus(T113, T114), T121), qc22(T115, X104, T121, T117)).
s2c1(plus(T195, plus(T196, T197)), T199) :- s2c1(plus(plus(T196, T197), T195), T199).
s2c1(plus(T230, plus(T231, T232)), T234) :- ','(s2c1(T230, T238), qc22(plus(T231, T232), X225, T238, T234)).
s2c1(plus(plus(T321, T322), T320), T324) :- s2c1(plus(plus(T320, T321), T322), T324).
s2c1(plus(T353, T352), T355) :- s2c1(plus(T353, T352), T355).
s2c1(plus(0, T364), T364).
s2c1(plus(T386, T385), T388) :- ','(s2c1(T385, T392), qc22(T386, X383, T392, T388)).
s2c1(plus(T439, s(T438)), s(T441)) :- ','(isNatc33(s(T438)), ','(isNatc33(T439), addc99(T438, T439, T441))).
s2c1(plus(T469, 0), T469) :- ','(isNatc33(0), isNatc33(T469)).
s2c1(plus(T474, 0), T474).
s2c1(plus(T495, T496), T498) :- ','(s2c1(T495, T502), qc22(T496, X509, T502, T498)).
s2c1(plus(T517, T518), T520) :- ','(isNatc33(T517), ','(isNatc33(T518), addc99(T517, T518, T520))).
isNatc33(s(T165)) :- isNatc33(T165).
isNatc33(0).
qc22(T115, T128, T121, T117) :- ','(s2c1(T115, T128), s2c1(plus(T121, T128), T117)).
addc60(s(T282), T283, T284, s(T286)) :- addc60(T282, T283, T284, T286).
addc60(0, T295, T296, plus(T295, T296)).
addc99(s(T457), T458, s(T460)) :- addc99(T457, T458, T460).
addc99(0, T466, T466).

Afs:

s21(x1, x2) = s21(x1)

(5) TriplesToPiDPProof (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)
s2c1_in: (b,f)
isNatc33_in: (b)
addc99_in: (b,b,f)
qc22_in: (b,f,b,f)
p22_in: (b,f,b,f)
isNat33_in: (b)
add99_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
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, s2c1_in_ga(plus(T113, T114), T121))
U10_GA(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaga(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → P22_IN_GAGA(T115, X104, T121, T117)
P22_IN_GAGA(T115, X104, T121, T117) → U2_GAGA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAGA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
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, s2c1_in_ga(T230, T238))
U14_GA(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaga(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → P22_IN_GAGA(plus(T231, T232), X225, T238, T234)
P22_IN_GAGA(T115, T128, T121, T117) → U3_GAGA(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U3_GAGA(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U4_GAGA(T115, T128, T121, T117, s21_in_ga(plus(T121, T128), T117))
U3_GAGA(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → S21_IN_GA(plus(T121, T128), T117)
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, s2c1_in_ga(T385, T392))
U20_GA(T386, T385, T388, s2c1_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaga(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s2c1_out_ga(T385, T392)) → P22_IN_GAGA(T386, X383, T392, T388)
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)
S21_IN_GA(plus(T416, T415), T418) → U23_GA(T416, T415, T418, isNatc33_in_g(T415))
U23_GA(T416, T415, T418, isNatc33_out_g(T415)) → U24_GA(T416, T415, T418, isNat33_in_g(T416))
U23_GA(T416, T415, T418, isNatc33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U25_GA(T439, T438, T441, isNatc33_in_g(s(T438)))
U25_GA(T439, T438, T441, isNatc33_out_g(s(T438))) → U26_GA(T439, T438, T441, isNatc33_in_g(T439))
U26_GA(T439, T438, T441, isNatc33_out_g(T439)) → U27_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_GA(T439, T438, T441, isNatc33_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(T495, T496), T498) → U28_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) → U29_GA(T495, T496, T498, s2c1_in_ga(T495, T502))
U29_GA(T495, T496, T498, s2c1_out_ga(T495, T502)) → U30_GA(T495, T496, T498, p22_in_gaga(T496, X509, T502, T498))
U29_GA(T495, T496, T498, s2c1_out_ga(T495, T502)) → P22_IN_GAGA(T496, X509, T502, T498)
S21_IN_GA(plus(T517, T518), T520) → U31_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNatc33_in_g(T517))
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → ISNAT33_IN_G(T518)
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → U34_GA(T517, T518, T520, isNatc33_in_g(T518))
U34_GA(T517, T518, T520, isNatc33_out_g(T518)) → U35_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_GA(T517, T518, T520, isNatc33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U37_ga(T32, T33, T34, T35, T37, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s2c1_in_ga(plus(T70, plus(T71, T72)), T74) → U38_ga(T70, T71, T72, T74, s2c1_in_ga(plus(T72, plus(T70, T71)), T74))
s2c1_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115)), T117) → U39_ga(T113, T114, T115, T117, s2c1_in_ga(plus(T113, T114), T121))
s2c1_in_ga(plus(T195, plus(T196, T197)), T199) → U41_ga(T195, T196, T197, T199, s2c1_in_ga(plus(plus(T196, T197), T195), T199))
s2c1_in_ga(plus(T230, plus(T231, T232)), T234) → U42_ga(T230, T231, T232, T234, s2c1_in_ga(T230, T238))
s2c1_in_ga(plus(plus(T321, T322), T320), T324) → U44_ga(T321, T322, T320, T324, s2c1_in_ga(plus(plus(T320, T321), T322), T324))
s2c1_in_ga(plus(T353, T352), T355) → U45_ga(T353, T352, T355, s2c1_in_ga(plus(T353, T352), T355))
s2c1_in_ga(plus(0, T364), T364) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385), T388) → U46_ga(T386, T385, T388, s2c1_in_ga(T385, T392))
s2c1_in_ga(plus(T439, s(T438)), s(T441)) → U48_ga(T439, T438, T441, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, T441, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, T441, isNatc33_in_g(T439))
U49_ga(T439, T438, T441, isNatc33_out_g(T439)) → U50_ga(T439, T438, T441, addc99_in_gga(T438, T439, T441))
addc99_in_gga(s(T457), T458, s(T460)) → U62_gga(T457, T458, T460, addc99_in_gga(T457, T458, T460))
addc99_in_gga(0, T466, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, T460, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, T441, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0), T469) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0), T474) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496), T498) → U53_ga(T495, T496, T498, s2c1_in_ga(T495, T502))
s2c1_in_ga(plus(T517, T518), T520) → U55_ga(T517, T518, T520, isNatc33_in_g(T517))
U55_ga(T517, T518, T520, isNatc33_out_g(T517)) → U56_ga(T517, T518, T520, isNatc33_in_g(T518))
U56_ga(T517, T518, T520, isNatc33_out_g(T518)) → U57_ga(T517, T518, T520, addc99_in_gga(T517, T518, T520))
U57_ga(T517, T518, T520, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, T498, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, T498, qc22_in_gaga(T496, X509, T502, T498))
qc22_in_gaga(T115, T128, T121, T117) → U59_gaga(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U59_gaga(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, T117, s2c1_in_ga(plus(T121, T128), T117))
U60_gaga(T115, T128, T121, T117, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, T498, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, T388, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, T388, qc22_in_gaga(T386, X383, T392, T388))
U47_ga(T386, T385, T388, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, T355, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, T324, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, T234, qc22_in_gaga(plus(T231, T232), X225, T238, T234))
U43_ga(T230, T231, T232, T234, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, T199, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, T117, qc22_in_gaga(T115, X104, T121, T117))
U40_ga(T113, T114, T115, T117, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, T74, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, T37, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_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)
s2c1_in_ga(x1, x2) = s2c1_in_ga(x1)
U37_ga(x1, x2, x3, x4, x5, x6) = U37_ga(x1, x2, x3, x4, x6)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
0 = 0
s2c1_out_ga(x1, x2) = s2c1_out_ga(x1, x2)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
U45_ga(x1, x2, x3, x4) = U45_ga(x1, x2, x4)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x2, x4)
s(x1) = s(x1)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
isNatc33_in_g(x1) = isNatc33_in_g(x1)
U58_g(x1, x2) = U58_g(x1, x2)
isNatc33_out_g(x1) = isNatc33_out_g(x1)
U49_ga(x1, x2, x3, x4) = U49_ga(x1, x2, x4)
U50_ga(x1, x2, x3, x4) = U50_ga(x1, x2, x4)
addc99_in_gga(x1, x2, x3) = addc99_in_gga(x1, x2)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
addc99_out_gga(x1, x2, x3) = addc99_out_gga(x1, x2, x3)
U51_ga(x1, x2) = U51_ga(x1, x2)
U52_ga(x1, x2) = U52_ga(x1, x2)
U53_ga(x1, x2, x3, x4) = U53_ga(x1, x2, x4)
U55_ga(x1, x2, x3, x4) = U55_ga(x1, x2, x4)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4) = U57_ga(x1, x2, x4)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x2, x4)
qc22_in_gaga(x1, x2, x3, x4) = qc22_in_gaga(x1, x3)
U59_gaga(x1, x2, x3, x4, x5) = U59_gaga(x1, x3, x5)
U60_gaga(x1, x2, x3, x4, x5) = U60_gaga(x1, x2, x3, x5)
qc22_out_gaga(x1, x2, x3, x4) = qc22_out_gaga(x1, x2, x3, x4)
U47_ga(x1, x2, x3, x4) = U47_ga(x1, x2, x4)
U43_ga(x1, x2, x3, x4, x5) = U43_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5) = U40_ga(x1, x2, x3, x5)
p22_in_gaga(x1, x2, x3, x4) = p22_in_gaga(x1, x3)
isNat33_in_g(x1) = isNat33_in_g(x1)
add99_in_gga(x1, x2, x3) = add99_in_gga(x1, x2)
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)
U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x3, x5)
P22_IN_GAGA(x1, x2, x3, x4) = P22_IN_GAGA(x1, x3)
U2_GAGA(x1, x2, x3, x4, x5) = U2_GAGA(x1, 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)
U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x2, x3, x5)
U3_GAGA(x1, x2, x3, x4, x5) = U3_GAGA(x1, x3, x5)
U4_GAGA(x1, x2, x3, x4, x5) = U4_GAGA(x1, 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)
U21_GA(x1, x2, x3, x4) = U21_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)
U27_GA(x1, x2, x3, x4) = U27_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)
U28_GA(x1, x2, x3, x4) = U28_GA(x1, x2, x4)
U29_GA(x1, x2, x3, x4) = U29_GA(x1, x2, x4)
U30_GA(x1, x2, x3, x4) = U30_GA(x1, x2, x4)
U31_GA(x1, x2, x3, x4) = U31_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)
U35_GA(x1, x2, x3, x4) = U35_GA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(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, s2c1_in_ga(plus(T113, T114), T121))
U10_GA(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U11_GA(T113, T114, T115, T117, p22_in_gaga(T115, X104, T121, T117))
U10_GA(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → P22_IN_GAGA(T115, X104, T121, T117)
P22_IN_GAGA(T115, X104, T121, T117) → U2_GAGA(T115, X104, T121, T117, s21_in_ga(T115, X104))
P22_IN_GAGA(T115, X104, T121, T117) → S21_IN_GA(T115, X104)
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, s2c1_in_ga(T230, T238))
U14_GA(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U15_GA(T230, T231, T232, T234, p22_in_gaga(plus(T231, T232), X225, T238, T234))
U14_GA(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → P22_IN_GAGA(plus(T231, T232), X225, T238, T234)
P22_IN_GAGA(T115, T128, T121, T117) → U3_GAGA(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U3_GAGA(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U4_GAGA(T115, T128, T121, T117, s21_in_ga(plus(T121, T128), T117))
U3_GAGA(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → S21_IN_GA(plus(T121, T128), T117)
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, s2c1_in_ga(T385, T392))
U20_GA(T386, T385, T388, s2c1_out_ga(T385, T392)) → U21_GA(T386, T385, T388, p22_in_gaga(T386, X383, T392, T388))
U20_GA(T386, T385, T388, s2c1_out_ga(T385, T392)) → P22_IN_GAGA(T386, X383, T392, T388)
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)
S21_IN_GA(plus(T416, T415), T418) → U23_GA(T416, T415, T418, isNatc33_in_g(T415))
U23_GA(T416, T415, T418, isNatc33_out_g(T415)) → U24_GA(T416, T415, T418, isNat33_in_g(T416))
U23_GA(T416, T415, T418, isNatc33_out_g(T415)) → ISNAT33_IN_G(T416)
S21_IN_GA(plus(T439, s(T438)), s(T441)) → U25_GA(T439, T438, T441, isNatc33_in_g(s(T438)))
U25_GA(T439, T438, T441, isNatc33_out_g(s(T438))) → U26_GA(T439, T438, T441, isNatc33_in_g(T439))
U26_GA(T439, T438, T441, isNatc33_out_g(T439)) → U27_GA(T439, T438, T441, add99_in_gga(T438, T439, T441))
U26_GA(T439, T438, T441, isNatc33_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(T495, T496), T498) → U28_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) → U29_GA(T495, T496, T498, s2c1_in_ga(T495, T502))
U29_GA(T495, T496, T498, s2c1_out_ga(T495, T502)) → U30_GA(T495, T496, T498, p22_in_gaga(T496, X509, T502, T498))
U29_GA(T495, T496, T498, s2c1_out_ga(T495, T502)) → P22_IN_GAGA(T496, X509, T502, T498)
S21_IN_GA(plus(T517, T518), T520) → U31_GA(T517, T518, T520, isNat33_in_g(T517))
S21_IN_GA(plus(T517, T518), T520) → ISNAT33_IN_G(T517)
S21_IN_GA(plus(T517, T518), T520) → U32_GA(T517, T518, T520, isNatc33_in_g(T517))
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → U33_GA(T517, T518, T520, isNat33_in_g(T518))
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → ISNAT33_IN_G(T518)
U32_GA(T517, T518, T520, isNatc33_out_g(T517)) → U34_GA(T517, T518, T520, isNatc33_in_g(T518))
U34_GA(T517, T518, T520, isNatc33_out_g(T518)) → U35_GA(T517, T518, T520, add99_in_gga(T517, T518, T520))
U34_GA(T517, T518, T520, isNatc33_out_g(T518)) → ADD99_IN_GGA(T517, T518, T520)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U37_ga(T32, T33, T34, T35, T37, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s2c1_in_ga(plus(T70, plus(T71, T72)), T74) → U38_ga(T70, T71, T72, T74, s2c1_in_ga(plus(T72, plus(T70, T71)), T74))
s2c1_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115)), T117) → U39_ga(T113, T114, T115, T117, s2c1_in_ga(plus(T113, T114), T121))
s2c1_in_ga(plus(T195, plus(T196, T197)), T199) → U41_ga(T195, T196, T197, T199, s2c1_in_ga(plus(plus(T196, T197), T195), T199))
s2c1_in_ga(plus(T230, plus(T231, T232)), T234) → U42_ga(T230, T231, T232, T234, s2c1_in_ga(T230, T238))
s2c1_in_ga(plus(plus(T321, T322), T320), T324) → U44_ga(T321, T322, T320, T324, s2c1_in_ga(plus(plus(T320, T321), T322), T324))
s2c1_in_ga(plus(T353, T352), T355) → U45_ga(T353, T352, T355, s2c1_in_ga(plus(T353, T352), T355))
s2c1_in_ga(plus(0, T364), T364) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385), T388) → U46_ga(T386, T385, T388, s2c1_in_ga(T385, T392))
s2c1_in_ga(plus(T439, s(T438)), s(T441)) → U48_ga(T439, T438, T441, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, T441, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, T441, isNatc33_in_g(T439))
U49_ga(T439, T438, T441, isNatc33_out_g(T439)) → U50_ga(T439, T438, T441, addc99_in_gga(T438, T439, T441))
addc99_in_gga(s(T457), T458, s(T460)) → U62_gga(T457, T458, T460, addc99_in_gga(T457, T458, T460))
addc99_in_gga(0, T466, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, T460, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, T441, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0), T469) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0), T474) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496), T498) → U53_ga(T495, T496, T498, s2c1_in_ga(T495, T502))
s2c1_in_ga(plus(T517, T518), T520) → U55_ga(T517, T518, T520, isNatc33_in_g(T517))
U55_ga(T517, T518, T520, isNatc33_out_g(T517)) → U56_ga(T517, T518, T520, isNatc33_in_g(T518))
U56_ga(T517, T518, T520, isNatc33_out_g(T518)) → U57_ga(T517, T518, T520, addc99_in_gga(T517, T518, T520))
U57_ga(T517, T518, T520, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, T498, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, T498, qc22_in_gaga(T496, X509, T502, T498))
qc22_in_gaga(T115, T128, T121, T117) → U59_gaga(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U59_gaga(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, T117, s2c1_in_ga(plus(T121, T128), T117))
U60_gaga(T115, T128, T121, T117, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, T498, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, T388, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, T388, qc22_in_gaga(T386, X383, T392, T388))
U47_ga(T386, T385, T388, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, T355, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, T324, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, T234, qc22_in_gaga(plus(T231, T232), X225, T238, T234))
U43_ga(T230, T231, T232, T234, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, T199, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, T117, qc22_in_gaga(T115, X104, T121, T117))
U40_ga(T113, T114, T115, T117, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, T74, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, T37, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 36 less nodes.

(8) Complex Obligation (AND)

(9) 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:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U37_ga(T32, T33, T34, T35, T37, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s2c1_in_ga(plus(T70, plus(T71, T72)), T74) → U38_ga(T70, T71, T72, T74, s2c1_in_ga(plus(T72, plus(T70, T71)), T74))
s2c1_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115)), T117) → U39_ga(T113, T114, T115, T117, s2c1_in_ga(plus(T113, T114), T121))
s2c1_in_ga(plus(T195, plus(T196, T197)), T199) → U41_ga(T195, T196, T197, T199, s2c1_in_ga(plus(plus(T196, T197), T195), T199))
s2c1_in_ga(plus(T230, plus(T231, T232)), T234) → U42_ga(T230, T231, T232, T234, s2c1_in_ga(T230, T238))
s2c1_in_ga(plus(plus(T321, T322), T320), T324) → U44_ga(T321, T322, T320, T324, s2c1_in_ga(plus(plus(T320, T321), T322), T324))
s2c1_in_ga(plus(T353, T352), T355) → U45_ga(T353, T352, T355, s2c1_in_ga(plus(T353, T352), T355))
s2c1_in_ga(plus(0, T364), T364) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385), T388) → U46_ga(T386, T385, T388, s2c1_in_ga(T385, T392))
s2c1_in_ga(plus(T439, s(T438)), s(T441)) → U48_ga(T439, T438, T441, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, T441, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, T441, isNatc33_in_g(T439))
U49_ga(T439, T438, T441, isNatc33_out_g(T439)) → U50_ga(T439, T438, T441, addc99_in_gga(T438, T439, T441))
addc99_in_gga(s(T457), T458, s(T460)) → U62_gga(T457, T458, T460, addc99_in_gga(T457, T458, T460))
addc99_in_gga(0, T466, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, T460, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, T441, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0), T469) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0), T474) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496), T498) → U53_ga(T495, T496, T498, s2c1_in_ga(T495, T502))
s2c1_in_ga(plus(T517, T518), T520) → U55_ga(T517, T518, T520, isNatc33_in_g(T517))
U55_ga(T517, T518, T520, isNatc33_out_g(T517)) → U56_ga(T517, T518, T520, isNatc33_in_g(T518))
U56_ga(T517, T518, T520, isNatc33_out_g(T518)) → U57_ga(T517, T518, T520, addc99_in_gga(T517, T518, T520))
U57_ga(T517, T518, T520, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, T498, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, T498, qc22_in_gaga(T496, X509, T502, T498))
qc22_in_gaga(T115, T128, T121, T117) → U59_gaga(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U59_gaga(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, T117, s2c1_in_ga(plus(T121, T128), T117))
U60_gaga(T115, T128, T121, T117, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, T498, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, T388, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, T388, qc22_in_gaga(T386, X383, T392, T388))
U47_ga(T386, T385, T388, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, T355, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, T324, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, T234, qc22_in_gaga(plus(T231, T232), X225, T238, T234))
U43_ga(T230, T231, T232, T234, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, T199, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, T117, qc22_in_gaga(T115, X104, T121, T117))
U40_ga(T113, T114, T115, T117, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, T74, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, T37, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
plus(x1, x2) = plus(x1, x2)
s2c1_in_ga(x1, x2) = s2c1_in_ga(x1)
U37_ga(x1, x2, x3, x4, x5, x6) = U37_ga(x1, x2, x3, x4, x6)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
0 = 0
s2c1_out_ga(x1, x2) = s2c1_out_ga(x1, x2)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
U45_ga(x1, x2, x3, x4) = U45_ga(x1, x2, x4)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x2, x4)
s(x1) = s(x1)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
isNatc33_in_g(x1) = isNatc33_in_g(x1)
U58_g(x1, x2) = U58_g(x1, x2)
isNatc33_out_g(x1) = isNatc33_out_g(x1)
U49_ga(x1, x2, x3, x4) = U49_ga(x1, x2, x4)
U50_ga(x1, x2, x3, x4) = U50_ga(x1, x2, x4)
addc99_in_gga(x1, x2, x3) = addc99_in_gga(x1, x2)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
addc99_out_gga(x1, x2, x3) = addc99_out_gga(x1, x2, x3)
U51_ga(x1, x2) = U51_ga(x1, x2)
U52_ga(x1, x2) = U52_ga(x1, x2)
U53_ga(x1, x2, x3, x4) = U53_ga(x1, x2, x4)
U55_ga(x1, x2, x3, x4) = U55_ga(x1, x2, x4)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4) = U57_ga(x1, x2, x4)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x2, x4)
qc22_in_gaga(x1, x2, x3, x4) = qc22_in_gaga(x1, x3)
U59_gaga(x1, x2, x3, x4, x5) = U59_gaga(x1, x3, x5)
U60_gaga(x1, x2, x3, x4, x5) = U60_gaga(x1, x2, x3, x5)
qc22_out_gaga(x1, x2, x3, x4) = qc22_out_gaga(x1, x2, x3, x4)
U47_ga(x1, x2, x3, x4) = U47_ga(x1, x2, x4)
U43_ga(x1, x2, x3, x4, x5) = U43_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5) = U40_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

(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:

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

(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:

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.

(14) 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

(15) YES

(16) 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:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U37_ga(T32, T33, T34, T35, T37, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s2c1_in_ga(plus(T70, plus(T71, T72)), T74) → U38_ga(T70, T71, T72, T74, s2c1_in_ga(plus(T72, plus(T70, T71)), T74))
s2c1_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115)), T117) → U39_ga(T113, T114, T115, T117, s2c1_in_ga(plus(T113, T114), T121))
s2c1_in_ga(plus(T195, plus(T196, T197)), T199) → U41_ga(T195, T196, T197, T199, s2c1_in_ga(plus(plus(T196, T197), T195), T199))
s2c1_in_ga(plus(T230, plus(T231, T232)), T234) → U42_ga(T230, T231, T232, T234, s2c1_in_ga(T230, T238))
s2c1_in_ga(plus(plus(T321, T322), T320), T324) → U44_ga(T321, T322, T320, T324, s2c1_in_ga(plus(plus(T320, T321), T322), T324))
s2c1_in_ga(plus(T353, T352), T355) → U45_ga(T353, T352, T355, s2c1_in_ga(plus(T353, T352), T355))
s2c1_in_ga(plus(0, T364), T364) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385), T388) → U46_ga(T386, T385, T388, s2c1_in_ga(T385, T392))
s2c1_in_ga(plus(T439, s(T438)), s(T441)) → U48_ga(T439, T438, T441, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, T441, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, T441, isNatc33_in_g(T439))
U49_ga(T439, T438, T441, isNatc33_out_g(T439)) → U50_ga(T439, T438, T441, addc99_in_gga(T438, T439, T441))
addc99_in_gga(s(T457), T458, s(T460)) → U62_gga(T457, T458, T460, addc99_in_gga(T457, T458, T460))
addc99_in_gga(0, T466, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, T460, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, T441, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0), T469) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0), T474) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496), T498) → U53_ga(T495, T496, T498, s2c1_in_ga(T495, T502))
s2c1_in_ga(plus(T517, T518), T520) → U55_ga(T517, T518, T520, isNatc33_in_g(T517))
U55_ga(T517, T518, T520, isNatc33_out_g(T517)) → U56_ga(T517, T518, T520, isNatc33_in_g(T518))
U56_ga(T517, T518, T520, isNatc33_out_g(T518)) → U57_ga(T517, T518, T520, addc99_in_gga(T517, T518, T520))
U57_ga(T517, T518, T520, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, T498, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, T498, qc22_in_gaga(T496, X509, T502, T498))
qc22_in_gaga(T115, T128, T121, T117) → U59_gaga(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U59_gaga(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, T117, s2c1_in_ga(plus(T121, T128), T117))
U60_gaga(T115, T128, T121, T117, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, T498, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, T388, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, T388, qc22_in_gaga(T386, X383, T392, T388))
U47_ga(T386, T385, T388, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, T355, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, T324, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, T234, qc22_in_gaga(plus(T231, T232), X225, T238, T234))
U43_ga(T230, T231, T232, T234, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, T199, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, T117, qc22_in_gaga(T115, X104, T121, T117))
U40_ga(T113, T114, T115, T117, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, T74, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, T37, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
plus(x1, x2) = plus(x1, x2)
s2c1_in_ga(x1, x2) = s2c1_in_ga(x1)
U37_ga(x1, x2, x3, x4, x5, x6) = U37_ga(x1, x2, x3, x4, x6)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
0 = 0
s2c1_out_ga(x1, x2) = s2c1_out_ga(x1, x2)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
U45_ga(x1, x2, x3, x4) = U45_ga(x1, x2, x4)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x2, x4)
s(x1) = s(x1)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
isNatc33_in_g(x1) = isNatc33_in_g(x1)
U58_g(x1, x2) = U58_g(x1, x2)
isNatc33_out_g(x1) = isNatc33_out_g(x1)
U49_ga(x1, x2, x3, x4) = U49_ga(x1, x2, x4)
U50_ga(x1, x2, x3, x4) = U50_ga(x1, x2, x4)
addc99_in_gga(x1, x2, x3) = addc99_in_gga(x1, x2)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
addc99_out_gga(x1, x2, x3) = addc99_out_gga(x1, x2, x3)
U51_ga(x1, x2) = U51_ga(x1, x2)
U52_ga(x1, x2) = U52_ga(x1, x2)
U53_ga(x1, x2, x3, x4) = U53_ga(x1, x2, x4)
U55_ga(x1, x2, x3, x4) = U55_ga(x1, x2, x4)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4) = U57_ga(x1, x2, x4)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x2, x4)
qc22_in_gaga(x1, x2, x3, x4) = qc22_in_gaga(x1, x3)
U59_gaga(x1, x2, x3, x4, x5) = U59_gaga(x1, x3, x5)
U60_gaga(x1, x2, x3, x4, x5) = U60_gaga(x1, x2, x3, x5)
qc22_out_gaga(x1, x2, x3, x4) = qc22_out_gaga(x1, x2, x3, x4)
U47_ga(x1, x2, x3, x4) = U47_ga(x1, x2, x4)
U43_ga(x1, x2, x3, x4, x5) = U43_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5) = U40_ga(x1, x2, x3, x5)
ISNAT33_IN_G(x1) = ISNAT33_IN_G(x1)

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:

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

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

(19) PiDPToQDPProof (EQUIVALENT 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:

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

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:

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

(22) YES

(23) 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, s2c1_in_ga(plus(T113, T114), T121))
U10_GA(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → P22_IN_GAGA(T115, X104, T121, T117)
P22_IN_GAGA(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, s2c1_in_ga(T230, T238))
U14_GA(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → P22_IN_GAGA(plus(T231, T232), X225, T238, T234)
P22_IN_GAGA(T115, T128, T121, T117) → U3_GAGA(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U3_GAGA(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → S21_IN_GA(plus(T121, T128), T117)
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, s2c1_in_ga(T385, T392))
U20_GA(T386, T385, T388, s2c1_out_ga(T385, T392)) → P22_IN_GAGA(T386, X383, T392, T388)
S21_IN_GA(plus(T495, T496), T498) → S21_IN_GA(T495, X508)
S21_IN_GA(plus(T495, T496), T498) → U29_GA(T495, T496, T498, s2c1_in_ga(T495, T502))
U29_GA(T495, T496, T498, s2c1_out_ga(T495, T502)) → P22_IN_GAGA(T496, X509, T502, T498)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35))), T37) → U37_ga(T32, T33, T34, T35, T37, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35), T37))
s2c1_in_ga(plus(T70, plus(T71, T72)), T74) → U38_ga(T70, T71, T72, T74, s2c1_in_ga(plus(T72, plus(T70, T71)), T74))
s2c1_in_ga(plus(T87, plus(T88, 0)), plus(T87, T88)) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115)), T117) → U39_ga(T113, T114, T115, T117, s2c1_in_ga(plus(T113, T114), T121))
s2c1_in_ga(plus(T195, plus(T196, T197)), T199) → U41_ga(T195, T196, T197, T199, s2c1_in_ga(plus(plus(T196, T197), T195), T199))
s2c1_in_ga(plus(T230, plus(T231, T232)), T234) → U42_ga(T230, T231, T232, T234, s2c1_in_ga(T230, T238))
s2c1_in_ga(plus(plus(T321, T322), T320), T324) → U44_ga(T321, T322, T320, T324, s2c1_in_ga(plus(plus(T320, T321), T322), T324))
s2c1_in_ga(plus(T353, T352), T355) → U45_ga(T353, T352, T355, s2c1_in_ga(plus(T353, T352), T355))
s2c1_in_ga(plus(0, T364), T364) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385), T388) → U46_ga(T386, T385, T388, s2c1_in_ga(T385, T392))
s2c1_in_ga(plus(T439, s(T438)), s(T441)) → U48_ga(T439, T438, T441, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, T441, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, T441, isNatc33_in_g(T439))
U49_ga(T439, T438, T441, isNatc33_out_g(T439)) → U50_ga(T439, T438, T441, addc99_in_gga(T438, T439, T441))
addc99_in_gga(s(T457), T458, s(T460)) → U62_gga(T457, T458, T460, addc99_in_gga(T457, T458, T460))
addc99_in_gga(0, T466, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, T460, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, T441, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0), T469) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0), T474) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496), T498) → U53_ga(T495, T496, T498, s2c1_in_ga(T495, T502))
s2c1_in_ga(plus(T517, T518), T520) → U55_ga(T517, T518, T520, isNatc33_in_g(T517))
U55_ga(T517, T518, T520, isNatc33_out_g(T517)) → U56_ga(T517, T518, T520, isNatc33_in_g(T518))
U56_ga(T517, T518, T520, isNatc33_out_g(T518)) → U57_ga(T517, T518, T520, addc99_in_gga(T517, T518, T520))
U57_ga(T517, T518, T520, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, T498, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, T498, qc22_in_gaga(T496, X509, T502, T498))
qc22_in_gaga(T115, T128, T121, T117) → U59_gaga(T115, T128, T121, T117, s2c1_in_ga(T115, T128))
U59_gaga(T115, T128, T121, T117, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, T117, s2c1_in_ga(plus(T121, T128), T117))
U60_gaga(T115, T128, T121, T117, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, T498, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, T388, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, T388, qc22_in_gaga(T386, X383, T392, T388))
U47_ga(T386, T385, T388, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, T355, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, T324, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, T234, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, T234, qc22_in_gaga(plus(T231, T232), X225, T238, T234))
U43_ga(T230, T231, T232, T234, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, T199, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, T117, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, T117, qc22_in_gaga(T115, X104, T121, T117))
U40_ga(T113, T114, T115, T117, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, T74, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, T37, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The argument filtering Pi contains the following mapping:
plus(x1, x2) = plus(x1, x2)
s2c1_in_ga(x1, x2) = s2c1_in_ga(x1)
U37_ga(x1, x2, x3, x4, x5, x6) = U37_ga(x1, x2, x3, x4, x6)
U38_ga(x1, x2, x3, x4, x5) = U38_ga(x1, x2, x3, x5)
0 = 0
s2c1_out_ga(x1, x2) = s2c1_out_ga(x1, x2)
U39_ga(x1, x2, x3, x4, x5) = U39_ga(x1, x2, x3, x5)
U41_ga(x1, x2, x3, x4, x5) = U41_ga(x1, x2, x3, x5)
U42_ga(x1, x2, x3, x4, x5) = U42_ga(x1, x2, x3, x5)
U44_ga(x1, x2, x3, x4, x5) = U44_ga(x1, x2, x3, x5)
U45_ga(x1, x2, x3, x4) = U45_ga(x1, x2, x4)
U46_ga(x1, x2, x3, x4) = U46_ga(x1, x2, x4)
s(x1) = s(x1)
U48_ga(x1, x2, x3, x4) = U48_ga(x1, x2, x4)
isNatc33_in_g(x1) = isNatc33_in_g(x1)
U58_g(x1, x2) = U58_g(x1, x2)
isNatc33_out_g(x1) = isNatc33_out_g(x1)
U49_ga(x1, x2, x3, x4) = U49_ga(x1, x2, x4)
U50_ga(x1, x2, x3, x4) = U50_ga(x1, x2, x4)
addc99_in_gga(x1, x2, x3) = addc99_in_gga(x1, x2)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
addc99_out_gga(x1, x2, x3) = addc99_out_gga(x1, x2, x3)
U51_ga(x1, x2) = U51_ga(x1, x2)
U52_ga(x1, x2) = U52_ga(x1, x2)
U53_ga(x1, x2, x3, x4) = U53_ga(x1, x2, x4)
U55_ga(x1, x2, x3, x4) = U55_ga(x1, x2, x4)
U56_ga(x1, x2, x3, x4) = U56_ga(x1, x2, x4)
U57_ga(x1, x2, x3, x4) = U57_ga(x1, x2, x4)
U54_ga(x1, x2, x3, x4) = U54_ga(x1, x2, x4)
qc22_in_gaga(x1, x2, x3, x4) = qc22_in_gaga(x1, x3)
U59_gaga(x1, x2, x3, x4, x5) = U59_gaga(x1, x3, x5)
U60_gaga(x1, x2, x3, x4, x5) = U60_gaga(x1, x2, x3, x5)
qc22_out_gaga(x1, x2, x3, x4) = qc22_out_gaga(x1, x2, x3, x4)
U47_ga(x1, x2, x3, x4) = U47_ga(x1, x2, x4)
U43_ga(x1, x2, x3, x4, x5) = U43_ga(x1, x2, x3, x5)
U40_ga(x1, x2, x3, x4, x5) = U40_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)
P22_IN_GAGA(x1, x2, x3, x4) = P22_IN_GAGA(x1, x3)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x3, x5)
U3_GAGA(x1, x2, x3, x4, x5) = U3_GAGA(x1, x3, x5)
U20_GA(x1, x2, x3, x4) = U20_GA(x1, x2, x4)
U29_GA(x1, x2, x3, x4) = U29_GA(x1, x2, x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) 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, s2c1_in_ga(plus(T113, T114)))
U10_GA(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → P22_IN_GAGA(T115, T121)
P22_IN_GAGA(T115, T121) → 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, s2c1_in_ga(T230))
U14_GA(T230, T231, T232, s2c1_out_ga(T230, T238)) → P22_IN_GAGA(plus(T231, T232), T238)
P22_IN_GAGA(T115, T121) → U3_GAGA(T115, T121, s2c1_in_ga(T115))
U3_GAGA(T115, T121, s2c1_out_ga(T115, T128)) → S21_IN_GA(plus(T121, T128))
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, s2c1_in_ga(T385))
U20_GA(T386, T385, s2c1_out_ga(T385, T392)) → P22_IN_GAGA(T386, T392)
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
S21_IN_GA(plus(T495, T496)) → U29_GA(T495, T496, s2c1_in_ga(T495))
U29_GA(T495, T496, s2c1_out_ga(T495, T502)) → P22_IN_GAGA(T496, T502)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U37_ga(T32, T33, T34, T35, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s2c1_in_ga(plus(T70, plus(T71, T72))) → U38_ga(T70, T71, T72, s2c1_in_ga(plus(T72, plus(T70, T71))))
s2c1_in_ga(plus(T87, plus(T88, 0))) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115))) → U39_ga(T113, T114, T115, s2c1_in_ga(plus(T113, T114)))
s2c1_in_ga(plus(T195, plus(T196, T197))) → U41_ga(T195, T196, T197, s2c1_in_ga(plus(plus(T196, T197), T195)))
s2c1_in_ga(plus(T230, plus(T231, T232))) → U42_ga(T230, T231, T232, s2c1_in_ga(T230))
s2c1_in_ga(plus(plus(T321, T322), T320)) → U44_ga(T321, T322, T320, s2c1_in_ga(plus(plus(T320, T321), T322)))
s2c1_in_ga(plus(T353, T352)) → U45_ga(T353, T352, s2c1_in_ga(plus(T353, T352)))
s2c1_in_ga(plus(0, T364)) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385)) → U46_ga(T386, T385, s2c1_in_ga(T385))
s2c1_in_ga(plus(T439, s(T438))) → U48_ga(T439, T438, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, isNatc33_in_g(T439))
U49_ga(T439, T438, isNatc33_out_g(T439)) → U50_ga(T439, T438, addc99_in_gga(T438, T439))
addc99_in_gga(s(T457), T458) → U62_gga(T457, T458, addc99_in_gga(T457, T458))
addc99_in_gga(0, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0)) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0)) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496)) → U53_ga(T495, T496, s2c1_in_ga(T495))
s2c1_in_ga(plus(T517, T518)) → U55_ga(T517, T518, isNatc33_in_g(T517))
U55_ga(T517, T518, isNatc33_out_g(T517)) → U56_ga(T517, T518, isNatc33_in_g(T518))
U56_ga(T517, T518, isNatc33_out_g(T518)) → U57_ga(T517, T518, addc99_in_gga(T517, T518))
U57_ga(T517, T518, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, qc22_in_gaga(T496, T502))
qc22_in_gaga(T115, T121) → U59_gaga(T115, T121, s2c1_in_ga(T115))
U59_gaga(T115, T121, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, s2c1_in_ga(plus(T121, T128)))
U60_gaga(T115, T128, T121, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, qc22_in_gaga(T386, T392))
U47_ga(T386, T385, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, qc22_in_gaga(plus(T231, T232), T238))
U43_ga(T230, T231, T232, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, qc22_in_gaga(T115, T121))
U40_ga(T113, T114, T115, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The set Q consists of the following terms:

s2c1_in_ga(x0)
isNatc33_in_g(x0)
U58_g(x0, x1)
U48_ga(x0, x1, x2)
U49_ga(x0, x1, x2)
addc99_in_gga(x0, x1)
U62_gga(x0, x1, x2)
U50_ga(x0, x1, x2)
U51_ga(x0, x1)
U52_ga(x0, x1)
U55_ga(x0, x1, x2)
U56_ga(x0, x1, x2)
U57_ga(x0, x1, x2)
U53_ga(x0, x1, x2)
qc22_in_gaga(x0, x1)
U59_gaga(x0, x1, x2)
U60_gaga(x0, x1, x2, x3)
U54_ga(x0, x1, x2)
U46_ga(x0, x1, x2)
U47_ga(x0, x1, x2)
U45_ga(x0, x1, x2)
U44_ga(x0, x1, x2, x3)
U42_ga(x0, x1, x2, x3)
U43_ga(x0, x1, x2, x3)
U41_ga(x0, x1, x2, x3)
U39_ga(x0, x1, x2, x3)
U40_ga(x0, x1, x2, x3)
U38_ga(x0, x1, x2, x3)
U37_ga(x0, x1, x2, x3, x4)

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

(26) 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))) → U10_GA(T113, T114, T115, s2c1_in_ga(plus(T113, T114)))
U14_GA(T230, T231, T232, s2c1_out_ga(T230, T238)) → P22_IN_GAGA(plus(T231, T232), T238)
U3_GAGA(T115, T121, s2c1_out_ga(T115, T128)) → S21_IN_GA(plus(T121, T128))
S21_IN_GA(plus(T386, T385)) → U20_GA(T386, T385, s2c1_in_ga(T385))
S21_IN_GA(plus(T495, T496)) → U29_GA(T495, T496, s2c1_in_ga(T495))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(P22_IN_GAGA(x₁, x₂)) = 1 + x₁ + x₂
POL(S21_IN_GA(x₁)) = 1 + x₁
POL(U10_GA(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(U14_GA(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(U20_GA(x₁, x₂, x₃)) = x₁ + x₃
POL(U29_GA(x₁, x₂, x₃)) = x₂ + x₃
POL(U37_ga(x₁, x₂, x₃, x₄, x₅)) = x₅
POL(U38_ga(x₁, x₂, x₃, x₄)) = x₄
POL(U39_ga(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(U3_GAGA(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U40_ga(x₁, x₂, x₃, x₄)) = x₄
POL(U41_ga(x₁, x₂, x₃, x₄)) = x₄
POL(U42_ga(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(U43_ga(x₁, x₂, x₃, x₄)) = x₄
POL(U44_ga(x₁, x₂, x₃, x₄)) = x₄
POL(U45_ga(x₁, x₂, x₃)) = x₃
POL(U46_ga(x₁, x₂, x₃)) = x₁ + x₃
POL(U47_ga(x₁, x₂, x₃)) = x₃
POL(U48_ga(x₁, x₂, x₃)) = x₁ + x₂
POL(U49_ga(x₁, x₂, x₃)) = x₁ + x₂
POL(U50_ga(x₁, x₂, x₃)) = x₃
POL(U51_ga(x₁, x₂)) = 1 + x₁
POL(U52_ga(x₁, x₂)) = 1 + x₁
POL(U53_ga(x₁, x₂, x₃)) = x₂ + x₃
POL(U54_ga(x₁, x₂, x₃)) = x₃
POL(U55_ga(x₁, x₂, x₃)) = x₁ + x₂
POL(U56_ga(x₁, x₂, x₃)) = x₁ + x₂
POL(U57_ga(x₁, x₂, x₃)) = x₃
POL(U58_g(x₁, x₂)) = 0
POL(U59_gaga(x₁, x₂, x₃)) = x₂ + x₃
POL(U60_gaga(x₁, x₂, x₃, x₄)) = x₄
POL(U62_gga(x₁, x₂, x₃)) = x₃
POL(addc99_in_gga(x₁, x₂)) = x₁ + x₂
POL(addc99_out_gga(x₁, x₂, x₃)) = 1 + x₃
POL(isNatc33_in_g(x₁)) = 0
POL(isNatc33_out_g(x₁)) = 0
POL(plus(x₁, x₂)) = x₁ + x₂
POL(qc22_in_gaga(x₁, x₂)) = 1 + x₁ + x₂
POL(qc22_out_gaga(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(s(x₁)) = x₁
POL(s2c1_in_ga(x₁)) = x₁
POL(s2c1_out_ga(x₁, x₂)) = 1 + x₂

The following usable rules [FROCOS05] were oriented:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U37_ga(T32, T33, T34, T35, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s2c1_in_ga(plus(T70, plus(T71, T72))) → U38_ga(T70, T71, T72, s2c1_in_ga(plus(T72, plus(T70, T71))))
s2c1_in_ga(plus(T87, plus(T88, 0))) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115))) → U39_ga(T113, T114, T115, s2c1_in_ga(plus(T113, T114)))
s2c1_in_ga(plus(T195, plus(T196, T197))) → U41_ga(T195, T196, T197, s2c1_in_ga(plus(plus(T196, T197), T195)))
s2c1_in_ga(plus(T230, plus(T231, T232))) → U42_ga(T230, T231, T232, s2c1_in_ga(T230))
s2c1_in_ga(plus(plus(T321, T322), T320)) → U44_ga(T321, T322, T320, s2c1_in_ga(plus(plus(T320, T321), T322)))
s2c1_in_ga(plus(T353, T352)) → U45_ga(T353, T352, s2c1_in_ga(plus(T353, T352)))
s2c1_in_ga(plus(0, T364)) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385)) → U46_ga(T386, T385, s2c1_in_ga(T385))
s2c1_in_ga(plus(T439, s(T438))) → U48_ga(T439, T438, isNatc33_in_g(s(T438)))
s2c1_in_ga(plus(T469, 0)) → U51_ga(T469, isNatc33_in_g(0))
s2c1_in_ga(plus(T474, 0)) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496)) → U53_ga(T495, T496, s2c1_in_ga(T495))
s2c1_in_ga(plus(T517, T518)) → U55_ga(T517, T518, isNatc33_in_g(T517))
U38_ga(T70, T71, T72, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)
U41_ga(T195, T196, T197, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U44_ga(T321, T322, T320, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U45_ga(T353, T352, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U53_ga(T495, T496, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, qc22_in_gaga(T496, T502))
U54_ga(T495, T496, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
qc22_in_gaga(T115, T121) → U59_gaga(T115, T121, s2c1_in_ga(T115))
U59_gaga(T115, T121, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, s2c1_in_ga(plus(T121, T128)))
U60_gaga(T115, T128, T121, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U46_ga(T386, T385, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, qc22_in_gaga(T386, T392))
U47_ga(T386, T385, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U42_ga(T230, T231, T232, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, qc22_in_gaga(plus(T231, T232), T238))
U43_ga(T230, T231, T232, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U39_ga(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, qc22_in_gaga(T115, T121))
U40_ga(T113, T114, T115, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U48_ga(T439, T438, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, isNatc33_in_g(T439))
U49_ga(T439, T438, isNatc33_out_g(T439)) → U50_ga(T439, T438, addc99_in_gga(T438, T439))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
U55_ga(T517, T518, isNatc33_out_g(T517)) → U56_ga(T517, T518, isNatc33_in_g(T518))
U56_ga(T517, T518, isNatc33_out_g(T518)) → U57_ga(T517, T518, addc99_in_gga(T517, T518))
addc99_in_gga(s(T457), T458) → U62_gga(T457, T458, addc99_in_gga(T457, T458))
addc99_in_gga(0, T466) → addc99_out_gga(0, T466, T466)
U50_ga(T439, T438, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
U57_ga(T517, T518, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U62_gga(T457, T458, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))

(27) 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))
U10_GA(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → P22_IN_GAGA(T115, T121)
P22_IN_GAGA(T115, T121) → 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, s2c1_in_ga(T230))
P22_IN_GAGA(T115, T121) → U3_GAGA(T115, T121, s2c1_in_ga(T115))
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)
U20_GA(T386, T385, s2c1_out_ga(T385, T392)) → P22_IN_GAGA(T386, T392)
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)
U29_GA(T495, T496, s2c1_out_ga(T495, T502)) → P22_IN_GAGA(T496, T502)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U37_ga(T32, T33, T34, T35, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s2c1_in_ga(plus(T70, plus(T71, T72))) → U38_ga(T70, T71, T72, s2c1_in_ga(plus(T72, plus(T70, T71))))
s2c1_in_ga(plus(T87, plus(T88, 0))) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115))) → U39_ga(T113, T114, T115, s2c1_in_ga(plus(T113, T114)))
s2c1_in_ga(plus(T195, plus(T196, T197))) → U41_ga(T195, T196, T197, s2c1_in_ga(plus(plus(T196, T197), T195)))
s2c1_in_ga(plus(T230, plus(T231, T232))) → U42_ga(T230, T231, T232, s2c1_in_ga(T230))
s2c1_in_ga(plus(plus(T321, T322), T320)) → U44_ga(T321, T322, T320, s2c1_in_ga(plus(plus(T320, T321), T322)))
s2c1_in_ga(plus(T353, T352)) → U45_ga(T353, T352, s2c1_in_ga(plus(T353, T352)))
s2c1_in_ga(plus(0, T364)) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385)) → U46_ga(T386, T385, s2c1_in_ga(T385))
s2c1_in_ga(plus(T439, s(T438))) → U48_ga(T439, T438, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, isNatc33_in_g(T439))
U49_ga(T439, T438, isNatc33_out_g(T439)) → U50_ga(T439, T438, addc99_in_gga(T438, T439))
addc99_in_gga(s(T457), T458) → U62_gga(T457, T458, addc99_in_gga(T457, T458))
addc99_in_gga(0, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0)) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0)) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496)) → U53_ga(T495, T496, s2c1_in_ga(T495))
s2c1_in_ga(plus(T517, T518)) → U55_ga(T517, T518, isNatc33_in_g(T517))
U55_ga(T517, T518, isNatc33_out_g(T517)) → U56_ga(T517, T518, isNatc33_in_g(T518))
U56_ga(T517, T518, isNatc33_out_g(T518)) → U57_ga(T517, T518, addc99_in_gga(T517, T518))
U57_ga(T517, T518, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, qc22_in_gaga(T496, T502))
qc22_in_gaga(T115, T121) → U59_gaga(T115, T121, s2c1_in_ga(T115))
U59_gaga(T115, T121, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, s2c1_in_ga(plus(T121, T128)))
U60_gaga(T115, T128, T121, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, qc22_in_gaga(T386, T392))
U47_ga(T386, T385, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, qc22_in_gaga(plus(T231, T232), T238))
U43_ga(T230, T231, T232, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, qc22_in_gaga(T115, T121))
U40_ga(T113, T114, T115, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The set Q consists of the following terms:

s2c1_in_ga(x0)
isNatc33_in_g(x0)
U58_g(x0, x1)
U48_ga(x0, x1, x2)
U49_ga(x0, x1, x2)
addc99_in_gga(x0, x1)
U62_gga(x0, x1, x2)
U50_ga(x0, x1, x2)
U51_ga(x0, x1)
U52_ga(x0, x1)
U55_ga(x0, x1, x2)
U56_ga(x0, x1, x2)
U57_ga(x0, x1, x2)
U53_ga(x0, x1, x2)
qc22_in_gaga(x0, x1)
U59_gaga(x0, x1, x2)
U60_gaga(x0, x1, x2, x3)
U54_ga(x0, x1, x2)
U46_ga(x0, x1, x2)
U47_ga(x0, x1, x2)
U45_ga(x0, x1, x2)
U44_ga(x0, x1, x2, x3)
U42_ga(x0, x1, x2, x3)
U43_ga(x0, x1, x2, x3)
U41_ga(x0, x1, x2, x3)
U39_ga(x0, x1, x2, x3)
U40_ga(x0, x1, x2, x3)
U38_ga(x0, x1, x2, x3)
U37_ga(x0, x1, x2, x3, x4)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(29) 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(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
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(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(T495, T496)) → S21_IN_GA(T495)

The TRS R consists of the following rules:

s2c1_in_ga(plus(T32, plus(T33, plus(T34, T35)))) → U37_ga(T32, T33, T34, T35, s2c1_in_ga(plus(plus(plus(T32, T33), T34), T35)))
s2c1_in_ga(plus(T70, plus(T71, T72))) → U38_ga(T70, T71, T72, s2c1_in_ga(plus(T72, plus(T70, T71))))
s2c1_in_ga(plus(T87, plus(T88, 0))) → s2c1_out_ga(plus(T87, plus(T88, 0)), plus(T87, T88))
s2c1_in_ga(plus(T113, plus(T114, T115))) → U39_ga(T113, T114, T115, s2c1_in_ga(plus(T113, T114)))
s2c1_in_ga(plus(T195, plus(T196, T197))) → U41_ga(T195, T196, T197, s2c1_in_ga(plus(plus(T196, T197), T195)))
s2c1_in_ga(plus(T230, plus(T231, T232))) → U42_ga(T230, T231, T232, s2c1_in_ga(T230))
s2c1_in_ga(plus(plus(T321, T322), T320)) → U44_ga(T321, T322, T320, s2c1_in_ga(plus(plus(T320, T321), T322)))
s2c1_in_ga(plus(T353, T352)) → U45_ga(T353, T352, s2c1_in_ga(plus(T353, T352)))
s2c1_in_ga(plus(0, T364)) → s2c1_out_ga(plus(0, T364), T364)
s2c1_in_ga(plus(T386, T385)) → U46_ga(T386, T385, s2c1_in_ga(T385))
s2c1_in_ga(plus(T439, s(T438))) → U48_ga(T439, T438, isNatc33_in_g(s(T438)))
isNatc33_in_g(s(T165)) → U58_g(T165, isNatc33_in_g(T165))
isNatc33_in_g(0) → isNatc33_out_g(0)
U58_g(T165, isNatc33_out_g(T165)) → isNatc33_out_g(s(T165))
U48_ga(T439, T438, isNatc33_out_g(s(T438))) → U49_ga(T439, T438, isNatc33_in_g(T439))
U49_ga(T439, T438, isNatc33_out_g(T439)) → U50_ga(T439, T438, addc99_in_gga(T438, T439))
addc99_in_gga(s(T457), T458) → U62_gga(T457, T458, addc99_in_gga(T457, T458))
addc99_in_gga(0, T466) → addc99_out_gga(0, T466, T466)
U62_gga(T457, T458, addc99_out_gga(T457, T458, T460)) → addc99_out_gga(s(T457), T458, s(T460))
U50_ga(T439, T438, addc99_out_gga(T438, T439, T441)) → s2c1_out_ga(plus(T439, s(T438)), s(T441))
s2c1_in_ga(plus(T469, 0)) → U51_ga(T469, isNatc33_in_g(0))
U51_ga(T469, isNatc33_out_g(0)) → U52_ga(T469, isNatc33_in_g(T469))
U52_ga(T469, isNatc33_out_g(T469)) → s2c1_out_ga(plus(T469, 0), T469)
s2c1_in_ga(plus(T474, 0)) → s2c1_out_ga(plus(T474, 0), T474)
s2c1_in_ga(plus(T495, T496)) → U53_ga(T495, T496, s2c1_in_ga(T495))
s2c1_in_ga(plus(T517, T518)) → U55_ga(T517, T518, isNatc33_in_g(T517))
U55_ga(T517, T518, isNatc33_out_g(T517)) → U56_ga(T517, T518, isNatc33_in_g(T518))
U56_ga(T517, T518, isNatc33_out_g(T518)) → U57_ga(T517, T518, addc99_in_gga(T517, T518))
U57_ga(T517, T518, addc99_out_gga(T517, T518, T520)) → s2c1_out_ga(plus(T517, T518), T520)
U53_ga(T495, T496, s2c1_out_ga(T495, T502)) → U54_ga(T495, T496, qc22_in_gaga(T496, T502))
qc22_in_gaga(T115, T121) → U59_gaga(T115, T121, s2c1_in_ga(T115))
U59_gaga(T115, T121, s2c1_out_ga(T115, T128)) → U60_gaga(T115, T128, T121, s2c1_in_ga(plus(T121, T128)))
U60_gaga(T115, T128, T121, s2c1_out_ga(plus(T121, T128), T117)) → qc22_out_gaga(T115, T128, T121, T117)
U54_ga(T495, T496, qc22_out_gaga(T496, X509, T502, T498)) → s2c1_out_ga(plus(T495, T496), T498)
U46_ga(T386, T385, s2c1_out_ga(T385, T392)) → U47_ga(T386, T385, qc22_in_gaga(T386, T392))
U47_ga(T386, T385, qc22_out_gaga(T386, X383, T392, T388)) → s2c1_out_ga(plus(T386, T385), T388)
U45_ga(T353, T352, s2c1_out_ga(plus(T353, T352), T355)) → s2c1_out_ga(plus(T353, T352), T355)
U44_ga(T321, T322, T320, s2c1_out_ga(plus(plus(T320, T321), T322), T324)) → s2c1_out_ga(plus(plus(T321, T322), T320), T324)
U42_ga(T230, T231, T232, s2c1_out_ga(T230, T238)) → U43_ga(T230, T231, T232, qc22_in_gaga(plus(T231, T232), T238))
U43_ga(T230, T231, T232, qc22_out_gaga(plus(T231, T232), X225, T238, T234)) → s2c1_out_ga(plus(T230, plus(T231, T232)), T234)
U41_ga(T195, T196, T197, s2c1_out_ga(plus(plus(T196, T197), T195), T199)) → s2c1_out_ga(plus(T195, plus(T196, T197)), T199)
U39_ga(T113, T114, T115, s2c1_out_ga(plus(T113, T114), T121)) → U40_ga(T113, T114, T115, qc22_in_gaga(T115, T121))
U40_ga(T113, T114, T115, qc22_out_gaga(T115, X104, T121, T117)) → s2c1_out_ga(plus(T113, plus(T114, T115)), T117)
U38_ga(T70, T71, T72, s2c1_out_ga(plus(T72, plus(T70, T71)), T74)) → s2c1_out_ga(plus(T70, plus(T71, T72)), T74)
U37_ga(T32, T33, T34, T35, s2c1_out_ga(plus(plus(plus(T32, T33), T34), T35), T37)) → s2c1_out_ga(plus(T32, plus(T33, plus(T34, T35))), T37)

The set Q consists of the following terms:

s2c1_in_ga(x0)
isNatc33_in_g(x0)
U58_g(x0, x1)
U48_ga(x0, x1, x2)
U49_ga(x0, x1, x2)
addc99_in_gga(x0, x1)
U62_gga(x0, x1, x2)
U50_ga(x0, x1, x2)
U51_ga(x0, x1)
U52_ga(x0, x1)
U55_ga(x0, x1, x2)
U56_ga(x0, x1, x2)
U57_ga(x0, x1, x2)
U53_ga(x0, x1, x2)
qc22_in_gaga(x0, x1)
U59_gaga(x0, x1, x2)
U60_gaga(x0, x1, x2, x3)
U54_ga(x0, x1, x2)
U46_ga(x0, x1, x2)
U47_ga(x0, x1, x2)
U45_ga(x0, x1, x2)
U44_ga(x0, x1, x2, x3)
U42_ga(x0, x1, x2, x3)
U43_ga(x0, x1, x2, x3)
U41_ga(x0, x1, x2, x3)
U39_ga(x0, x1, x2, x3)
U40_ga(x0, x1, x2, x3)
U38_ga(x0, x1, x2, x3)
U37_ga(x0, x1, x2, x3, x4)

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

(30) 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.

(31) 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(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
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(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(T495, T496)) → S21_IN_GA(T495)

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

s2c1_in_ga(x0)
isNatc33_in_g(x0)
U58_g(x0, x1)
U48_ga(x0, x1, x2)
U49_ga(x0, x1, x2)
addc99_in_gga(x0, x1)
U62_gga(x0, x1, x2)
U50_ga(x0, x1, x2)
U51_ga(x0, x1)
U52_ga(x0, x1)
U55_ga(x0, x1, x2)
U56_ga(x0, x1, x2)
U57_ga(x0, x1, x2)
U53_ga(x0, x1, x2)
qc22_in_gaga(x0, x1)
U59_gaga(x0, x1, x2)
U60_gaga(x0, x1, x2, x3)
U54_ga(x0, x1, x2)
U46_ga(x0, x1, x2)
U47_ga(x0, x1, x2)
U45_ga(x0, x1, x2)
U44_ga(x0, x1, x2, x3)
U42_ga(x0, x1, x2, x3)
U43_ga(x0, x1, x2, x3)
U41_ga(x0, x1, x2, x3)
U39_ga(x0, x1, x2, x3)
U40_ga(x0, x1, x2, x3)
U38_ga(x0, x1, x2, x3)
U37_ga(x0, x1, x2, x3, x4)

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

(32) 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].

s2c1_in_ga(x0)
isNatc33_in_g(x0)
U58_g(x0, x1)
U48_ga(x0, x1, x2)
U49_ga(x0, x1, x2)
addc99_in_gga(x0, x1)
U62_gga(x0, x1, x2)
U50_ga(x0, x1, x2)
U51_ga(x0, x1)
U52_ga(x0, x1)
U55_ga(x0, x1, x2)
U56_ga(x0, x1, x2)
U57_ga(x0, x1, x2)
U53_ga(x0, x1, x2)
qc22_in_gaga(x0, x1)
U59_gaga(x0, x1, x2)
U60_gaga(x0, x1, x2, x3)
U54_ga(x0, x1, x2)
U46_ga(x0, x1, x2)
U47_ga(x0, x1, x2)
U45_ga(x0, x1, x2)
U44_ga(x0, x1, x2, x3)
U42_ga(x0, x1, x2, x3)
U43_ga(x0, x1, x2, x3)
U41_ga(x0, x1, x2, x3)
U39_ga(x0, x1, x2, x3)
U40_ga(x0, x1, x2, x3)
U38_ga(x0, x1, x2, x3)
U37_ga(x0, x1, x2, x3, x4)

(33) 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(T113, plus(T114, T115))) → S21_IN_GA(plus(T113, T114))
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(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(T495, T496)) → S21_IN_GA(T495)

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

(34) 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(T230, plus(T231, T232))) → S21_IN_GA(T230)
S21_IN_GA(plus(T386, T385)) → S21_IN_GA(T385)
S21_IN_GA(plus(T495, T496)) → S21_IN_GA(T495)

Used ordering: Polynomial interpretation [POLO]:

POL(S21_IN_GA(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂

(35) 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.

(36) 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))).

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

(4) Obligation:

(5) TriplesToPiDPProof (SOUND transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) Obligation:

(30) UsableRulesProof (EQUIVALENT transformation)

(31) Obligation:

(32) QReductionProof (EQUIVALENT transformation)

(33) Obligation:

(34) MRRProof (EQUIVALENT transformation)

(35) Obligation:

(36) NonTerminationProof (EQUIVALENT transformation)

(37) NO