Left Termination of the query pattern
parse(g,a)
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

parse(Xs, T) :- ','(app(As, .(a, .(s(A, B, C), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B, C), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(s(A, B), .(b, Bs))), Xs), ','(app(As, .(s(a, s(A, B), b), Bs), Ys), parse(Ys, T))).
parse(Xs, T) :- ','(app(As, .(a, .(b, Bs)), Xs), ','(app(As, .(s(a, b), Bs), Ys), parse(Ys, T))).
parse(.(s(A, B), []), s(A, B)).
parse(.(s(A, B, C), []), s(A, B, C)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: parse(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: parse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: parse(T1, T2), SCOPE: 1, CLAUSE: 0 | parse(T1, T2), SCOPE: 1, CLAUSE: 1 | parse(T1, T2), SCOPE: 1, CLAUSE: 2 | parse(T1, T2), SCOPE: 1, CLAUSE: 3 | parse(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))) | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 5 | ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6 | ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6 | ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app([], .(s(a, s(T24, T25, T26), b), T27), X10), parse(X10, T7))\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="B: app([], .(s(a, s(T24, T25, T26), b), T27), X10)\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: parse(T33, T7)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 5 | app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app([], .(s(a, s(T24, T25, T26), b), T27), X10), SCOPE: 3, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), T5), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, T7))), SCOPE: 2, CLAUSE: 6\n\nT5 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ?_2 | parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), ','(app(.(T68, X109), .(s(a, s(X110, X111, X112), b), X113), X10), parse(X10, T7)))\n\nT68 is ground\nT69 is ground\nX10 is free\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="C: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69)\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), parse(X10, T7))\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 5 | app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 6\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 5\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(X109, .(a, .(s(X110, X111, X112), .(b, X113))), T69), SCOPE: 4, CLAUSE: 6\n\nT69 is ground\nX109 is free\nX110 is free\nX111 is free\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(X184, .(a, .(s(X185, X186, X187), .(b, X188))), T116)\n\nT116 is ground\nX184 is free\nX185 is free\nX186 is free\nX187 is free\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="D: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10)\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: parse(T125, T7)\n\nT125 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 5 | app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 6\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(.(T68, T78), .(s(a, s(T79, T80, T81), b), T82), X10), SCOPE: 5, CLAUSE: 6\n\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="I: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220)\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 5 | app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 6\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 5\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(T147, .(s(a, s(T148, T149, T150), b), T151), X220), SCOPE: 6, CLAUSE: 6\n\nT147 is ground\nT148 is ground\nT149 is ground\nT150 is ground\nT151 is ground\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app(T193, .(s(a, s(T194, T195, T196), b), T197), X242)\n\nT193 is ground\nT194 is ground\nT195 is ground\nT196 is ground\nT197 is ground\nX242 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: parse(T5, T2), SCOPE: 1, CLAUSE: 1 | parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: parse(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: parse(T5, T2), SCOPE: 1, CLAUSE: 2 | parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), ','(app(X289, .(s(a, s(X290, X291), b), X292), X293), parse(X293, T232)))\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="E: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230)\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(app(T239, .(s(a, s(T240, T241), b), T242), X293), parse(X293, T232))\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 5 | app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 6\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 5\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: app(X289, .(a, .(s(X290, X291), .(b, X292))), T230), SCOPE: 7, CLAUSE: 6\n\nT230 is ground\nX289 is free\nX290 is free\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: app(X353, .(a, .(s(X354, X355), .(b, X356))), T269)\n\nT269 is ground\nX353 is free\nX354 is free\nX355 is free\nX356 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="F: app(T239, .(s(a, s(T240, T241), b), T242), X293)\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: parse(T282, T232)\n\nT282 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 5 | app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 6\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 5\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app(T239, .(s(a, s(T240, T241), b), T242), X293), SCOPE: 8, CLAUSE: 6\n\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nX293 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app(T315, .(s(a, s(T316, T317), b), T318), X388)\n\nT315 is ground\nT316 is ground\nT317 is ground\nT318 is ground\nX388 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: parse(T5, T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: parse(T5, T2), SCOPE: 1, CLAUSE: 3 | parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(app(X423, .(a, .(b, X424)), T341), ','(app(X423, .(s(a, b), X424), X425), parse(X425, T343)))\n\nT341 is ground\nX423 is free\nX424 is free\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="G: app(X423, .(a, .(b, X424)), T341)\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(app(T346, .(s(a, b), T347), X425), parse(X425, T343))\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 5 | app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 6\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 5\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app(X423, .(a, .(b, X424)), T341), SCOPE: 9, CLAUSE: 6\n\nT341 is ground\nX423 is free\nX424 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app(X463, .(a, .(b, X464)), T360)\n\nT360 is ground\nX463 is free\nX464 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="H: app(T346, .(s(a, b), T347), X425)\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: parse(T365, T343)\n\nT365 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 5 | app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 6\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 5\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: app(T346, .(s(a, b), T347), X425), SCOPE: 10, CLAUSE: 6\n\nT346 is ground\nT347 is ground\nX425 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: app(T380, .(s(a, b), T381), X492)\n\nT380 is ground\nT381 is ground\nX492 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: parse(T5, T2), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: parse(T5, T2), SCOPE: 1, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 99 [arrowhead = none ];
99 -> 2;

100 [label="ONLY EVAL with clause\nparse(X3, X4) :- ','(app(X5, .(a, .(s(X6, X7, X8), .(b, X9))), X3), ','(app(X5, .(s(a, s(X6, X7, X8), b), X9), X10), parse(X10, X4))).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 100 [arrowhead = none ];
100 -> 3;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 101 [arrowhead = none ];
101 -> 4;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 102 [arrowhead = none ];
102 -> 5;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 103 [arrowhead = none ];
103 -> 6;

104 [label="EVAL with clause\napp([], X31, X31).\nand substitutionX5 -> [],\nX6 -> T24,\nX7 -> T25,\nX8 -> T26,\nX9 -> T27,\nX31 -> .(a, .(s(T24, T25, T26), .(b, T27))),\nX32 -> T24,\nX33 -> T25,\nX34 -> T26,\nX35 -> T27,\nT5 -> .(a, .(s(T24, T25, T26), .(b, T27)))", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 104 [arrowhead = none ];
104 -> 7;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 105 [arrowhead = none ];
105 -> 8;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 106 [arrowhead = none ];
106 -> 17;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 107 [arrowhead = none ];
107 -> 18;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 108 [arrowhead = none ];
108 -> 9;

109 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT25 is ground\nT26 is ground\nT27 is ground\nT33 is ground\nreplacements:X10 -> T33", fontsize=14, style = filled, fillcolor = violet];
7 -> 109 [arrowhead = none ];
109 -> 10;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 110 [arrowhead = none ];
110 -> 11;

111 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
10 -> 111 [arrowhead = none , style=dashed];
111 -> 1[style=dashed];

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 112 [arrowhead = none ];
112 -> 12;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 113 [arrowhead = none ];
113 -> 13;

114 [label="ONLY EVAL with clause\napp([], X51, X51).\nand substitutionT24 -> T54,\nT25 -> T55,\nT26 -> T56,\nT27 -> T57,\nX51 -> .(s(a, s(T54, T55, T56), b), T57),\nX10 -> .(s(a, s(T54, T55, T56), b), T57)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 114 [arrowhead = none ];
114 -> 14;

115 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 115 [arrowhead = none ];
115 -> 16;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 116 [arrowhead = none ];
116 -> 15;

117 [label="EVAL with clause\napp(.(X104, X105), X106, .(X104, X107)) :- app(X105, X106, X107).\nand substitutionX104 -> T68,\nX105 -> X109,\nX5 -> .(T68, X109),\nX6 -> X110,\nX7 -> X111,\nX8 -> X112,\nX9 -> X113,\nX106 -> .(a, .(s(X110, X111, X112), .(b, X113))),\nX108 -> T68,\nX107 -> T69,\nT5 -> .(T68, T69)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 117 [arrowhead = none ];
117 -> 19;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 118 [arrowhead = none ];
118 -> 20;

119 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 120 [arrowhead = none ];
120 -> 21;

121 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nT69 is ground\nreplacements:X109 -> T78,\nX110 -> T79,\nX111 -> T80,\nX112 -> T81,\nX113 -> T82", fontsize=14, style = filled, fillcolor = violet];
19 -> 121 [arrowhead = none ];
121 -> 22;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 122 [arrowhead = none ];
122 -> 23;

123 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 123 [arrowhead = none ];
123 -> 31;

124 [label="SPLIT 2\nnew knowledge:\nT68 is ground\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground\nT82 is ground\nT125 is ground\nreplacements:X10 -> T125", fontsize=14, style = filled, fillcolor = violet];
22 -> 124 [arrowhead = none ];
124 -> 32;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 125 [arrowhead = none ];
125 -> 24;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 126 [arrowhead = none ];
126 -> 25;

127 [label="EVAL with clause\napp([], X154, X154).\nand substitutionX109 -> [],\nX110 -> T107,\nX111 -> T108,\nX112 -> T109,\nX113 -> T110,\nX154 -> .(a, .(s(T107, T108, T109), .(b, T110))),\nX155 -> T107,\nX156 -> T108,\nX157 -> T109,\nX158 -> T110,\nT69 -> .(a, .(s(T107, T108, T109), .(b, T110)))", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 127 [arrowhead = none ];
127 -> 26;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 128 [arrowhead = none ];
128 -> 27;

129 [label="EVAL with clause\napp(.(X179, X180), X181, .(X179, X182)) :- app(X180, X181, X182).\nand substitutionX179 -> T115,\nX180 -> X184,\nX109 -> .(T115, X184),\nX110 -> X185,\nX111 -> X186,\nX112 -> X187,\nX113 -> X188,\nX181 -> .(a, .(s(X185, X186, X187), .(b, X188))),\nX183 -> T115,\nX182 -> T116,\nT69 -> .(T115, T116)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 129 [arrowhead = none ];
129 -> 29;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 130 [arrowhead = none ];
130 -> 30;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 131 [arrowhead = none ];
131 -> 28;

132 [label="INSTANCE with matching:\nX109 -> X184\nX110 -> X185\nX111 -> X186\nX112 -> X187\nX113 -> X188\nT69 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 132 [arrowhead = none , style=dashed];
132 -> 21[style=dashed];

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 133 [arrowhead = none ];
133 -> 33;

134 [label="INSTANCE with matching:\nT1 -> T125\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 134 [arrowhead = none , style=dashed];
134 -> 1[style=dashed];

135 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 135 [arrowhead = none ];
135 -> 34;

136 [label="ONLY EVAL with clause\napp(.(X216, X217), X218, .(X216, X219)) :- app(X217, X218, X219).\nand substitutionT68 -> T146,\nX216 -> T146,\nT78 -> T147,\nX217 -> T147,\nT79 -> T148,\nT80 -> T149,\nT81 -> T150,\nT82 -> T151,\nX218 -> .(s(a, s(T148, T149, T150), b), T151),\nX219 -> X220,\nX10 -> .(T146, X220)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 136 [arrowhead = none ];
136 -> 35;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 137 [arrowhead = none ];
137 -> 36;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 138 [arrowhead = none ];
138 -> 37;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 139 [arrowhead = none ];
139 -> 38;

140 [label="EVAL with clause\napp([], X227, X227).\nand substitutionT147 -> [],\nT148 -> T176,\nT149 -> T177,\nT150 -> T178,\nT151 -> T179,\nX227 -> .(s(a, s(T176, T177, T178), b), T179),\nX220 -> .(s(a, s(T176, T177, T178), b), T179)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 140 [arrowhead = none ];
140 -> 39;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 141 [arrowhead = none ];
141 -> 40;

142 [label="EVAL with clause\napp(.(X238, X239), X240, .(X238, X241)) :- app(X239, X240, X241).\nand substitutionX238 -> T192,\nX239 -> T193,\nT147 -> .(T192, T193),\nT148 -> T194,\nT149 -> T195,\nT150 -> T196,\nT151 -> T197,\nX240 -> .(s(a, s(T194, T195, T196), b), T197),\nX241 -> X242,\nX220 -> .(T192, X242)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 142 [arrowhead = none ];
142 -> 42;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 143 [arrowhead = none ];
143 -> 43;

144 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 144 [arrowhead = none ];
144 -> 41;

145 [label="INSTANCE with matching:\nT147 -> T193\nT148 -> T194\nT149 -> T195\nT150 -> T196\nT151 -> T197\nX220 -> X242", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 145 [arrowhead = none , style=dashed];
145 -> 35[style=dashed];

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 146 [arrowhead = none ];
146 -> 45;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 147 [arrowhead = none ];
147 -> 46;

148 [label="ONLY EVAL with clause\nparse(X287, X288) :- ','(app(X289, .(a, .(s(X290, X291), .(b, X292))), X287), ','(app(X289, .(s(a, s(X290, X291), b), X292), X293), parse(X293, X288))).\nand substitutionT5 -> T230,\nX287 -> T230,\nT2 -> T232,\nX288 -> T232,\nT231 -> T232", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 148 [arrowhead = none ];
148 -> 47;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 149 [arrowhead = none ];
149 -> 68;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 150 [arrowhead = none ];
150 -> 69;

151 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 151 [arrowhead = none ];
151 -> 48;

152 [label="SPLIT 2\nnew knowledge:\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nT230 is ground\nreplacements:X289 -> T239,\nX290 -> T240,\nX291 -> T241,\nX292 -> T242", fontsize=14, style = filled, fillcolor = violet];
47 -> 152 [arrowhead = none ];
152 -> 49;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 153 [arrowhead = none ];
153 -> 50;

154 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 154 [arrowhead = none ];
154 -> 58;

155 [label="SPLIT 2\nnew knowledge:\nT239 is ground\nT240 is ground\nT241 is ground\nT242 is ground\nT282 is ground\nreplacements:X293 -> T282", fontsize=14, style = filled, fillcolor = violet];
49 -> 155 [arrowhead = none ];
155 -> 59;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 156 [arrowhead = none ];
156 -> 51;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 157 [arrowhead = none ];
157 -> 52;

158 [label="EVAL with clause\napp([], X326, X326).\nand substitutionX289 -> [],\nX290 -> T261,\nX291 -> T262,\nX292 -> T263,\nX326 -> .(a, .(s(T261, T262), .(b, T263))),\nX327 -> T261,\nX328 -> T262,\nX329 -> T263,\nT230 -> .(a, .(s(T261, T262), .(b, T263)))", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 158 [arrowhead = none ];
158 -> 53;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 159 [arrowhead = none ];
159 -> 54;

160 [label="EVAL with clause\napp(.(X348, X349), X350, .(X348, X351)) :- app(X349, X350, X351).\nand substitutionX348 -> T268,\nX349 -> X353,\nX289 -> .(T268, X353),\nX290 -> X354,\nX291 -> X355,\nX292 -> X356,\nX350 -> .(a, .(s(X354, X355), .(b, X356))),\nX352 -> T268,\nX351 -> T269,\nT230 -> .(T268, T269)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 160 [arrowhead = none ];
160 -> 56;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 161 [arrowhead = none ];
161 -> 57;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 162 [arrowhead = none ];
162 -> 55;

163 [label="INSTANCE with matching:\nX289 -> X353\nX290 -> X354\nX291 -> X355\nX292 -> X356\nT230 -> T269", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 163 [arrowhead = none , style=dashed];
163 -> 48[style=dashed];

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 164 [arrowhead = none ];
164 -> 60;

165 [label="INSTANCE with matching:\nT1 -> T282\nT2 -> T232", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 165 [arrowhead = none , style=dashed];
165 -> 1[style=dashed];

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 166 [arrowhead = none ];
166 -> 61;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 167 [arrowhead = none ];
167 -> 62;

168 [label="EVAL with clause\napp([], X373, X373).\nand substitutionT239 -> [],\nT240 -> T301,\nT241 -> T302,\nT242 -> T303,\nX373 -> .(s(a, s(T301, T302), b), T303),\nX293 -> .(s(a, s(T301, T302), b), T303)", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 168 [arrowhead = none ];
168 -> 63;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 169 [arrowhead = none ];
169 -> 64;

170 [label="EVAL with clause\napp(.(X384, X385), X386, .(X384, X387)) :- app(X385, X386, X387).\nand substitutionX384 -> T314,\nX385 -> T315,\nT239 -> .(T314, T315),\nT240 -> T316,\nT241 -> T317,\nT242 -> T318,\nX386 -> .(s(a, s(T316, T317), b), T318),\nX387 -> X388,\nX293 -> .(T314, X388)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 170 [arrowhead = none ];
170 -> 66;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 171 [arrowhead = none ];
171 -> 67;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 172 [arrowhead = none ];
172 -> 65;

173 [label="INSTANCE with matching:\nT239 -> T315\nT240 -> T316\nT241 -> T317\nT242 -> T318\nX293 -> X388", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 173 [arrowhead = none , style=dashed];
173 -> 58[style=dashed];

174 [label="ONLY EVAL with clause\nparse(X421, X422) :- ','(app(X423, .(a, .(b, X424)), X421), ','(app(X423, .(s(a, b), X424), X425), parse(X425, X422))).\nand substitutionT5 -> T341,\nX421 -> T341,\nT2 -> T343,\nX422 -> T343,\nT342 -> T343", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 174 [arrowhead = none ];
174 -> 70;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 175 [arrowhead = none ];
175 -> 91;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 176 [arrowhead = none ];
176 -> 92;

177 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
70 -> 177 [arrowhead = none ];
177 -> 71;

178 [label="SPLIT 2\nnew knowledge:\nT346 is ground\nT347 is ground\nT341 is ground\nreplacements:X423 -> T346,\nX424 -> T347", fontsize=14, style = filled, fillcolor = violet];
70 -> 178 [arrowhead = none ];
178 -> 72;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 179 [arrowhead = none ];
179 -> 73;

180 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 180 [arrowhead = none ];
180 -> 81;

181 [label="SPLIT 2\nnew knowledge:\nT346 is ground\nT347 is ground\nT365 is ground\nreplacements:X425 -> T365", fontsize=14, style = filled, fillcolor = violet];
72 -> 181 [arrowhead = none ];
181 -> 82;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 182 [arrowhead = none ];
182 -> 74;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 183 [arrowhead = none ];
183 -> 75;

184 [label="EVAL with clause\napp([], X442, X442).\nand substitutionX423 -> [],\nX424 -> T354,\nX442 -> .(a, .(b, T354)),\nX443 -> T354,\nT341 -> .(a, .(b, T354))", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 184 [arrowhead = none ];
184 -> 76;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 185 [arrowhead = none ];
185 -> 77;

186 [label="EVAL with clause\napp(.(X458, X459), X460, .(X458, X461)) :- app(X459, X460, X461).\nand substitutionX458 -> T359,\nX459 -> X463,\nX423 -> .(T359, X463),\nX424 -> X464,\nX460 -> .(a, .(b, X464)),\nX462 -> T359,\nX461 -> T360,\nT341 -> .(T359, T360)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 186 [arrowhead = none ];
186 -> 79;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 187 [arrowhead = none ];
187 -> 80;

188 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 188 [arrowhead = none ];
188 -> 78;

189 [label="INSTANCE with matching:\nX423 -> X463\nX424 -> X464\nT341 -> T360", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 189 [arrowhead = none , style=dashed];
189 -> 71[style=dashed];

190 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 190 [arrowhead = none ];
190 -> 83;

191 [label="INSTANCE with matching:\nT1 -> T365\nT2 -> T343", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 191 [arrowhead = none , style=dashed];
191 -> 1[style=dashed];

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 192 [arrowhead = none ];
192 -> 84;

193 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 193 [arrowhead = none ];
193 -> 85;

194 [label="EVAL with clause\napp([], X477, X477).\nand substitutionT346 -> [],\nT347 -> T372,\nX477 -> .(s(a, b), T372),\nX425 -> .(s(a, b), T372)", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 194 [arrowhead = none ];
194 -> 86;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 195 [arrowhead = none ];
195 -> 87;

196 [label="EVAL with clause\napp(.(X488, X489), X490, .(X488, X491)) :- app(X489, X490, X491).\nand substitutionX488 -> T379,\nX489 -> T380,\nT346 -> .(T379, T380),\nT347 -> T381,\nX490 -> .(s(a, b), T381),\nX491 -> X492,\nX425 -> .(T379, X492)", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 196 [arrowhead = none ];
196 -> 89;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 197 [arrowhead = none ];
197 -> 90;

198 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 198 [arrowhead = none ];
198 -> 88;

199 [label="INSTANCE with matching:\nT346 -> T380\nT347 -> T381\nX425 -> X492", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 199 [arrowhead = none , style=dashed];
199 -> 81[style=dashed];

200 [label="EVAL with clause\nparse(.(s(X511, X512), []), s(X511, X512)).\nand substitutionX511 -> T394,\nX512 -> T395,\nT5 -> .(s(T394, T395), []),\nT2 -> s(T394, T395)", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 200 [arrowhead = none ];
200 -> 93;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 201 [arrowhead = none ];
201 -> 94;

202 [label="EVAL with clause\nparse(.(s(X519, X520, X521), []), s(X519, X520, X521)).\nand substitutionX519 -> T402,\nX520 -> T403,\nX521 -> T404,\nT5 -> .(s(T402, T403, T404), []),\nT2 -> s(T402, T403, T404)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 202 [arrowhead = none ];
202 -> 96;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 203 [arrowhead = none ];
203 -> 97;

204 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 204 [arrowhead = none ];
204 -> 95;

205 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 205 [arrowhead = none ];
205 -> 98;

}

(2) Obligation:

Triples:

appC(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) :- appC(X2, X3, X4, X5, X6, X7).
appI(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) :- appI(X2, X3, X4, X5, X6, X7).
appE(.(X1, X2), X3, X4, X5, .(X1, X6)) :- appE(X2, X3, X4, X5, X6).
appF(.(X1, X2), X3, X4, X5, .(X1, X6)) :- appF(X2, X3, X4, X5, X6).
appG(.(X1, X2), X3, .(X1, X4)) :- appG(X2, X3, X4).
appH(.(X1, X2), X3, .(X1, X4)) :- appH(X2, X3, X4).
parseA(.(a, .(s(X1, X2, X3), .(b, X4))), X5) :- ','(appcB(X1, X2, X3, X4, X6), parseA(X6, X5)).
parseA(.(X1, X2), X3) :- appC(X4, X5, X6, X7, X8, X2).
parseA(.(X1, X2), X3) :- ','(appcC(X4, X5, X6, X7, X8, X2), appI(X4, X5, X6, X7, X8, X9)).
parseA(.(X1, X2), X3) :- ','(appcC(X4, X5, X6, X7, X8, X2), ','(appcD(X1, X4, X5, X6, X7, X8, X9), parseA(X9, X3))).
parseA(X1, X2) :- appE(X3, X4, X5, X6, X1).
parseA(X1, X2) :- ','(appcE(X3, X4, X5, X6, X1), appF(X3, X4, X5, X6, X7)).
parseA(X1, X2) :- ','(appcE(X3, X4, X5, X6, X1), ','(appcF(X3, X4, X5, X6, X7), parseA(X7, X2))).
parseA(X1, X2) :- appG(X3, X4, X1).
parseA(X1, X2) :- ','(appcG(X3, X4, X1), appH(X3, X4, X5)).
parseA(X1, X2) :- ','(appcG(X3, X4, X1), ','(appcH(X3, X4, X5), parseA(X5, X2))).

Clauses:

parsecA(.(a, .(s(X1, X2, X3), .(b, X4))), X5) :- ','(appcB(X1, X2, X3, X4, X6), parsecA(X6, X5)).
parsecA(.(X1, X2), X3) :- ','(appcC(X4, X5, X6, X7, X8, X2), ','(appcD(X1, X4, X5, X6, X7, X8, X9), parsecA(X9, X3))).
parsecA(X1, X2) :- ','(appcE(X3, X4, X5, X6, X1), ','(appcF(X3, X4, X5, X6, X7), parsecA(X7, X2))).
parsecA(X1, X2) :- ','(appcG(X3, X4, X1), ','(appcH(X3, X4, X5), parsecA(X5, X2))).
parsecA(.(s(X1, X2), []), s(X1, X2)).
parsecA(.(s(X1, X2, X3), []), s(X1, X2, X3)).
appcC([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))).
appcC(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) :- appcC(X2, X3, X4, X5, X6, X7).
appcI([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)).
appcI(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) :- appcI(X2, X3, X4, X5, X6, X7).
appcE([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))).
appcE(.(X1, X2), X3, X4, X5, .(X1, X6)) :- appcE(X2, X3, X4, X5, X6).
appcF([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)).
appcF(.(X1, X2), X3, X4, X5, .(X1, X6)) :- appcF(X2, X3, X4, X5, X6).
appcG([], X1, .(a, .(b, X1))).
appcG(.(X1, X2), X3, .(X1, X4)) :- appcG(X2, X3, X4).
appcH([], X1, .(s(a, b), X1)).
appcH(.(X1, X2), X3, .(X1, X4)) :- appcH(X2, X3, X4).
appcB(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)).
appcD(X1, X2, X3, X4, X5, X6, .(X1, X7)) :- appcI(X2, X3, X4, X5, X6, X7).

Afs:

parseA(x1, x2)  =  parseA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
parseA_in: (b,f)
appC_in: (f,f,f,f,f,b)
appcC_in: (f,f,f,f,f,b)
appI_in: (b,b,b,b,b,f)
appcD_in: (b,b,b,b,b,b,f)
appcI_in: (b,b,b,b,b,f)
appE_in: (f,f,f,f,b)
appcE_in: (f,f,f,f,b)
appF_in: (b,b,b,b,f)
appcF_in: (b,b,b,b,f)
appG_in: (f,f,b)
appcG_in: (f,f,b)
appH_in: (b,b,f)
appcH_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4))), X5) → U7_GA(X1, X2, X3, X4, X5, appcB_in_gggga(X1, X2, X3, X4, X6))
U7_GA(X1, X2, X3, X4, X5, appcB_out_gggga(X1, X2, X3, X4, X6)) → U8_GA(X1, X2, X3, X4, X5, parseA_in_ga(X6, X5))
U7_GA(X1, X2, X3, X4, X5, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6, X5)
PARSEA_IN_GA(.(X1, X2), X3) → U9_GA(X1, X2, X3, appC_in_aaaaag(X4, X5, X6, X7, X8, X2))
PARSEA_IN_GA(.(X1, X2), X3) → APPC_IN_AAAAAG(X4, X5, X6, X7, X8, X2)
APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U1_AAAAAG(X1, X2, X3, X4, X5, X6, X7, appC_in_aaaaag(X2, X3, X4, X5, X6, X7))
APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPC_IN_AAAAAG(X2, X3, X4, X5, X6, X7)
PARSEA_IN_GA(.(X1, X2), X3) → U10_GA(X1, X2, X3, appcC_in_aaaaag(X4, X5, X6, X7, X8, X2))
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U11_GA(X1, X2, X3, appI_in_ggggga(X4, X5, X6, X7, X8, X9))
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → APPI_IN_GGGGGA(X4, X5, X6, X7, X8, X9)
APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U2_GGGGGA(X1, X2, X3, X4, X5, X6, X7, appI_in_ggggga(X2, X3, X4, X5, X6, X7))
APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6, X7)
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, X3, appcD_in_gggggga(X1, X4, X5, X6, X7, X8, X9))
U12_GA(X1, X2, X3, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → U13_GA(X1, X2, X3, parseA_in_ga(X9, X3))
U12_GA(X1, X2, X3, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9, X3)
PARSEA_IN_GA(X1, X2) → U14_GA(X1, X2, appE_in_aaaag(X3, X4, X5, X6, X1))
PARSEA_IN_GA(X1, X2) → APPE_IN_AAAAG(X3, X4, X5, X6, X1)
APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → U3_AAAAG(X1, X2, X3, X4, X5, X6, appE_in_aaaag(X2, X3, X4, X5, X6))
APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPE_IN_AAAAG(X2, X3, X4, X5, X6)
PARSEA_IN_GA(X1, X2) → U15_GA(X1, X2, appcE_in_aaaag(X3, X4, X5, X6, X1))
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U16_GA(X1, X2, appF_in_gggga(X3, X4, X5, X6, X7))
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → APPF_IN_GGGGA(X3, X4, X5, X6, X7)
APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → U4_GGGGA(X1, X2, X3, X4, X5, X6, appF_in_gggga(X2, X3, X4, X5, X6))
APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPF_IN_GGGGA(X2, X3, X4, X5, X6)
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, X2, appcF_in_gggga(X3, X4, X5, X6, X7))
U17_GA(X1, X2, appcF_out_gggga(X3, X4, X5, X6, X7)) → U18_GA(X1, X2, parseA_in_ga(X7, X2))
U17_GA(X1, X2, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7, X2)
PARSEA_IN_GA(X1, X2) → U19_GA(X1, X2, appG_in_aag(X3, X4, X1))
PARSEA_IN_GA(X1, X2) → APPG_IN_AAG(X3, X4, X1)
APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → U5_AAG(X1, X2, X3, X4, appG_in_aag(X2, X3, X4))
APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPG_IN_AAG(X2, X3, X4)
PARSEA_IN_GA(X1, X2) → U20_GA(X1, X2, appcG_in_aag(X3, X4, X1))
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → U21_GA(X1, X2, appH_in_gga(X3, X4, X5))
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → APPH_IN_GGA(X3, X4, X5)
APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U6_GGA(X1, X2, X3, X4, appH_in_gga(X2, X3, X4))
APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPH_IN_GGA(X2, X3, X4)
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, X2, appcH_in_gga(X3, X4, X5))
U22_GA(X1, X2, appcH_out_gga(X3, X4, X5)) → U23_GA(X1, X2, parseA_in_ga(X5, X2))
U22_GA(X1, X2, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5, X2)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appC_in_aaaaag(x6)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appI_in_ggggga(x1, x2, x3, x4, x5)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appE_in_aaaag(x1, x2, x3, x4, x5)  =  appE_in_aaaag(x5)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appF_in_gggga(x1, x2, x3, x4, x5)  =  appF_in_gggga(x1, x2, x3, x4)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appG_in_aag(x1, x2, x3)  =  appG_in_aag(x3)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appH_in_gga(x1, x2, x3)  =  appH_in_gga(x1, x2)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
PARSEA_IN_GA(x1, x2)  =  PARSEA_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, x6)  =  U8_GA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)
APPC_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPC_IN_AAAAAG(x6)
U1_AAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAAAG(x1, x7, x8)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
APPI_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPI_IN_GGGGGA(x1, x2, x3, x4, x5)
U2_GGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GGGGGA(x1, x2, x3, x4, x5, x6, x8)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x2, x4)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
APPE_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPE_IN_AAAAG(x5)
U3_AAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAAG(x1, x6, x7)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
APPF_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPF_IN_GGGGA(x1, x2, x3, x4)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x1, x2, x3, x4, x5, x7)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U18_GA(x1, x2, x3)  =  U18_GA(x1, x3)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
APPG_IN_AAG(x1, x2, x3)  =  APPG_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x1, x4, x5)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
APPH_IN_GGA(x1, x2, x3)  =  APPH_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4))), X5) → U7_GA(X1, X2, X3, X4, X5, appcB_in_gggga(X1, X2, X3, X4, X6))
U7_GA(X1, X2, X3, X4, X5, appcB_out_gggga(X1, X2, X3, X4, X6)) → U8_GA(X1, X2, X3, X4, X5, parseA_in_ga(X6, X5))
U7_GA(X1, X2, X3, X4, X5, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6, X5)
PARSEA_IN_GA(.(X1, X2), X3) → U9_GA(X1, X2, X3, appC_in_aaaaag(X4, X5, X6, X7, X8, X2))
PARSEA_IN_GA(.(X1, X2), X3) → APPC_IN_AAAAAG(X4, X5, X6, X7, X8, X2)
APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U1_AAAAAG(X1, X2, X3, X4, X5, X6, X7, appC_in_aaaaag(X2, X3, X4, X5, X6, X7))
APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPC_IN_AAAAAG(X2, X3, X4, X5, X6, X7)
PARSEA_IN_GA(.(X1, X2), X3) → U10_GA(X1, X2, X3, appcC_in_aaaaag(X4, X5, X6, X7, X8, X2))
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U11_GA(X1, X2, X3, appI_in_ggggga(X4, X5, X6, X7, X8, X9))
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → APPI_IN_GGGGGA(X4, X5, X6, X7, X8, X9)
APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U2_GGGGGA(X1, X2, X3, X4, X5, X6, X7, appI_in_ggggga(X2, X3, X4, X5, X6, X7))
APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6, X7)
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, X3, appcD_in_gggggga(X1, X4, X5, X6, X7, X8, X9))
U12_GA(X1, X2, X3, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → U13_GA(X1, X2, X3, parseA_in_ga(X9, X3))
U12_GA(X1, X2, X3, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9, X3)
PARSEA_IN_GA(X1, X2) → U14_GA(X1, X2, appE_in_aaaag(X3, X4, X5, X6, X1))
PARSEA_IN_GA(X1, X2) → APPE_IN_AAAAG(X3, X4, X5, X6, X1)
APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → U3_AAAAG(X1, X2, X3, X4, X5, X6, appE_in_aaaag(X2, X3, X4, X5, X6))
APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPE_IN_AAAAG(X2, X3, X4, X5, X6)
PARSEA_IN_GA(X1, X2) → U15_GA(X1, X2, appcE_in_aaaag(X3, X4, X5, X6, X1))
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U16_GA(X1, X2, appF_in_gggga(X3, X4, X5, X6, X7))
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → APPF_IN_GGGGA(X3, X4, X5, X6, X7)
APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → U4_GGGGA(X1, X2, X3, X4, X5, X6, appF_in_gggga(X2, X3, X4, X5, X6))
APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPF_IN_GGGGA(X2, X3, X4, X5, X6)
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, X2, appcF_in_gggga(X3, X4, X5, X6, X7))
U17_GA(X1, X2, appcF_out_gggga(X3, X4, X5, X6, X7)) → U18_GA(X1, X2, parseA_in_ga(X7, X2))
U17_GA(X1, X2, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7, X2)
PARSEA_IN_GA(X1, X2) → U19_GA(X1, X2, appG_in_aag(X3, X4, X1))
PARSEA_IN_GA(X1, X2) → APPG_IN_AAG(X3, X4, X1)
APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → U5_AAG(X1, X2, X3, X4, appG_in_aag(X2, X3, X4))
APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPG_IN_AAG(X2, X3, X4)
PARSEA_IN_GA(X1, X2) → U20_GA(X1, X2, appcG_in_aag(X3, X4, X1))
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → U21_GA(X1, X2, appH_in_gga(X3, X4, X5))
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → APPH_IN_GGA(X3, X4, X5)
APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U6_GGA(X1, X2, X3, X4, appH_in_gga(X2, X3, X4))
APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPH_IN_GGA(X2, X3, X4)
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, X2, appcH_in_gga(X3, X4, X5))
U22_GA(X1, X2, appcH_out_gga(X3, X4, X5)) → U23_GA(X1, X2, parseA_in_ga(X5, X2))
U22_GA(X1, X2, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5, X2)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
parseA_in_ga(x1, x2)  =  parseA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appC_in_aaaaag(x6)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appI_in_ggggga(x1, x2, x3, x4, x5)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appE_in_aaaag(x1, x2, x3, x4, x5)  =  appE_in_aaaag(x5)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appF_in_gggga(x1, x2, x3, x4, x5)  =  appF_in_gggga(x1, x2, x3, x4)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appG_in_aag(x1, x2, x3)  =  appG_in_aag(x3)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appH_in_gga(x1, x2, x3)  =  appH_in_gga(x1, x2)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
PARSEA_IN_GA(x1, x2)  =  PARSEA_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, x6)  =  U8_GA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)
APPC_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPC_IN_AAAAAG(x6)
U1_AAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAAAG(x1, x7, x8)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
APPI_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPI_IN_GGGGGA(x1, x2, x3, x4, x5)
U2_GGGGGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GGGGGA(x1, x2, x3, x4, x5, x6, x8)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x2, x4)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
APPE_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPE_IN_AAAAG(x5)
U3_AAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAAG(x1, x6, x7)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
APPF_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPF_IN_GGGGA(x1, x2, x3, x4)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x1, x2, x3, x4, x5, x7)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U18_GA(x1, x2, x3)  =  U18_GA(x1, x3)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
APPG_IN_AAG(x1, x2, x3)  =  APPG_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x1, x4, x5)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
APPH_IN_GGA(x1, x2, x3)  =  APPH_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPH_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPH_IN_GGA(x1, x2, x3)  =  APPH_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPH_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPH_IN_GGA(X2, X3, X4)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPH_IN_GGA(.(X1, X2), X3) → APPH_IN_GGA(X2, X3)

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

(12) 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:

• APPH_IN_GGA(.(X1, X2), X3) → APPH_IN_GGA(X2, X3)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPG_IN_AAG(X2, X3, X4)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPG_IN_AAG(x1, x2, x3)  =  APPG_IN_AAG(x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPG_IN_AAG(.(X1, X2), X3, .(X1, X4)) → APPG_IN_AAG(X2, X3, X4)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPG_IN_AAG(.(X1, X4)) → APPG_IN_AAG(X4)

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

(19) 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:

• APPG_IN_AAG(.(X1, X4)) → APPG_IN_AAG(X4)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPF_IN_GGGGA(X2, X3, X4, X5, X6)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPF_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPF_IN_GGGGA(x1, x2, x3, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

APPF_IN_GGGGA(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPF_IN_GGGGA(X2, X3, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPF_IN_GGGGA(x1, x2, x3, x4, x5)  =  APPF_IN_GGGGA(x1, x2, x3, 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:

APPF_IN_GGGGA(.(X1, X2), X3, X4, X5) → APPF_IN_GGGGA(X2, X3, X4, X5)

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

(26) 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:

• APPF_IN_GGGGA(.(X1, X2), X3, X4, X5) → APPF_IN_GGGGA(X2, X3, X4, X5)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

(27) YES

(28) Obligation:

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

APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPE_IN_AAAAG(X2, X3, X4, X5, X6)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPE_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPE_IN_AAAAG(x5)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

APPE_IN_AAAAG(.(X1, X2), X3, X4, X5, .(X1, X6)) → APPE_IN_AAAAG(X2, X3, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPE_IN_AAAAG(x1, x2, x3, x4, x5)  =  APPE_IN_AAAAG(x5)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

APPE_IN_AAAAG(.(X1, X6)) → APPE_IN_AAAAG(X6)

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

(33) 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:

• APPE_IN_AAAAG(.(X1, X6)) → APPE_IN_AAAAG(X6)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6, X7)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPI_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPI_IN_GGGGGA(x1, x2, x3, x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6, X7)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPI_IN_GGGGGA(x1, x2, x3, x4, x5, x6)  =  APPI_IN_GGGGGA(x1, x2, x3, x4, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6)

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

(40) 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:

• APPI_IN_GGGGGA(.(X1, X2), X3, X4, X5, X6) → APPI_IN_GGGGGA(X2, X3, X4, X5, X6)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5

(41) YES

(42) Obligation:

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

APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPC_IN_AAAAAG(X2, X3, X4, X5, X6, X7)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
APPC_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPC_IN_AAAAAG(x6)

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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) Obligation:

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

APPC_IN_AAAAAG(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → APPC_IN_AAAAAG(X2, X3, X4, X5, X6, X7)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPC_IN_AAAAAG(x1, x2, x3, x4, x5, x6)  =  APPC_IN_AAAAAG(x6)

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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) Obligation:

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

APPC_IN_AAAAAG(.(X1, X7)) → APPC_IN_AAAAAG(X7)

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

(47) 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:

• APPC_IN_AAAAAG(.(X1, X7)) → APPC_IN_AAAAAG(X7)
  The graph contains the following edges 1 > 1

(48) YES

(49) Obligation:

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

U7_GA(X1, X2, X3, X4, X5, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6, X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4))), X5) → U7_GA(X1, X2, X3, X4, X5, appcB_in_gggga(X1, X2, X3, X4, X6))
PARSEA_IN_GA(.(X1, X2), X3) → U10_GA(X1, X2, X3, appcC_in_aaaaag(X4, X5, X6, X7, X8, X2))
U10_GA(X1, X2, X3, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, X3, appcD_in_gggggga(X1, X4, X5, X6, X7, X8, X9))
U12_GA(X1, X2, X3, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9, X3)
PARSEA_IN_GA(X1, X2) → U15_GA(X1, X2, appcE_in_aaaag(X3, X4, X5, X6, X1))
U15_GA(X1, X2, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, X2, appcF_in_gggga(X3, X4, X5, X6, X7))
U17_GA(X1, X2, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7, X2)
PARSEA_IN_GA(X1, X2) → U20_GA(X1, X2, appcG_in_aag(X3, X4, X1))
U20_GA(X1, X2, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, X2, appcH_in_gga(X3, X4, X5))
U22_GA(X1, X2, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5, X2)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_in_aaaaag(X2, X3, X4, X5, X6, X7))
U36_aaaaag(X1, X2, X3, X4, X5, X6, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7)) → U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
appcI_in_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7)) → U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_in_ggggga(X2, X3, X4, X5, X6, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, X7, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6)) → U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_in_aaaag(X2, X3, X4, X5, X6))
U38_aaaag(X1, X2, X3, X4, X5, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3)) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5, .(X1, X6)) → U39_gggga(X1, X2, X3, X4, X5, X6, appcF_in_gggga(X2, X3, X4, X5, X6))
U39_gggga(X1, X2, X3, X4, X5, X6, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag([], X1, .(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X2), X3, .(X1, X4)) → U40_aag(X1, X2, X3, X4, appcG_in_aag(X2, X3, X4))
U40_aag(X1, X2, X3, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1, .(s(a, b), X1)) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3, .(X1, X4)) → U41_gga(X1, X2, X3, X4, appcH_in_gga(X2, X3, X4))
U41_gga(X1, X2, X3, X4, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
a  =  a
s(x1, x2, x3)  =  s(x1, x2, x3)
b  =  b
appcB_in_gggga(x1, x2, x3, x4, x5)  =  appcB_in_gggga(x1, x2, x3, x4)
appcB_out_gggga(x1, x2, x3, x4, x5)  =  appcB_out_gggga(x1, x2, x3, x4, x5)
appcC_in_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_in_aaaaag(x6)
appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)  =  appcC_out_aaaaag(x1, x2, x3, x4, x5, x6)
U36_aaaaag(x1, x2, x3, x4, x5, x6, x7, x8)  =  U36_aaaaag(x1, x7, x8)
appcD_in_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_in_gggggga(x1, x2, x3, x4, x5, x6)
U42_gggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_gggggga(x1, x2, x3, x4, x5, x6, x8)
appcI_in_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_in_ggggga(x1, x2, x3, x4, x5)
[]  =  []
appcI_out_ggggga(x1, x2, x3, x4, x5, x6)  =  appcI_out_ggggga(x1, x2, x3, x4, x5, x6)
U37_ggggga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U37_ggggga(x1, x2, x3, x4, x5, x6, x8)
appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)  =  appcD_out_gggggga(x1, x2, x3, x4, x5, x6, x7)
appcE_in_aaaag(x1, x2, x3, x4, x5)  =  appcE_in_aaaag(x5)
s(x1, x2)  =  s(x1, x2)
appcE_out_aaaag(x1, x2, x3, x4, x5)  =  appcE_out_aaaag(x1, x2, x3, x4, x5)
U38_aaaag(x1, x2, x3, x4, x5, x6, x7)  =  U38_aaaag(x1, x6, x7)
appcF_in_gggga(x1, x2, x3, x4, x5)  =  appcF_in_gggga(x1, x2, x3, x4)
appcF_out_gggga(x1, x2, x3, x4, x5)  =  appcF_out_gggga(x1, x2, x3, x4, x5)
U39_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U39_gggga(x1, x2, x3, x4, x5, x7)
appcG_in_aag(x1, x2, x3)  =  appcG_in_aag(x3)
appcG_out_aag(x1, x2, x3)  =  appcG_out_aag(x1, x2, x3)
U40_aag(x1, x2, x3, x4, x5)  =  U40_aag(x1, x4, x5)
appcH_in_gga(x1, x2, x3)  =  appcH_in_gga(x1, x2)
appcH_out_gga(x1, x2, x3)  =  appcH_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4, x5)  =  U41_gga(x1, x2, x3, x5)
PARSEA_IN_GA(x1, x2)  =  PARSEA_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x1, x2, x3, x4, x6)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)

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

(50) PiDPToQDPProof (SOUND transformation)

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

(51) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_in_gggga(X1, X2, X3, X4))
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, appcD_in_gggggga(X1, X4, X5, X6, X7, X8))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6) → U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gggga(x0, x1, x2, x3)
appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(52) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_in_gggga(X1, X2, X3, X4)) at position [4] we obtained the following new rules [LPAR04]:

PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))

(53) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, appcD_in_gggggga(X1, X4, X5, X6, X7, X8))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))

The TRS R consists of the following rules:

appcB_in_gggga(X1, X2, X3, X4) → appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6) → U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gggga(x0, x1, x2, x3)
appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(54) 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.

(55) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, appcD_in_gggggga(X1, X4, X5, X6, X7, X8))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6) → U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcB_in_gggga(x0, x1, x2, x3)
appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(56) 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].

appcB_in_gggga(x0, x1, x2, x3)

(57) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, appcD_in_gggggga(X1, X4, X5, X6, X7, X8))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6) → U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(58) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, appcD_in_gggggga(X1, X4, X5, X6, X7, X8)) at position [2] we obtained the following new rules [LPAR04]:

U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))

(59) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcD_in_gggggga(X1, X2, X3, X4, X5, X6) → U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(60) 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.

(61) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))

The TRS R consists of the following rules:

appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcD_in_gggggga(x0, x1, x2, x3, x4, x5)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(62) 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].

appcD_in_gggggga(x0, x1, x2, x3, x4, x5)

(63) Obligation:

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

U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6)
PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))

The TRS R consists of the following rules:

appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(64) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, X6)) → PARSEA_IN_GA(X6) we obtained the following new rules [LPAR04]:

U7_GA(z0, z1, z2, z3, appcB_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

(65) Obligation:

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

PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))
U7_GA(z0, z1, z2, z3, appcB_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

The TRS R consists of the following rules:

appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

PARSEA_IN_GA(.(X1, X2)) → U10_GA(X1, X2, appcC_in_aaaaag(X2))
U15_GA(X1, appcE_out_aaaag(X3, X4, X5, X6, X1)) → U17_GA(X1, appcF_in_gggga(X3, X4, X5, X6))
PARSEA_IN_GA(.(a, .(s(X1, X2, X3), .(b, X4)))) → U7_GA(X1, X2, X3, X4, appcB_out_gggga(X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4)))
U7_GA(z0, z1, z2, z3, appcB_out_gggga(z0, z1, z2, z3, .(s(a, s(z0, z1, z2), b), z3))) → PARSEA_IN_GA(.(s(a, s(z0, z1, z2), b), z3))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PARSEA_IN_GA(x₁)) = x₁   
POL(U10_GA(x₁, x₂, x₃)) = x₃   
POL(U12_GA(x₁, x₂, x₃)) = x₃   
POL(U15_GA(x₁, x₂)) = x₂   
POL(U17_GA(x₁, x₂)) = x₂   
POL(U20_GA(x₁, x₂)) = x₂   
POL(U22_GA(x₁, x₂)) = x₂   
POL(U36_aaaaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U37_ggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 1 + x₇   
POL(U38_aaaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U39_gggga(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₆   
POL(U40_aag(x₁, x₂, x₃)) = 1 + x₃   
POL(U41_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U42_gggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 1 + x₇   
POL(U7_GA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL([]) = 1   
POL(a) = 0   
POL(appcB_out_gggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(appcC_in_aaaaag(x₁)) = x₁   
POL(appcC_out_aaaaag(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₁ + x₅   
POL(appcD_out_gggggga(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = x₇   
POL(appcE_in_aaaag(x₁)) = x₁   
POL(appcE_out_aaaag(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₄   
POL(appcF_in_gggga(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL(appcF_out_gggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(appcG_in_aag(x₁)) = x₁   
POL(appcG_out_aag(x₁, x₂, x₃)) = x₁ + x₂   
POL(appcH_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appcH_out_gga(x₁, x₂, x₃)) = x₃   
POL(appcI_in_ggggga(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₅   
POL(appcI_out_ggggga(x₁, x₂, x₃, x₄, x₅, x₆)) = x₆   
POL(b) = 0   
POL(s(x₁, x₂)) = 0   
POL(s(x₁, x₂, x₃)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

(67) Obligation:

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

U12_GA(X1, X2, appcD_out_gggggga(X1, X4, X5, X6, X7, X8, X9)) → PARSEA_IN_GA(X9)
PARSEA_IN_GA(X1) → U15_GA(X1, appcE_in_aaaag(X1))
U17_GA(X1, appcF_out_gggga(X3, X4, X5, X6, X7)) → PARSEA_IN_GA(X7)
PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)
U10_GA(X1, X2, appcC_out_aaaaag(X4, X5, X6, X7, X8, X2)) → U12_GA(X1, X2, U42_gggggga(X1, X4, X5, X6, X7, X8, appcI_in_ggggga(X4, X5, X6, X7, X8)))

The TRS R consists of the following rules:

appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(68) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(69) Obligation:

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

PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)

The TRS R consists of the following rules:

appcI_in_ggggga([], X1, X2, X3, X4) → appcI_out_ggggga([], X1, X2, X3, X4, .(s(a, s(X1, X2, X3), b), X4))
appcI_in_ggggga(.(X1, X2), X3, X4, X5, X6) → U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_in_ggggga(X2, X3, X4, X5, X6))
U42_gggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcD_out_gggggga(X1, X2, X3, X4, X5, X6, .(X1, X7))
U37_ggggga(X1, X2, X3, X4, X5, X6, appcI_out_ggggga(X2, X3, X4, X5, X6, X7)) → appcI_out_ggggga(.(X1, X2), X3, X4, X5, X6, .(X1, X7))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))
appcF_in_gggga([], X1, X2, X3) → appcF_out_gggga([], X1, X2, X3, .(s(a, s(X1, X2), b), X3))
appcF_in_gggga(.(X1, X2), X3, X4, X5) → U39_gggga(X1, X2, X3, X4, X5, appcF_in_gggga(X2, X3, X4, X5))
U39_gggga(X1, X2, X3, X4, X5, appcF_out_gggga(X2, X3, X4, X5, X6)) → appcF_out_gggga(.(X1, X2), X3, X4, X5, .(X1, X6))
appcE_in_aaaag(.(a, .(s(X1, X2), .(b, X3)))) → appcE_out_aaaag([], X1, X2, X3, .(a, .(s(X1, X2), .(b, X3))))
appcE_in_aaaag(.(X1, X6)) → U38_aaaag(X1, X6, appcE_in_aaaag(X6))
U38_aaaag(X1, X6, appcE_out_aaaag(X2, X3, X4, X5, X6)) → appcE_out_aaaag(.(X1, X2), X3, X4, X5, .(X1, X6))
appcC_in_aaaaag(.(a, .(s(X1, X2, X3), .(b, X4)))) → appcC_out_aaaaag([], X1, X2, X3, X4, .(a, .(s(X1, X2, X3), .(b, X4))))
appcC_in_aaaaag(.(X1, X7)) → U36_aaaaag(X1, X7, appcC_in_aaaaag(X7))
U36_aaaaag(X1, X7, appcC_out_aaaaag(X2, X3, X4, X5, X6, X7)) → appcC_out_aaaaag(.(X1, X2), X3, X4, X5, X6, .(X1, X7))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(70) 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.

(71) Obligation:

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

PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)
appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(72) 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].

appcC_in_aaaaag(x0)
U36_aaaaag(x0, x1, x2)
appcI_in_ggggga(x0, x1, x2, x3, x4)
U37_ggggga(x0, x1, x2, x3, x4, x5, x6)
U42_gggggga(x0, x1, x2, x3, x4, x5, x6)
appcE_in_aaaag(x0)
U38_aaaag(x0, x1, x2)
appcF_in_gggga(x0, x1, x2, x3)
U39_gggga(x0, x1, x2, x3, x4, x5)

(73) Obligation:

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

PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U20_GA(X1, appcG_out_aag(X3, X4, X1)) → U22_GA(X1, appcH_in_gga(X3, X4))

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(PARSEA_IN_GA(x₁)) = 1 + x₁   
POL(U20_GA(x₁, x₂)) = 1 + x₂   
POL(U22_GA(x₁, x₂)) = x₂   
POL(U40_aag(x₁, x₂, x₃)) = x₁ + x₃   
POL(U41_gga(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL([]) = 1   
POL(a) = 0   
POL(appcG_in_aag(x₁)) = x₁   
POL(appcG_out_aag(x₁, x₂, x₃)) = x₁ + x₂   
POL(appcH_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appcH_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(b) = 1   
POL(s(x₁, x₂)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))

(75) Obligation:

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

PARSEA_IN_GA(X1) → U20_GA(X1, appcG_in_aag(X1))
U22_GA(X1, appcH_out_gga(X3, X4, X5)) → PARSEA_IN_GA(X5)

The TRS R consists of the following rules:

appcH_in_gga([], X1) → appcH_out_gga([], X1, .(s(a, b), X1))
appcH_in_gga(.(X1, X2), X3) → U41_gga(X1, X2, X3, appcH_in_gga(X2, X3))
U41_gga(X1, X2, X3, appcH_out_gga(X2, X3, X4)) → appcH_out_gga(.(X1, X2), X3, .(X1, X4))
appcG_in_aag(.(a, .(b, X1))) → appcG_out_aag([], X1, .(a, .(b, X1)))
appcG_in_aag(.(X1, X4)) → U40_aag(X1, X4, appcG_in_aag(X4))
U40_aag(X1, X4, appcG_out_aag(X2, X3, X4)) → appcG_out_aag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcG_in_aag(x0)
U40_aag(x0, x1, x2)
appcH_in_gga(x0, x1)
U41_gga(x0, x1, x2, x3)

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

(76) DependencyGraphProof (EQUIVALENT transformation)

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

(77) TRUE