Left Termination proof of /tmp/tmpIWoB7n/hidden

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 188 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 81 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 187 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 6 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 49 ms)

↳10 QDP

↳11 UsableRulesReductionPairsProof (⇔, 436 ms)

↳12 QDP

↳13 UsableRulesProof (⇔, 0 ms)

↳14 QDP

↳15 QReductionProof (⇔, 18 ms)

↳16 QDP

↳17 MRRProof (⇔, 204 ms)

↳18 QDP

↳19 DependencyGraphProof (⇔, 0 ms)

↳20 TRUE

(0) Obligation:

Clauses:

hidden_flatten([], L, L).
hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F))).
hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L).

Query: hidden_flatten(g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: hidden_flatten(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | hidden_flatten([], T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nhidden_flatten(T1, T2, T3) !=? hidden_flatten([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: hidden_flatten([], T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: hidden_flatten([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_1, ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17))) | hidden_flatten(.(.(T11, T12), T13), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nhidden_flatten(T1, T2, T3) !=? hidden_flatten([], X2, X2)\nhidden_flatten(T1, T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17))\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 0 | ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 1 | ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: hidden_flatten(.(T11, T12), T21, T22) | ','(hidden_flatten([], T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 1 | ','(hidden_flatten([], T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 1 | ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free\nX25 is free\nhidden_flatten(T13, T16, X22) !=? hidden_flatten([], X25, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: hidden_flatten(.(T11, T12), T21, T22)\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(hidden_flatten([], T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 1 | ','(hidden_flatten([], T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(hidden_flatten([], T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free\nX25 is free\nhidden_flatten(T13, T16, X22) !=? hidden_flatten([], X25, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(hidden_flatten(T13, T16, X22), hidden_flatten(.(T11, T12), X22, T17)), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground\nX22 is free\nX25 is free\nhidden_flatten(T13, T16, X22) !=? hidden_flatten([], X25, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(','(!_2, ','(hidden_flatten(T41, T43, X70), hidden_flatten(.(T39, T40), X70, X71))), hidden_flatten(.(T11, T12), X71, T44))\n\nT11 is ground\nT12 is ground\nT39 is ground\nT40 is ground\nT41 is ground\nX71 is free\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(hidden_flatten(T41, T43, X70), hidden_flatten(.(T39, T40), X70, X71)), hidden_flatten(.(T11, T12), X71, T44))\n\nT11 is ground\nT12 is ground\nT39 is ground\nT40 is ground\nT41 is ground\nX71 is free\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: hidden_flatten(T41, T43, X70)\n\nT41 is ground\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(hidden_flatten(.(T39, T40), T47, X71), hidden_flatten(.(T11, T12), X71, T48))\n\nT11 is ground\nT12 is ground\nT39 is ground\nT40 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 0 | hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 1 | hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 2\n\nT41 is ground\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | hidden_flatten([], T43, X70), SCOPE: 3, CLAUSE: 1 | hidden_flatten([], T43, X70), SCOPE: 3, CLAUSE: 2\n\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 1 | hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 2\n\nT41 is ground\nX70 is free\nX78 is free\nhidden_flatten(T41, T43, X70) !=? hidden_flatten([], X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: hidden_flatten([], T43, X70), SCOPE: 3, CLAUSE: 1 | hidden_flatten([], T43, X70), SCOPE: 3, CLAUSE: 2\n\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: hidden_flatten([], T43, X70), SCOPE: 3, CLAUSE: 2\n\nX70 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 1\n\nT41 is ground\nX70 is free\nX78 is free\nhidden_flatten(T41, T43, X70) !=? hidden_flatten([], X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: hidden_flatten(T41, T43, X70), SCOPE: 3, CLAUSE: 2\n\nT41 is ground\nX70 is free\nX78 is free\nhidden_flatten(T41, T43, X70) !=? hidden_flatten([], X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(!_3, ','(hidden_flatten(T72, T74, X119), hidden_flatten(.(T70, T71), X119, X120)))\n\nT70 is ground\nT71 is ground\nT72 is ground\nX120 is free\nX119 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(hidden_flatten(T72, T74, X119), hidden_flatten(.(T70, T71), X119, X120))\n\nT70 is ground\nT71 is ground\nT72 is ground\nX120 is free\nX119 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: hidden_flatten(T72, T74, X119)\n\nT72 is ground\nX119 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: hidden_flatten(.(T70, T71), T77, X120)\n\nT70 is ground\nT71 is ground\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: hidden_flatten(T87, T89, X141)\n\nT87 is ground\nX141 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(hidden_flatten(T99, T101, X160), hidden_flatten(.(T11, T12), .(T98, X160), T102))\n\nT11 is ground\nT12 is ground\nT98 is ground\nT99 is ground\nX160 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: hidden_flatten(T99, T101, X160)\n\nT99 is ground\nX160 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: hidden_flatten(.(T11, T12), .(T98, T105), T106)\n\nT11 is ground\nT12 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: hidden_flatten(T114, T117, T118)\n\nT113 is ground\nT114 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nhidden_flatten(.(T113, T114), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 0 | hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 1 | hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 2\n\nT113 is ground\nT114 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nhidden_flatten(.(T113, T114), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true | hidden_flatten([], T117, T118), SCOPE: 4, CLAUSE: 1 | hidden_flatten([], T117, T118), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 1 | hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 2\n\nT113 is ground\nT114 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nX177 is free\nhidden_flatten(.(T113, T114), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)\nhidden_flatten(T114, T117, T118) !=? hidden_flatten([], X177, X177)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: hidden_flatten([], T117, T118), SCOPE: 4, CLAUSE: 1 | hidden_flatten([], T117, T118), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: hidden_flatten([], T117, T118), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 1\n\nT113 is ground\nT114 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nX177 is free\nhidden_flatten(.(T113, T114), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)\nhidden_flatten(T114, T117, T118) !=? hidden_flatten([], X177, X177)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: hidden_flatten(T114, T117, T118), SCOPE: 4, CLAUSE: 2\n\nT113 is ground\nT114 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nX177 is free\nhidden_flatten(.(T113, T114), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)\nhidden_flatten(T114, T117, T118) !=? hidden_flatten([], X177, X177)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(!_4, ','(hidden_flatten(T144, T147, X214), hidden_flatten(.(T142, T143), X214, T148)))\n\nT113 is ground\nT142 is ground\nT143 is ground\nT144 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nX214 is free\nhidden_flatten(.(T113, .(.(T142, T143), T144)), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(hidden_flatten(T144, T147, X214), hidden_flatten(.(T142, T143), X214, T148))\n\nT113 is ground\nT142 is ground\nT143 is ground\nT144 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nX214 is free\nhidden_flatten(.(T113, .(.(T142, T143), T144)), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: hidden_flatten(T160, T163, T164)\n\nT113 is ground\nT159 is ground\nT160 is ground\nX17 is free\nX18 is free\nX19 is free\nX20 is free\nX21 is free\nhidden_flatten(.(T113, .(T159, T160)), T2, T3) !=? hidden_flatten(.(.(X17, X18), X19), X20, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 59 [arrowhead = none ];
59 -> 2;

60 [label="EVAL with clause\nhidden_flatten([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 60 [arrowhead = none ];
60 -> 3;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 61 [arrowhead = none ];
61 -> 4;

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 62 [arrowhead = none ];
62 -> 5;

63 [label="EVAL with clause\nhidden_flatten(.(.(X17, X18), X19), X20, X21) :- ','(!_1, ','(hidden_flatten(X19, X20, X22), hidden_flatten(.(X17, X18), X22, X21))).\nand substitutionX17 -> T11,\nX18 -> T12,\nX19 -> T13,\nT1 -> .(.(T11, T12), T13),\nT2 -> T16,\nX20 -> T16,\nT3 -> T17,\nX21 -> T17,\nT14 -> T16,\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 63 [arrowhead = none ];
63 -> 8;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 64 [arrowhead = none ];
64 -> 9;

65 [label="BACKTRACK\nfor clause: hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 65 [arrowhead = none ];
65 -> 6;

66 [label="BACKTRACK\nfor clause: hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 66 [arrowhead = none ];
66 -> 7;

67 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 67 [arrowhead = none ];
67 -> 10;

68 [label="EVAL with clause\nhidden_flatten(.(X171, X172), X173, .(X171, X174)) :- hidden_flatten(X172, X173, X174).\nand substitutionX171 -> T113,\nX172 -> T114,\nT1 -> .(T113, T114),\nT2 -> T117,\nX173 -> T117,\nX174 -> T118,\nT3 -> .(T113, T118),\nT115 -> T117,\nT116 -> T118", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 68 [arrowhead = none ];
68 -> 44;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 69 [arrowhead = none ];
69 -> 45;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 70 [arrowhead = none ];
70 -> 11;

71 [label="EVAL with clause\nhidden_flatten([], X25, X25).\nand substitutionT13 -> [],\nT16 -> T21,\nX25 -> T21,\nX22 -> T21,\nT20 -> T21,\nT17 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 71 [arrowhead = none ];
71 -> 12;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 72 [arrowhead = none ];
72 -> 13;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 74 [arrowhead = none ];
74 -> 15;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 75 [arrowhead = none ];
75 -> 18;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="INSTANCE with matching:\nT1 -> .(T11, T12)\nT2 -> T21\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 77 [arrowhead = none , style=dashed];
77 -> 1[style=dashed];

78 [label="BACKTRACK\nfor clause: hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 78 [arrowhead = none ];
78 -> 16;

79 [label="BACKTRACK\nfor clause: hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="EVAL with clause\nhidden_flatten(.(.(X65, X66), X67), X68, X69) :- ','(!_2, ','(hidden_flatten(X67, X68, X70), hidden_flatten(.(X65, X66), X70, X69))).\nand substitutionX65 -> T39,\nX66 -> T40,\nX67 -> T41,\nT13 -> .(.(T39, T40), T41),\nT16 -> T43,\nX68 -> T43,\nX22 -> X71,\nX69 -> X71,\nT42 -> T43,\nT17 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 80 [arrowhead = none ];
80 -> 20;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 81 [arrowhead = none ];
81 -> 21;

82 [label="EVAL with clause\nhidden_flatten(.(X156, X157), X158, .(X156, X159)) :- hidden_flatten(X157, X158, X159).\nand substitutionX156 -> T98,\nX157 -> T99,\nT13 -> .(T98, T99),\nT16 -> T101,\nX158 -> T101,\nX159 -> X160,\nX22 -> .(T98, X160),\nT100 -> T101,\nT17 -> T102", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 82 [arrowhead = none ];
82 -> 40;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 83 [arrowhead = none ];
83 -> 41;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 84 [arrowhead = none ];
84 -> 22;

85 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="SPLIT 2\nnew knowledge:\nT41 is ground\nreplacements:X70 -> T47,\nT44 -> T48", fontsize=14, style = filled, fillcolor = violet];
22 -> 86 [arrowhead = none ];
86 -> 24;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="INSTANCE with matching:\nT13 -> .(T39, T40)\nT16 -> T47\nX22 -> X71\nT17 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 88 [arrowhead = none , style=dashed];
88 -> 10[style=dashed];

89 [label="EVAL with clause\nhidden_flatten([], X78, X78).\nand substitutionT41 -> [],\nT43 -> T53,\nX78 -> T53,\nX70 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 89 [arrowhead = none ];
89 -> 26;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 90 [arrowhead = none ];
90 -> 27;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 91 [arrowhead = none ];
91 -> 28;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="BACKTRACK\nfor clause: hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 94 [arrowhead = none ];
94 -> 29;

95 [label="BACKTRACK\nfor clause: hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 95 [arrowhead = none ];
95 -> 30;

96 [label="EVAL with clause\nhidden_flatten(.(.(X114, X115), X116), X117, X118) :- ','(!_3, ','(hidden_flatten(X116, X117, X119), hidden_flatten(.(X114, X115), X119, X118))).\nand substitutionX114 -> T70,\nX115 -> T71,\nX116 -> T72,\nT41 -> .(.(T70, T71), T72),\nT43 -> T74,\nX117 -> T74,\nX70 -> X120,\nX118 -> X120,\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 96 [arrowhead = none ];
96 -> 33;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="EVAL with clause\nhidden_flatten(.(X137, X138), X139, .(X137, X140)) :- hidden_flatten(X138, X139, X140).\nand substitutionX137 -> T86,\nX138 -> T87,\nT41 -> .(T86, T87),\nT43 -> T89,\nX139 -> T89,\nX140 -> X141,\nX70 -> .(T86, X141),\nT88 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 98 [arrowhead = none ];
98 -> 38;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 100 [arrowhead = none ];
100 -> 35;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 101 [arrowhead = none ];
101 -> 36;

102 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nreplacements:X119 -> T77", fontsize=14, style = filled, fillcolor = violet];
35 -> 102 [arrowhead = none ];
102 -> 37;

103 [label="INSTANCE with matching:\nT41 -> T72\nT43 -> T74\nX70 -> X119", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 103 [arrowhead = none , style=dashed];
103 -> 23[style=dashed];

104 [label="INSTANCE with matching:\nT1 -> .(T70, T71)\nT2 -> T77\nT3 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 104 [arrowhead = none , style=dashed];
104 -> 1[style=dashed];

105 [label="INSTANCE with matching:\nT41 -> T87\nT43 -> T89\nX70 -> X141", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 105 [arrowhead = none , style=dashed];
105 -> 23[style=dashed];

106 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 106 [arrowhead = none ];
106 -> 42;

107 [label="SPLIT 2\nnew knowledge:\nT99 is ground\nreplacements:X160 -> T105,\nT102 -> T106", fontsize=14, style = filled, fillcolor = violet];
40 -> 107 [arrowhead = none ];
107 -> 43;

108 [label="INSTANCE with matching:\nT41 -> T99\nT43 -> T101\nX70 -> X160", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 108 [arrowhead = none , style=dashed];
108 -> 23[style=dashed];

109 [label="INSTANCE with matching:\nT1 -> .(T11, T12)\nT2 -> .(T98, T105)\nT3 -> T106", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 109 [arrowhead = none , style=dashed];
109 -> 1[style=dashed];

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 110 [arrowhead = none ];
110 -> 46;

111 [label="EVAL with clause\nhidden_flatten([], X177, X177).\nand substitutionT114 -> [],\nT117 -> T121,\nX177 -> T121,\nT118 -> T121", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 111 [arrowhead = none ];
111 -> 47;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 113 [arrowhead = none ];
113 -> 49;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 114 [arrowhead = none ];
114 -> 52;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 115 [arrowhead = none ];
115 -> 53;

116 [label="BACKTRACK\nfor clause: hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 116 [arrowhead = none ];
116 -> 50;

117 [label="BACKTRACK\nfor clause: hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 117 [arrowhead = none ];
117 -> 51;

118 [label="EVAL with clause\nhidden_flatten(.(.(X209, X210), X211), X212, X213) :- ','(!_4, ','(hidden_flatten(X211, X212, X214), hidden_flatten(.(X209, X210), X214, X213))).\nand substitutionX209 -> T142,\nX210 -> T143,\nX211 -> T144,\nT114 -> .(.(T142, T143), T144),\nT117 -> T147,\nX212 -> T147,\nT118 -> T148,\nX213 -> T148,\nT145 -> T147,\nT146 -> T148", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 118 [arrowhead = none ];
118 -> 54;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 119 [arrowhead = none ];
119 -> 55;

120 [label="EVAL with clause\nhidden_flatten(.(X225, X226), X227, .(X225, X228)) :- hidden_flatten(X226, X227, X228).\nand substitutionX225 -> T159,\nX226 -> T160,\nT114 -> .(T159, T160),\nT117 -> T163,\nX227 -> T163,\nX228 -> T164,\nT118 -> .(T159, T164),\nT161 -> T163,\nT162 -> T164", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 120 [arrowhead = none ];
120 -> 57;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 121 [arrowhead = none ];
121 -> 58;

122 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 122 [arrowhead = none ];
122 -> 56;

123 [label="INSTANCE with matching:\nT13 -> T144\nT16 -> T147\nX22 -> X214\nT11 -> T142\nT12 -> T143\nT17 -> T148", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 123 [arrowhead = none , style=dashed];
123 -> 10[style=dashed];

124 [label="INSTANCE with matching:\nT1 -> T160\nT2 -> T163\nT3 -> T164", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 124 [arrowhead = none , style=dashed];
124 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

pA([], T21, T21, T11, T12, T22) :- hidden_flattenB(.(T11, T12), T21, T22).
pA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) :- hidden_flattenC(T41, T43, X70).
pA(.(.(T39, T40), T41), T43, X71, T11, T12, T48) :- ','(hidden_flattenC(T41, T43, T47), pA(.(T39, T40), T47, X71, T11, T12, T48)).
pA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) :- hidden_flattenC(T99, T101, X160).
pA(.(T98, T99), T101, .(T98, T105), T11, T12, T106) :- ','(hidden_flattenC(T99, T101, T105), hidden_flattenB(.(T11, T12), .(T98, T105), T106)).
hidden_flattenC([], T53, T53).
hidden_flattenC(.(.(T70, T71), T72), T74, X120) :- hidden_flattenC(T72, T74, X119).
hidden_flattenC(.(.(T70, T71), T72), T74, X120) :- ','(hidden_flattenC(T72, T74, T77), hidden_flattenB(.(T70, T71), T77, X120)).
hidden_flattenC(.(T86, T87), T89, .(T86, X141)) :- hidden_flattenC(T87, T89, X141).
hidden_flattenB([], T5, T5).
hidden_flattenB(.(.(T11, T12), T13), T16, T17) :- pA(T13, T16, X22, T11, T12, T17).
hidden_flattenB(.(T113, []), T121, .(T113, T121)).
hidden_flattenB(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) :- pA(T144, T147, X214, T142, T143, T148).
hidden_flattenB(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) :- hidden_flattenB(T160, T163, T164).

Query: hidden_flattenB(g,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
hidden_flattenB_in: (b,f,f)
pA_in: (b,f,f,b,b,f)
hidden_flattenC_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

hidden_flattenB_in_gaa([], T5, T5) → hidden_flattenB_out_gaa([], T5, T5)
hidden_flattenB_in_gaa(.(.(T11, T12), T13), T16, T17) → U12_gaa(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
pA_in_gaagga([], T21, T21, T11, T12, T22) → U1_gaagga(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
hidden_flattenB_in_gaa(.(T113, []), T121, .(T113, T121)) → hidden_flattenB_out_gaa(.(T113, []), T121, .(T113, T121))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_gaa(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
hidden_flattenC_in_gaa([], T53, T53) → hidden_flattenC_out_gaa([], T53, T53)
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
hidden_flattenC_in_gaa(.(T86, T87), T89, .(T86, X141)) → U11_gaa(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
U11_gaa(T86, T87, T89, X141, hidden_flattenC_out_gaa(T87, T89, X141)) → hidden_flattenC_out_gaa(.(T86, T87), T89, .(T86, X141))
U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
hidden_flattenB_in_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_out_gaa(T160, T163, T164)) → hidden_flattenB_out_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164)))
U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_out_gaa(.(T70, T71), T77, X120)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, X119)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_out_gaa(T41, T43, X70)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44)
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
pA_in_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_out_gaa(T99, T101, X160)) → pA_out_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102)
pA_in_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_out_gaa(.(T11, T12), .(T98, T105), T106)) → pA_out_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106)
U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_out_gaagga(.(T39, T40), T47, X71, T11, T12, T48)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48)
U13_gaa(T113, T142, T143, T144, T147, T148, pA_out_gaagga(T144, T147, X214, T142, T143, T148)) → hidden_flattenB_out_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148))
U1_gaagga(T21, T11, T12, T22, hidden_flattenB_out_gaa(.(T11, T12), T21, T22)) → pA_out_gaagga([], T21, T21, T11, T12, T22)
U12_gaa(T11, T12, T13, T16, T17, pA_out_gaagga(T13, T16, X22, T11, T12, T17)) → hidden_flattenB_out_gaa(.(.(T11, T12), T13), T16, T17)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

hidden_flattenB_in_gaa([], T5, T5) → hidden_flattenB_out_gaa([], T5, T5)
hidden_flattenB_in_gaa(.(.(T11, T12), T13), T16, T17) → U12_gaa(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
pA_in_gaagga([], T21, T21, T11, T12, T22) → U1_gaagga(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
hidden_flattenB_in_gaa(.(T113, []), T121, .(T113, T121)) → hidden_flattenB_out_gaa(.(T113, []), T121, .(T113, T121))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_gaa(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
hidden_flattenC_in_gaa([], T53, T53) → hidden_flattenC_out_gaa([], T53, T53)
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
hidden_flattenC_in_gaa(.(T86, T87), T89, .(T86, X141)) → U11_gaa(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
U11_gaa(T86, T87, T89, X141, hidden_flattenC_out_gaa(T87, T89, X141)) → hidden_flattenC_out_gaa(.(T86, T87), T89, .(T86, X141))
U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
hidden_flattenB_in_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_out_gaa(T160, T163, T164)) → hidden_flattenB_out_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164)))
U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_out_gaa(.(T70, T71), T77, X120)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, X119)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_out_gaa(T41, T43, X70)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44)
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
pA_in_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_out_gaa(T99, T101, X160)) → pA_out_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102)
pA_in_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_out_gaa(.(T11, T12), .(T98, T105), T106)) → pA_out_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106)
U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_out_gaagga(.(T39, T40), T47, X71, T11, T12, T48)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48)
U13_gaa(T113, T142, T143, T144, T147, T148, pA_out_gaagga(T144, T147, X214, T142, T143, T148)) → hidden_flattenB_out_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148))
U1_gaagga(T21, T11, T12, T22, hidden_flattenB_out_gaa(.(T11, T12), T21, T22)) → pA_out_gaagga([], T21, T21, T11, T12, T22)
U12_gaa(T11, T12, T13, T16, T17, pA_out_gaagga(T13, T16, X22, T11, T12, T17)) → hidden_flattenB_out_gaa(.(.(T11, T12), T13), T16, T17)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13), T16, T17) → U12_GAA(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13), T16, T17) → PA_IN_GAAGGA(T13, T16, X22, T11, T12, T17)
PA_IN_GAAGGA([], T21, T21, T11, T12, T22) → U1_GAAGGA(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
PA_IN_GAAGGA([], T21, T21, T11, T12, T22) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), T21, T22)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_GAA(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → PA_IN_GAAGGA(T144, T147, X214, T142, T143, T148)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → HIDDEN_FLATTENC_IN_GAA(T41, T43, X70)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → U8_GAA(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → HIDDEN_FLATTENC_IN_GAA(T72, T74, X119)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
HIDDEN_FLATTENC_IN_GAA(.(T86, T87), T89, .(T86, X141)) → U11_GAA(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
HIDDEN_FLATTENC_IN_GAA(.(T86, T87), T89, .(T86, X141)) → HIDDEN_FLATTENC_IN_GAA(T87, T89, X141)
U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_GAA(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71), T77, X120)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_GAA(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → HIDDEN_FLATTENB_IN_GAA(T160, T163, T164)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → PA_IN_GAAGGA(.(T39, T40), T47, X71, T11, T12, T48)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_GAAGGA(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → HIDDEN_FLATTENC_IN_GAA(T99, T101, X160)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), .(T98, T105), T106)

The TRS R consists of the following rules:

hidden_flattenB_in_gaa([], T5, T5) → hidden_flattenB_out_gaa([], T5, T5)
hidden_flattenB_in_gaa(.(.(T11, T12), T13), T16, T17) → U12_gaa(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
pA_in_gaagga([], T21, T21, T11, T12, T22) → U1_gaagga(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
hidden_flattenB_in_gaa(.(T113, []), T121, .(T113, T121)) → hidden_flattenB_out_gaa(.(T113, []), T121, .(T113, T121))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_gaa(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
hidden_flattenC_in_gaa([], T53, T53) → hidden_flattenC_out_gaa([], T53, T53)
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
hidden_flattenC_in_gaa(.(T86, T87), T89, .(T86, X141)) → U11_gaa(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
U11_gaa(T86, T87, T89, X141, hidden_flattenC_out_gaa(T87, T89, X141)) → hidden_flattenC_out_gaa(.(T86, T87), T89, .(T86, X141))
U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
hidden_flattenB_in_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_out_gaa(T160, T163, T164)) → hidden_flattenB_out_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164)))
U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_out_gaa(.(T70, T71), T77, X120)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, X119)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_out_gaa(T41, T43, X70)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44)
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
pA_in_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_out_gaa(T99, T101, X160)) → pA_out_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102)
pA_in_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_out_gaa(.(T11, T12), .(T98, T105), T106)) → pA_out_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106)
U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_out_gaagga(.(T39, T40), T47, X71, T11, T12, T48)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48)
U13_gaa(T113, T142, T143, T144, T147, T148, pA_out_gaagga(T144, T147, X214, T142, T143, T148)) → hidden_flattenB_out_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148))
U1_gaagga(T21, T11, T12, T22, hidden_flattenB_out_gaa(.(T11, T12), T21, T22)) → pA_out_gaagga([], T21, T21, T11, T12, T22)
U12_gaa(T11, T12, T13, T16, T17, pA_out_gaagga(T13, T16, X22, T11, T12, T17)) → hidden_flattenB_out_gaa(.(.(T11, T12), T13), T16, T17)

The argument filtering Pi contains the following mapping:
hidden_flattenB_in_gaa(x1, x2, x3) = hidden_flattenB_in_gaa(x1)
[] = []
hidden_flattenB_out_gaa(x1, x2, x3) = hidden_flattenB_out_gaa
.(x1, x2) = .(x1, x2)
U12_gaa(x1, x2, x3, x4, x5, x6) = U12_gaa(x6)
pA_in_gaagga(x1, x2, x3, x4, x5, x6) = pA_in_gaagga(x1, x4, x5)
U1_gaagga(x1, x2, x3, x4, x5) = U1_gaagga(x5)
U13_gaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaa(x7)
U2_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U2_gaagga(x9)
hidden_flattenC_in_gaa(x1, x2, x3) = hidden_flattenC_in_gaa(x1)
hidden_flattenC_out_gaa(x1, x2, x3) = hidden_flattenC_out_gaa
U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x6)
U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x6)
U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x5)
U10_gaa(x1, x2, x3, x4, x5, x6) = U10_gaa(x6)
U14_gaa(x1, x2, x3, x4, x5, x6) = U14_gaa(x6)
pA_out_gaagga(x1, x2, x3, x4, x5, x6) = pA_out_gaagga
U3_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_gaagga(x1, x2, x6, x7, x9)
U4_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_gaagga(x9)
U5_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaagga(x8)
U6_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U6_gaagga(x5, x6, x8)
U7_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaagga(x8)
HIDDEN_FLATTENB_IN_GAA(x1, x2, x3) = HIDDEN_FLATTENB_IN_GAA(x1)
U12_GAA(x1, x2, x3, x4, x5, x6) = U12_GAA(x6)
PA_IN_GAAGGA(x1, x2, x3, x4, x5, x6) = PA_IN_GAAGGA(x1, x4, x5)
U1_GAAGGA(x1, x2, x3, x4, x5) = U1_GAAGGA(x5)
U13_GAA(x1, x2, x3, x4, x5, x6, x7) = U13_GAA(x7)
U2_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U2_GAAGGA(x9)
HIDDEN_FLATTENC_IN_GAA(x1, x2, x3) = HIDDEN_FLATTENC_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x6)
U9_GAA(x1, x2, x3, x4, x5, x6) = U9_GAA(x1, x2, x6)
U11_GAA(x1, x2, x3, x4, x5) = U11_GAA(x5)
U10_GAA(x1, x2, x3, x4, x5, x6) = U10_GAA(x6)
U14_GAA(x1, x2, x3, x4, x5, x6) = U14_GAA(x6)
U3_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAGGA(x1, x2, x6, x7, x9)
U4_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_GAAGGA(x9)
U5_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GAAGGA(x8)
U6_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GAAGGA(x5, x6, x8)
U7_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GAAGGA(x8)

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

(6) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13), T16, T17) → U12_GAA(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13), T16, T17) → PA_IN_GAAGGA(T13, T16, X22, T11, T12, T17)
PA_IN_GAAGGA([], T21, T21, T11, T12, T22) → U1_GAAGGA(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
PA_IN_GAAGGA([], T21, T21, T11, T12, T22) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), T21, T22)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_GAA(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → PA_IN_GAAGGA(T144, T147, X214, T142, T143, T148)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → HIDDEN_FLATTENC_IN_GAA(T41, T43, X70)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → U8_GAA(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → HIDDEN_FLATTENC_IN_GAA(T72, T74, X119)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
HIDDEN_FLATTENC_IN_GAA(.(T86, T87), T89, .(T86, X141)) → U11_GAA(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
HIDDEN_FLATTENC_IN_GAA(.(T86, T87), T89, .(T86, X141)) → HIDDEN_FLATTENC_IN_GAA(T87, T89, X141)
U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_GAA(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71), T77, X120)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_GAA(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → HIDDEN_FLATTENB_IN_GAA(T160, T163, T164)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → PA_IN_GAAGGA(.(T39, T40), T47, X71, T11, T12, T48)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_GAAGGA(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → HIDDEN_FLATTENC_IN_GAA(T99, T101, X160)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), .(T98, T105), T106)

The TRS R consists of the following rules:

hidden_flattenB_in_gaa([], T5, T5) → hidden_flattenB_out_gaa([], T5, T5)
hidden_flattenB_in_gaa(.(.(T11, T12), T13), T16, T17) → U12_gaa(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
pA_in_gaagga([], T21, T21, T11, T12, T22) → U1_gaagga(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
hidden_flattenB_in_gaa(.(T113, []), T121, .(T113, T121)) → hidden_flattenB_out_gaa(.(T113, []), T121, .(T113, T121))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_gaa(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
hidden_flattenC_in_gaa([], T53, T53) → hidden_flattenC_out_gaa([], T53, T53)
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
hidden_flattenC_in_gaa(.(T86, T87), T89, .(T86, X141)) → U11_gaa(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
U11_gaa(T86, T87, T89, X141, hidden_flattenC_out_gaa(T87, T89, X141)) → hidden_flattenC_out_gaa(.(T86, T87), T89, .(T86, X141))
U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
hidden_flattenB_in_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_out_gaa(T160, T163, T164)) → hidden_flattenB_out_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164)))
U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_out_gaa(.(T70, T71), T77, X120)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, X119)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_out_gaa(T41, T43, X70)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44)
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
pA_in_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_out_gaa(T99, T101, X160)) → pA_out_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102)
pA_in_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_out_gaa(.(T11, T12), .(T98, T105), T106)) → pA_out_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106)
U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_out_gaagga(.(T39, T40), T47, X71, T11, T12, T48)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48)
U13_gaa(T113, T142, T143, T144, T147, T148, pA_out_gaagga(T144, T147, X214, T142, T143, T148)) → hidden_flattenB_out_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148))
U1_gaagga(T21, T11, T12, T22, hidden_flattenB_out_gaa(.(T11, T12), T21, T22)) → pA_out_gaagga([], T21, T21, T11, T12, T22)
U12_gaa(T11, T12, T13, T16, T17, pA_out_gaagga(T13, T16, X22, T11, T12, T17)) → hidden_flattenB_out_gaa(.(.(T11, T12), T13), T16, T17)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 11 less nodes.

(8) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13), T16, T17) → PA_IN_GAAGGA(T13, T16, X22, T11, T12, T17)
PA_IN_GAAGGA([], T21, T21, T11, T12, T22) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), T21, T22)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → PA_IN_GAAGGA(T144, T147, X214, T142, T143, T148)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → HIDDEN_FLATTENC_IN_GAA(T41, T43, X70)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → HIDDEN_FLATTENC_IN_GAA(T72, T74, X119)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72), T74, X120) → U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
U9_GAA(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71), T77, X120)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → HIDDEN_FLATTENB_IN_GAA(T160, T163, T164)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87), T89, .(T86, X141)) → HIDDEN_FLATTENC_IN_GAA(T87, T89, X141)
PA_IN_GAAGGA(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_GAAGGA(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → PA_IN_GAAGGA(.(T39, T40), T47, X71, T11, T12, T48)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → HIDDEN_FLATTENC_IN_GAA(T99, T101, X160)
PA_IN_GAAGGA(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_GAAGGA(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12), .(T98, T105), T106)

The TRS R consists of the following rules:

hidden_flattenB_in_gaa([], T5, T5) → hidden_flattenB_out_gaa([], T5, T5)
hidden_flattenB_in_gaa(.(.(T11, T12), T13), T16, T17) → U12_gaa(T11, T12, T13, T16, T17, pA_in_gaagga(T13, T16, X22, T11, T12, T17))
pA_in_gaagga([], T21, T21, T11, T12, T22) → U1_gaagga(T21, T11, T12, T22, hidden_flattenB_in_gaa(.(T11, T12), T21, T22))
hidden_flattenB_in_gaa(.(T113, []), T121, .(T113, T121)) → hidden_flattenB_out_gaa(.(T113, []), T121, .(T113, T121))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148)) → U13_gaa(T113, T142, T143, T144, T147, T148, pA_in_gaagga(T144, T147, X214, T142, T143, T148))
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44) → U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_in_gaa(T41, T43, X70))
hidden_flattenC_in_gaa([], T53, T53) → hidden_flattenC_out_gaa([], T53, T53)
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, X119))
hidden_flattenC_in_gaa(.(.(T70, T71), T72), T74, X120) → U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_in_gaa(T72, T74, T77))
hidden_flattenC_in_gaa(.(T86, T87), T89, .(T86, X141)) → U11_gaa(T86, T87, T89, X141, hidden_flattenC_in_gaa(T87, T89, X141))
U11_gaa(T86, T87, T89, X141, hidden_flattenC_out_gaa(T87, T89, X141)) → hidden_flattenC_out_gaa(.(T86, T87), T89, .(T86, X141))
U9_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, T77)) → U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_in_gaa(.(T70, T71), T77, X120))
hidden_flattenB_in_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164))) → U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_in_gaa(T160, T163, T164))
U14_gaa(T113, T159, T160, T163, T164, hidden_flattenB_out_gaa(T160, T163, T164)) → hidden_flattenB_out_gaa(.(T113, .(T159, T160)), T163, .(T113, .(T159, T164)))
U10_gaa(T70, T71, T72, T74, X120, hidden_flattenB_out_gaa(.(T70, T71), T77, X120)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U8_gaa(T70, T71, T72, T74, X120, hidden_flattenC_out_gaa(T72, T74, X119)) → hidden_flattenC_out_gaa(.(.(T70, T71), T72), T74, X120)
U2_gaagga(T39, T40, T41, T43, X71, T11, T12, T44, hidden_flattenC_out_gaa(T41, T43, X70)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T44)
pA_in_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48) → U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_in_gaa(T41, T43, T47))
U3_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, hidden_flattenC_out_gaa(T41, T43, T47)) → U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_in_gaagga(.(T39, T40), T47, X71, T11, T12, T48))
pA_in_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102) → U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_in_gaa(T99, T101, X160))
U5_gaagga(T98, T99, T101, X160, T11, T12, T102, hidden_flattenC_out_gaa(T99, T101, X160)) → pA_out_gaagga(.(T98, T99), T101, .(T98, X160), T11, T12, T102)
pA_in_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106) → U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_in_gaa(T99, T101, T105))
U6_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenC_out_gaa(T99, T101, T105)) → U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_in_gaa(.(T11, T12), .(T98, T105), T106))
U7_gaagga(T98, T99, T101, T105, T11, T12, T106, hidden_flattenB_out_gaa(.(T11, T12), .(T98, T105), T106)) → pA_out_gaagga(.(T98, T99), T101, .(T98, T105), T11, T12, T106)
U4_gaagga(T39, T40, T41, T43, X71, T11, T12, T48, pA_out_gaagga(.(T39, T40), T47, X71, T11, T12, T48)) → pA_out_gaagga(.(.(T39, T40), T41), T43, X71, T11, T12, T48)
U13_gaa(T113, T142, T143, T144, T147, T148, pA_out_gaagga(T144, T147, X214, T142, T143, T148)) → hidden_flattenB_out_gaa(.(T113, .(.(T142, T143), T144)), T147, .(T113, T148))
U1_gaagga(T21, T11, T12, T22, hidden_flattenB_out_gaa(.(T11, T12), T21, T22)) → pA_out_gaagga([], T21, T21, T11, T12, T22)
U12_gaa(T11, T12, T13, T16, T17, pA_out_gaagga(T13, T16, X22, T11, T12, T17)) → hidden_flattenB_out_gaa(.(.(T11, T12), T13), T16, T17)

The argument filtering Pi contains the following mapping:
hidden_flattenB_in_gaa(x1, x2, x3) = hidden_flattenB_in_gaa(x1)
[] = []
hidden_flattenB_out_gaa(x1, x2, x3) = hidden_flattenB_out_gaa
.(x1, x2) = .(x1, x2)
U12_gaa(x1, x2, x3, x4, x5, x6) = U12_gaa(x6)
pA_in_gaagga(x1, x2, x3, x4, x5, x6) = pA_in_gaagga(x1, x4, x5)
U1_gaagga(x1, x2, x3, x4, x5) = U1_gaagga(x5)
U13_gaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaa(x7)
U2_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U2_gaagga(x9)
hidden_flattenC_in_gaa(x1, x2, x3) = hidden_flattenC_in_gaa(x1)
hidden_flattenC_out_gaa(x1, x2, x3) = hidden_flattenC_out_gaa
U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x6)
U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x6)
U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x5)
U10_gaa(x1, x2, x3, x4, x5, x6) = U10_gaa(x6)
U14_gaa(x1, x2, x3, x4, x5, x6) = U14_gaa(x6)
pA_out_gaagga(x1, x2, x3, x4, x5, x6) = pA_out_gaagga
U3_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_gaagga(x1, x2, x6, x7, x9)
U4_gaagga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_gaagga(x9)
U5_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U5_gaagga(x8)
U6_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U6_gaagga(x5, x6, x8)
U7_gaagga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaagga(x8)
HIDDEN_FLATTENB_IN_GAA(x1, x2, x3) = HIDDEN_FLATTENB_IN_GAA(x1)
PA_IN_GAAGGA(x1, x2, x3, x4, x5, x6) = PA_IN_GAAGGA(x1, x4, x5)
HIDDEN_FLATTENC_IN_GAA(x1, x2, x3) = HIDDEN_FLATTENC_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5, x6) = U9_GAA(x1, x2, x6)
U3_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAGGA(x1, x2, x6, x7, x9)
U6_GAAGGA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GAAGGA(x5, x6, x8)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13)) → PA_IN_GAAGGA(T13, T11, T12)
PA_IN_GAAGGA([], T11, T12) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144))) → PA_IN_GAAGGA(T144, T142, T143)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T41)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → HIDDEN_FLATTENC_IN_GAA(T72)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → U9_GAA(T70, T71, hidden_flattenC_in_gaa(T72))
U9_GAA(T70, T71, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160))) → HIDDEN_FLATTENB_IN_GAA(T160)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87)) → HIDDEN_FLATTENC_IN_GAA(T87)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_out_gaa) → PA_IN_GAAGGA(.(T39, T40), T11, T12)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T99)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → U6_GAAGGA(T11, T12, hidden_flattenC_in_gaa(T99))
U6_GAAGGA(T11, T12, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

The TRS R consists of the following rules:

hidden_flattenB_in_gaa([]) → hidden_flattenB_out_gaa
hidden_flattenB_in_gaa(.(.(T11, T12), T13)) → U12_gaa(pA_in_gaagga(T13, T11, T12))
pA_in_gaagga([], T11, T12) → U1_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
hidden_flattenB_in_gaa(.(T113, [])) → hidden_flattenB_out_gaa
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144))) → U13_gaa(pA_in_gaagga(T144, T142, T143))
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U2_gaagga(hidden_flattenC_in_gaa(T41))
hidden_flattenC_in_gaa([]) → hidden_flattenC_out_gaa
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U8_gaa(hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U9_gaa(T70, T71, hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(T86, T87)) → U11_gaa(hidden_flattenC_in_gaa(T87))
U11_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
U9_gaa(T70, T71, hidden_flattenC_out_gaa) → U10_gaa(hidden_flattenB_in_gaa(.(T70, T71)))
hidden_flattenB_in_gaa(.(T113, .(T159, T160))) → U14_gaa(hidden_flattenB_in_gaa(T160))
U14_gaa(hidden_flattenB_out_gaa) → hidden_flattenB_out_gaa
U10_gaa(hidden_flattenB_out_gaa) → hidden_flattenC_out_gaa
U8_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
U2_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U3_gaagga(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
U3_gaagga(T39, T40, T11, T12, hidden_flattenC_out_gaa) → U4_gaagga(pA_in_gaagga(.(T39, T40), T11, T12))
pA_in_gaagga(.(T98, T99), T11, T12) → U5_gaagga(hidden_flattenC_in_gaa(T99))
U5_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
pA_in_gaagga(.(T98, T99), T11, T12) → U6_gaagga(T11, T12, hidden_flattenC_in_gaa(T99))
U6_gaagga(T11, T12, hidden_flattenC_out_gaa) → U7_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
U7_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U4_gaagga(pA_out_gaagga) → pA_out_gaagga
U13_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U1_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U12_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa

The set Q consists of the following terms:

hidden_flattenB_in_gaa(x0)
pA_in_gaagga(x0, x1, x2)
hidden_flattenC_in_gaa(x0)
U11_gaa(x0)
U9_gaa(x0, x1, x2)
U14_gaa(x0)
U10_gaa(x0)
U8_gaa(x0)
U2_gaagga(x0)
U3_gaagga(x0, x1, x2, x3, x4)
U5_gaagga(x0)
U6_gaagga(x0, x1, x2)
U7_gaagga(x0)
U4_gaagga(x0)
U13_gaa(x0)
U1_gaagga(x0)
U12_gaa(x0)

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

(11) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

PA_IN_GAAGGA([], T11, T12) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

The following rules are removed from R:

hidden_flattenB_in_gaa([]) → hidden_flattenB_out_gaa
pA_in_gaagga([], T11, T12) → U1_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
hidden_flattenB_in_gaa(.(T113, [])) → hidden_flattenB_out_gaa
hidden_flattenC_in_gaa([]) → hidden_flattenC_out_gaa

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(HIDDEN_FLATTENB_IN_GAA(x₁)) = x₁
POL(HIDDEN_FLATTENC_IN_GAA(x₁)) = 2·x₁
POL(PA_IN_GAAGGA(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U10_gaa(x₁)) = x₁
POL(U11_gaa(x₁)) = x₁
POL(U12_gaa(x₁)) = x₁
POL(U13_gaa(x₁)) = x₁
POL(U14_gaa(x₁)) = x₁
POL(U1_gaagga(x₁)) = x₁
POL(U2_gaagga(x₁)) = x₁
POL(U3_GAAGGA(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U3_gaagga(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U4_gaagga(x₁)) = x₁
POL(U5_gaagga(x₁)) = x₁
POL(U6_GAAGGA(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U6_gaagga(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U7_gaagga(x₁)) = x₁
POL(U8_gaa(x₁)) = x₁
POL(U9_GAA(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U9_gaa(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL([]) = 0
POL(hidden_flattenB_in_gaa(x₁)) = x₁
POL(hidden_flattenB_out_gaa) = 0
POL(hidden_flattenC_in_gaa(x₁)) = 2·x₁
POL(hidden_flattenC_out_gaa) = 0
POL(pA_in_gaagga(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(pA_out_gaagga) = 0

(12) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13)) → PA_IN_GAAGGA(T13, T11, T12)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144))) → PA_IN_GAAGGA(T144, T142, T143)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T41)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → HIDDEN_FLATTENC_IN_GAA(T72)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → U9_GAA(T70, T71, hidden_flattenC_in_gaa(T72))
U9_GAA(T70, T71, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160))) → HIDDEN_FLATTENB_IN_GAA(T160)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87)) → HIDDEN_FLATTENC_IN_GAA(T87)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_out_gaa) → PA_IN_GAAGGA(.(T39, T40), T11, T12)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T99)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → U6_GAAGGA(T11, T12, hidden_flattenC_in_gaa(T99))
U6_GAAGGA(T11, T12, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

The TRS R consists of the following rules:

hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U8_gaa(hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U9_gaa(T70, T71, hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(T86, T87)) → U11_gaa(hidden_flattenC_in_gaa(T87))
U11_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
U9_gaa(T70, T71, hidden_flattenC_out_gaa) → U10_gaa(hidden_flattenB_in_gaa(.(T70, T71)))
hidden_flattenB_in_gaa(.(.(T11, T12), T13)) → U12_gaa(pA_in_gaagga(T13, T11, T12))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144))) → U13_gaa(pA_in_gaagga(T144, T142, T143))
hidden_flattenB_in_gaa(.(T113, .(T159, T160))) → U14_gaa(hidden_flattenB_in_gaa(T160))
U10_gaa(hidden_flattenB_out_gaa) → hidden_flattenC_out_gaa
U14_gaa(hidden_flattenB_out_gaa) → hidden_flattenB_out_gaa
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U2_gaagga(hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U3_gaagga(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(T98, T99), T11, T12) → U5_gaagga(hidden_flattenC_in_gaa(T99))
pA_in_gaagga(.(T98, T99), T11, T12) → U6_gaagga(T11, T12, hidden_flattenC_in_gaa(T99))
U13_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U6_gaagga(T11, T12, hidden_flattenC_out_gaa) → U7_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
U7_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U5_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U3_gaagga(T39, T40, T11, T12, hidden_flattenC_out_gaa) → U4_gaagga(pA_in_gaagga(.(T39, T40), T11, T12))
U4_gaagga(pA_out_gaagga) → pA_out_gaagga
U2_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U1_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U12_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U8_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa

The set Q consists of the following terms:

hidden_flattenB_in_gaa(x0)
pA_in_gaagga(x0, x1, x2)
hidden_flattenC_in_gaa(x0)
U11_gaa(x0)
U9_gaa(x0, x1, x2)
U14_gaa(x0)
U10_gaa(x0)
U8_gaa(x0)
U2_gaagga(x0)
U3_gaagga(x0, x1, x2, x3, x4)
U5_gaagga(x0)
U6_gaagga(x0, x1, x2)
U7_gaagga(x0)
U4_gaagga(x0)
U13_gaa(x0)
U1_gaagga(x0)
U12_gaa(x0)

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

(13) 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.

(14) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13)) → PA_IN_GAAGGA(T13, T11, T12)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144))) → PA_IN_GAAGGA(T144, T142, T143)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T41)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → HIDDEN_FLATTENC_IN_GAA(T72)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → U9_GAA(T70, T71, hidden_flattenC_in_gaa(T72))
U9_GAA(T70, T71, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160))) → HIDDEN_FLATTENB_IN_GAA(T160)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87)) → HIDDEN_FLATTENC_IN_GAA(T87)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_out_gaa) → PA_IN_GAAGGA(.(T39, T40), T11, T12)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T99)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → U6_GAAGGA(T11, T12, hidden_flattenC_in_gaa(T99))
U6_GAAGGA(T11, T12, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

The TRS R consists of the following rules:

hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U8_gaa(hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U9_gaa(T70, T71, hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(T86, T87)) → U11_gaa(hidden_flattenC_in_gaa(T87))
U11_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
U9_gaa(T70, T71, hidden_flattenC_out_gaa) → U10_gaa(hidden_flattenB_in_gaa(.(T70, T71)))
hidden_flattenB_in_gaa(.(.(T11, T12), T13)) → U12_gaa(pA_in_gaagga(T13, T11, T12))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144))) → U13_gaa(pA_in_gaagga(T144, T142, T143))
hidden_flattenB_in_gaa(.(T113, .(T159, T160))) → U14_gaa(hidden_flattenB_in_gaa(T160))
U10_gaa(hidden_flattenB_out_gaa) → hidden_flattenC_out_gaa
U14_gaa(hidden_flattenB_out_gaa) → hidden_flattenB_out_gaa
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U2_gaagga(hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U3_gaagga(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(T98, T99), T11, T12) → U5_gaagga(hidden_flattenC_in_gaa(T99))
pA_in_gaagga(.(T98, T99), T11, T12) → U6_gaagga(T11, T12, hidden_flattenC_in_gaa(T99))
U13_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U6_gaagga(T11, T12, hidden_flattenC_out_gaa) → U7_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
U7_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U5_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U3_gaagga(T39, T40, T11, T12, hidden_flattenC_out_gaa) → U4_gaagga(pA_in_gaagga(.(T39, T40), T11, T12))
U4_gaagga(pA_out_gaagga) → pA_out_gaagga
U2_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U12_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U8_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa

The set Q consists of the following terms:

hidden_flattenB_in_gaa(x0)
pA_in_gaagga(x0, x1, x2)
hidden_flattenC_in_gaa(x0)
U11_gaa(x0)
U9_gaa(x0, x1, x2)
U14_gaa(x0)
U10_gaa(x0)
U8_gaa(x0)
U2_gaagga(x0)
U3_gaagga(x0, x1, x2, x3, x4)
U5_gaagga(x0)
U6_gaagga(x0, x1, x2)
U7_gaagga(x0)
U4_gaagga(x0)
U13_gaa(x0)
U1_gaagga(x0)
U12_gaa(x0)

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

(15) 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].

U1_gaagga(x0)

(16) Obligation:

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13)) → PA_IN_GAAGGA(T13, T11, T12)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144))) → PA_IN_GAAGGA(T144, T142, T143)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T41)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → HIDDEN_FLATTENC_IN_GAA(T72)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → U9_GAA(T70, T71, hidden_flattenC_in_gaa(T72))
U9_GAA(T70, T71, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160))) → HIDDEN_FLATTENB_IN_GAA(T160)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87)) → HIDDEN_FLATTENC_IN_GAA(T87)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_out_gaa) → PA_IN_GAAGGA(.(T39, T40), T11, T12)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T99)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → U6_GAAGGA(T11, T12, hidden_flattenC_in_gaa(T99))
U6_GAAGGA(T11, T12, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

The TRS R consists of the following rules:

hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U8_gaa(hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U9_gaa(T70, T71, hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(T86, T87)) → U11_gaa(hidden_flattenC_in_gaa(T87))
U11_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
U9_gaa(T70, T71, hidden_flattenC_out_gaa) → U10_gaa(hidden_flattenB_in_gaa(.(T70, T71)))
hidden_flattenB_in_gaa(.(.(T11, T12), T13)) → U12_gaa(pA_in_gaagga(T13, T11, T12))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144))) → U13_gaa(pA_in_gaagga(T144, T142, T143))
hidden_flattenB_in_gaa(.(T113, .(T159, T160))) → U14_gaa(hidden_flattenB_in_gaa(T160))
U10_gaa(hidden_flattenB_out_gaa) → hidden_flattenC_out_gaa
U14_gaa(hidden_flattenB_out_gaa) → hidden_flattenB_out_gaa
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U2_gaagga(hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U3_gaagga(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(T98, T99), T11, T12) → U5_gaagga(hidden_flattenC_in_gaa(T99))
pA_in_gaagga(.(T98, T99), T11, T12) → U6_gaagga(T11, T12, hidden_flattenC_in_gaa(T99))
U13_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U6_gaagga(T11, T12, hidden_flattenC_out_gaa) → U7_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
U7_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U5_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U3_gaagga(T39, T40, T11, T12, hidden_flattenC_out_gaa) → U4_gaagga(pA_in_gaagga(.(T39, T40), T11, T12))
U4_gaagga(pA_out_gaagga) → pA_out_gaagga
U2_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U12_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U8_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa

The set Q consists of the following terms:

hidden_flattenB_in_gaa(x0)
pA_in_gaagga(x0, x1, x2)
hidden_flattenC_in_gaa(x0)
U11_gaa(x0)
U9_gaa(x0, x1, x2)
U14_gaa(x0)
U10_gaa(x0)
U8_gaa(x0)
U2_gaagga(x0)
U3_gaagga(x0, x1, x2, x3, x4)
U5_gaagga(x0)
U6_gaagga(x0, x1, x2)
U7_gaagga(x0)
U4_gaagga(x0)
U13_gaa(x0)
U12_gaa(x0)

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

(17) MRRProof (EQUIVALENT transformation)

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

HIDDEN_FLATTENB_IN_GAA(.(.(T11, T12), T13)) → PA_IN_GAAGGA(T13, T11, T12)
HIDDEN_FLATTENB_IN_GAA(.(T113, .(.(T142, T143), T144))) → PA_IN_GAAGGA(T144, T142, T143)
PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T41)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → HIDDEN_FLATTENC_IN_GAA(T72)
HIDDEN_FLATTENC_IN_GAA(.(.(T70, T71), T72)) → U9_GAA(T70, T71, hidden_flattenC_in_gaa(T72))
U9_GAA(T70, T71, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T70, T71))
HIDDEN_FLATTENB_IN_GAA(.(T113, .(T159, T160))) → HIDDEN_FLATTENB_IN_GAA(T160)
HIDDEN_FLATTENC_IN_GAA(.(T86, T87)) → HIDDEN_FLATTENC_IN_GAA(T87)
U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_out_gaa) → PA_IN_GAAGGA(.(T39, T40), T11, T12)
PA_IN_GAAGGA(.(T98, T99), T11, T12) → U6_GAAGGA(T11, T12, hidden_flattenC_in_gaa(T99))
U6_GAAGGA(T11, T12, hidden_flattenC_out_gaa) → HIDDEN_FLATTENB_IN_GAA(.(T11, T12))

Strictly oriented rules of the TRS R:

hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U8_gaa(hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(.(T70, T71), T72)) → U9_gaa(T70, T71, hidden_flattenC_in_gaa(T72))
hidden_flattenC_in_gaa(.(T86, T87)) → U11_gaa(hidden_flattenC_in_gaa(T87))
U9_gaa(T70, T71, hidden_flattenC_out_gaa) → U10_gaa(hidden_flattenB_in_gaa(.(T70, T71)))
hidden_flattenB_in_gaa(.(T113, .(.(T142, T143), T144))) → U13_gaa(pA_in_gaagga(T144, T142, T143))
hidden_flattenB_in_gaa(.(T113, .(T159, T160))) → U14_gaa(hidden_flattenB_in_gaa(T160))
U14_gaa(hidden_flattenB_out_gaa) → hidden_flattenB_out_gaa
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U2_gaagga(hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(.(T39, T40), T41), T11, T12) → U3_gaagga(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
pA_in_gaagga(.(T98, T99), T11, T12) → U5_gaagga(hidden_flattenC_in_gaa(T99))
pA_in_gaagga(.(T98, T99), T11, T12) → U6_gaagga(T11, T12, hidden_flattenC_in_gaa(T99))
U6_gaagga(T11, T12, hidden_flattenC_out_gaa) → U7_gaagga(hidden_flattenB_in_gaa(.(T11, T12)))
U3_gaagga(T39, T40, T11, T12, hidden_flattenC_out_gaa) → U4_gaagga(pA_in_gaagga(.(T39, T40), T11, T12))
U2_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(HIDDEN_FLATTENB_IN_GAA(x₁)) = x₁
POL(HIDDEN_FLATTENC_IN_GAA(x₁)) = 2 + x₁
POL(PA_IN_GAAGGA(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U10_gaa(x₁)) = x₁
POL(U11_gaa(x₁)) = x₁
POL(U12_gaa(x₁)) = x₁
POL(U13_gaa(x₁)) = x₁
POL(U14_gaa(x₁)) = 1 + x₁
POL(U2_gaagga(x₁)) = 1 + x₁
POL(U3_GAAGGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + x₂ + x₃ + x₄ + x₅
POL(U3_gaagga(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + x₂ + x₃ + x₄ + x₅
POL(U4_gaagga(x₁)) = x₁
POL(U5_gaagga(x₁)) = x₁
POL(U6_GAAGGA(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U6_gaagga(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U7_gaagga(x₁)) = x₁
POL(U8_gaa(x₁)) = x₁
POL(U9_GAA(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U9_gaa(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(hidden_flattenB_in_gaa(x₁)) = x₁
POL(hidden_flattenB_out_gaa) = 2
POL(hidden_flattenC_in_gaa(x₁)) = 1 + x₁
POL(hidden_flattenC_out_gaa) = 2
POL(pA_in_gaagga(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(pA_out_gaagga) = 2

(18) Obligation:

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

PA_IN_GAAGGA(.(.(T39, T40), T41), T11, T12) → U3_GAAGGA(T39, T40, T11, T12, hidden_flattenC_in_gaa(T41))
PA_IN_GAAGGA(.(T98, T99), T11, T12) → HIDDEN_FLATTENC_IN_GAA(T99)

The TRS R consists of the following rules:

U11_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa
hidden_flattenB_in_gaa(.(.(T11, T12), T13)) → U12_gaa(pA_in_gaagga(T13, T11, T12))
U10_gaa(hidden_flattenB_out_gaa) → hidden_flattenC_out_gaa
U13_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U7_gaagga(hidden_flattenB_out_gaa) → pA_out_gaagga
U5_gaagga(hidden_flattenC_out_gaa) → pA_out_gaagga
U4_gaagga(pA_out_gaagga) → pA_out_gaagga
U12_gaa(pA_out_gaagga) → hidden_flattenB_out_gaa
U8_gaa(hidden_flattenC_out_gaa) → hidden_flattenC_out_gaa

The set Q consists of the following terms:

hidden_flattenB_in_gaa(x0)
pA_in_gaagga(x0, x1, x2)
hidden_flattenC_in_gaa(x0)
U11_gaa(x0)
U9_gaa(x0, x1, x2)
U14_gaa(x0)
U10_gaa(x0)
U8_gaa(x0)
U2_gaagga(x0)
U3_gaagga(x0, x1, x2, x3, x4)
U5_gaagga(x0)
U6_gaagga(x0, x1, x2)
U7_gaagga(x0)
U4_gaagga(x0)
U13_gaa(x0)
U12_gaa(x0)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(12) Obligation:

(13) UsableRulesProof (EQUIVALENT transformation)

(14) Obligation:

(15) QReductionProof (EQUIVALENT transformation)

(16) Obligation:

(17) MRRProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) TRUE