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

(0) Obligation:

Clauses:

flatten(atom(X), .(X, [])).
flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).
count(atom(X), s(0)).
count(cons(atom(X), Y), s(Z)) :- count(Y, Z).
count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z)).

Query: count(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: count(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: count(T1, T2), SCOPE: 1, CLAUSE: 3 | count(T1, T2), SCOPE: 1, CLAUSE: 4 | count(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | count(atom(T4), T2), SCOPE: 1, CLAUSE: 4 | count(atom(T4), T2), SCOPE: 1, CLAUSE: 5\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: count(T1, T2), SCOPE: 1, CLAUSE: 4 | count(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nX2 is free\ncount(T1, T2) !=? count(atom(X2), s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: count(atom(T4), T2), SCOPE: 1, CLAUSE: 4 | count(atom(T4), T2), SCOPE: 1, CLAUSE: 5\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: count(atom(T4), T2), SCOPE: 1, CLAUSE: 5\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: count(T9, T11) | count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: count(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nX2 is free\nX13 is free\nX14 is free\nX15 is free\ncount(T1, T2) !=? count(atom(X2), s(0))\ncount(T1, T2) !=? count(cons(atom(X13), X14), s(X15))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: count(T9, T11), SCOPE: 2, CLAUSE: 3 | count(T9, T11), SCOPE: 2, CLAUSE: 4 | count(T9, T11), SCOPE: 2, CLAUSE: 5 | ?_2 | count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: count(T9, T11), SCOPE: 2, CLAUSE: 3\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: count(T9, T11), SCOPE: 2, CLAUSE: 4 | count(T9, T11), SCOPE: 2, CLAUSE: 5 | ?_2 | count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: count(T9, T11), SCOPE: 2, CLAUSE: 4\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: count(T9, T11), SCOPE: 2, CLAUSE: 5 | ?_2 | count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: count(T30, T32)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: count(T9, T11), SCOPE: 2, CLAUSE: 5\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(flatten(cons(cons(T51, T52), T53), X60), count(X60, T55))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="B: flatten(cons(cons(T51, T52), T53), X60)\n\nT51 is ground\nT52 is ground\nT53 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: count(T56, T55)\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 0 | flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 1 | flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 1 | flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: flatten(cons(cons(T51, T52), T53), X60), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="C: flatten(cons(T63, cons(T64, T65)), X85)\n\nT63 is ground\nT64 is ground\nT65 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 0 | flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 1 | flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 2\n\nT63 is ground\nT64 is ground\nT65 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 1 | flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 2\n\nT63 is ground\nT64 is ground\nT65 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 1\n\nT63 is ground\nT64 is ground\nT65 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: flatten(cons(T63, cons(T64, T65)), X85), SCOPE: 4, CLAUSE: 2\n\nT63 is ground\nT64 is ground\nT65 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flatten(cons(T79, T80), X108)\n\nT79 is ground\nT80 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 0 | flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 1 | flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 2\n\nT79 is ground\nT80 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 1 | flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 2\n\nT79 is ground\nT80 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 1\n\nT79 is ground\nT80 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: flatten(cons(T79, T80), X108), SCOPE: 5, CLAUSE: 2\n\nT79 is ground\nT80 is ground\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="D: flatten(T90, X131)\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: flatten(T90, X131), SCOPE: 6, CLAUSE: 0 | flatten(T90, X131), SCOPE: 6, CLAUSE: 1 | flatten(T90, X131), SCOPE: 6, CLAUSE: 2\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: flatten(T90, X131), SCOPE: 6, CLAUSE: 0\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: flatten(T90, X131), SCOPE: 6, CLAUSE: 1 | flatten(T90, X131), SCOPE: 6, CLAUSE: 2\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: flatten(T90, X131), SCOPE: 6, CLAUSE: 1\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: flatten(T90, X131), SCOPE: 6, CLAUSE: 2\n\nT90 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: flatten(T107, X158)\n\nT107 is ground\nX158 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: flatten(cons(T116, cons(T117, T118)), X175)\n\nT116 is ground\nT117 is ground\nT118 is ground\nX175 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: flatten(cons(T125, cons(T126, T127)), X192)\n\nT125 is ground\nT126 is ground\nT127 is ground\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: flatten(cons(T136, cons(T137, cons(T138, T139))), X209)\n\nT136 is ground\nT137 is ground\nT138 is ground\nT139 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: count(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150))\n\nT146 is ground\nT147 is ground\nT148 is ground\nX226 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 0 | ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 1 | ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 2\n\nT146 is ground\nT147 is ground\nT148 is ground\nX226 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 1 | ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 2\n\nT146 is ground\nT147 is ground\nT148 is ground\nX226 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(flatten(cons(cons(T146, T147), T148), X226), count(X226, T150)), SCOPE: 7, CLAUSE: 2\n\nT146 is ground\nT147 is ground\nT148 is ground\nX226 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(flatten(cons(T157, cons(T158, T159)), X247), count(X247, T150))\n\nT157 is ground\nT158 is ground\nT159 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: flatten(cons(T157, cons(T158, T159)), X247)\n\nT157 is ground\nT158 is ground\nT159 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: count(T160, T150)\n\nT160 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 68 [arrowhead = none ];
68 -> 2;

69 [label="EVAL with clause\ncount(atom(X2), s(0)).\nand substitutionX2 -> T4,\nT1 -> atom(T4),\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 3;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 70 [arrowhead = none ];
70 -> 4;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 71 [arrowhead = none ];
71 -> 5;

72 [label="EVAL with clause\ncount(cons(atom(X13), X14), s(X15)) :- count(X14, X15).\nand substitutionX13 -> T8,\nX14 -> T9,\nT1 -> cons(atom(T8), T9),\nX15 -> T11,\nT2 -> s(T11),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 72 [arrowhead = none ];
72 -> 8;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 73 [arrowhead = none ];
73 -> 9;

74 [label="BACKTRACK\nfor clause: count(cons(atom(X), Y), s(Z)) :- count(Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 74 [arrowhead = none ];
74 -> 6;

75 [label="BACKTRACK\nfor clause: count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 75 [arrowhead = none ];
75 -> 7;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="EVAL with clause\ncount(cons(cons(X222, X223), X224), X225) :- ','(flatten(cons(cons(X222, X223), X224), X226), count(X226, X225)).\nand substitutionX222 -> T146,\nX223 -> T147,\nX224 -> T148,\nT1 -> cons(cons(T146, T147), T148),\nT2 -> T150,\nX225 -> T150,\nT149 -> T150", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 77 [arrowhead = none ];
77 -> 60;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 78 [arrowhead = none ];
78 -> 61;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 79 [arrowhead = none ];
79 -> 11;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="EVAL with clause\ncount(atom(X20), s(0)).\nand substitutionX20 -> T16,\nT9 -> atom(T16),\nT11 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 82 [arrowhead = none ];
82 -> 14;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 83 [arrowhead = none ];
83 -> 16;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="EVAL with clause\ncount(cons(atom(X33), X34), s(X35)) :- count(X34, X35).\nand substitutionX33 -> T29,\nX34 -> T30,\nT9 -> cons(atom(T29), T30),\nX35 -> T32,\nT11 -> s(T32),\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 87 [arrowhead = none ];
87 -> 19;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 88 [arrowhead = none ];
88 -> 20;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 89 [arrowhead = none ];
89 -> 21;

90 [label="INSTANCE with matching:\nT1 -> T30\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 90 [arrowhead = none , style=dashed];
90 -> 1[style=dashed];

91 [label="EVAL with clause\ncount(cons(cons(X56, X57), X58), X59) :- ','(flatten(cons(cons(X56, X57), X58), X60), count(X60, X59)).\nand substitutionX56 -> T51,\nX57 -> T52,\nX58 -> T53,\nT9 -> cons(cons(T51, T52), T53),\nT11 -> T55,\nX59 -> T55,\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 91 [arrowhead = none ];
91 -> 22;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 92 [arrowhead = none ];
92 -> 23;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 93 [arrowhead = none ];
93 -> 58;

94 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 94 [arrowhead = none ];
94 -> 24;

95 [label="SPLIT 2\nnew knowledge:\nT51 is ground\nT52 is ground\nT53 is ground\nT56 is ground\nreplacements:X60 -> T56", fontsize=14, style = filled, fillcolor = violet];
22 -> 95 [arrowhead = none ];
95 -> 25;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 96 [arrowhead = none ];
96 -> 26;

97 [label="INSTANCE with matching:\nT1 -> T56\nT2 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 97 [arrowhead = none , style=dashed];
97 -> 1[style=dashed];

98 [label="BACKTRACK\nfor clause: flatten(atom(X), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 98 [arrowhead = none ];
98 -> 27;

99 [label="BACKTRACK\nfor clause: flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="ONLY EVAL with clause\nflatten(cons(cons(X81, X82), X83), X84) :- flatten(cons(X81, cons(X82, X83)), X84).\nand substitutionT51 -> T63,\nX81 -> T63,\nT52 -> T64,\nX82 -> T64,\nT53 -> T65,\nX83 -> T65,\nX60 -> X85,\nX84 -> X85", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 100 [arrowhead = none ];
100 -> 29;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 101 [arrowhead = none ];
101 -> 30;

102 [label="BACKTRACK\nfor clause: flatten(atom(X), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 103 [arrowhead = none ];
103 -> 32;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 104 [arrowhead = none ];
104 -> 33;

105 [label="EVAL with clause\nflatten(cons(atom(X105), X106), .(X105, X107)) :- flatten(X106, X107).\nand substitutionX105 -> T78,\nT63 -> atom(T78),\nT64 -> T79,\nT65 -> T80,\nX106 -> cons(T79, T80),\nX107 -> X108,\nX85 -> .(T78, X108)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 105 [arrowhead = none ];
105 -> 34;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 106 [arrowhead = none ];
106 -> 35;

107 [label="EVAL with clause\nflatten(cons(cons(X205, X206), X207), X208) :- flatten(cons(X205, cons(X206, X207)), X208).\nand substitutionX205 -> T136,\nX206 -> T137,\nT63 -> cons(T136, T137),\nT64 -> T138,\nT65 -> T139,\nX207 -> cons(T138, T139),\nX85 -> X209,\nX208 -> X209", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 107 [arrowhead = none ];
107 -> 56;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 108 [arrowhead = none ];
108 -> 57;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 109 [arrowhead = none ];
109 -> 36;

110 [label="BACKTRACK\nfor clause: flatten(atom(X), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 110 [arrowhead = none ];
110 -> 37;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 111 [arrowhead = none ];
111 -> 38;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 112 [arrowhead = none ];
112 -> 39;

113 [label="EVAL with clause\nflatten(cons(atom(X128), X129), .(X128, X130)) :- flatten(X129, X130).\nand substitutionX128 -> T89,\nT79 -> atom(T89),\nT80 -> T90,\nX129 -> T90,\nX130 -> X131,\nX108 -> .(T89, X131)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 113 [arrowhead = none ];
113 -> 40;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 114 [arrowhead = none ];
114 -> 41;

115 [label="EVAL with clause\nflatten(cons(cons(X188, X189), X190), X191) :- flatten(cons(X188, cons(X189, X190)), X191).\nand substitutionX188 -> T125,\nX189 -> T126,\nT79 -> cons(T125, T126),\nT80 -> T127,\nX190 -> T127,\nX108 -> X192,\nX191 -> X192", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 115 [arrowhead = none ];
115 -> 54;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 116 [arrowhead = none ];
116 -> 55;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 117 [arrowhead = none ];
117 -> 42;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 118 [arrowhead = none ];
118 -> 43;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="EVAL with clause\nflatten(atom(X138), .(X138, [])).\nand substitutionX138 -> T97,\nT90 -> atom(T97),\nX131 -> .(T97, [])", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 122 [arrowhead = none ];
122 -> 48;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 123 [arrowhead = none ];
123 -> 49;

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 124 [arrowhead = none ];
124 -> 47;

125 [label="EVAL with clause\nflatten(cons(atom(X155), X156), .(X155, X157)) :- flatten(X156, X157).\nand substitutionX155 -> T106,\nX156 -> T107,\nT90 -> cons(atom(T106), T107),\nX157 -> X158,\nX131 -> .(T106, X158)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 125 [arrowhead = none ];
125 -> 50;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 126 [arrowhead = none ];
126 -> 51;

127 [label="EVAL with clause\nflatten(cons(cons(X171, X172), X173), X174) :- flatten(cons(X171, cons(X172, X173)), X174).\nand substitutionX171 -> T116,\nX172 -> T117,\nX173 -> T118,\nT90 -> cons(cons(T116, T117), T118),\nX131 -> X175,\nX174 -> X175", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 127 [arrowhead = none ];
127 -> 52;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 128 [arrowhead = none ];
128 -> 53;

129 [label="INSTANCE with matching:\nT90 -> T107\nX131 -> X158", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 129 [arrowhead = none , style=dashed];
129 -> 40[style=dashed];

130 [label="INSTANCE with matching:\nT63 -> T116\nT64 -> T117\nT65 -> T118\nX85 -> X175", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 130 [arrowhead = none , style=dashed];
130 -> 29[style=dashed];

131 [label="INSTANCE with matching:\nT63 -> T125\nT64 -> T126\nT65 -> T127\nX85 -> X192", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 131 [arrowhead = none , style=dashed];
131 -> 29[style=dashed];

132 [label="INSTANCE with matching:\nT63 -> T136\nT64 -> T137\nT65 -> cons(T138, T139)\nX85 -> X209", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 132 [arrowhead = none , style=dashed];
132 -> 29[style=dashed];

133 [label="BACKTRACK\nfor clause: count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 133 [arrowhead = none ];
133 -> 59;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 134 [arrowhead = none ];
134 -> 62;

135 [label="BACKTRACK\nfor clause: flatten(atom(X), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 135 [arrowhead = none ];
135 -> 63;

136 [label="BACKTRACK\nfor clause: flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 136 [arrowhead = none ];
136 -> 64;

137 [label="ONLY EVAL with clause\nflatten(cons(cons(X243, X244), X245), X246) :- flatten(cons(X243, cons(X244, X245)), X246).\nand substitutionT146 -> T157,\nX243 -> T157,\nT147 -> T158,\nX244 -> T158,\nT148 -> T159,\nX245 -> T159,\nX226 -> X247,\nX246 -> X247", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 137 [arrowhead = none ];
137 -> 65;

138 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 138 [arrowhead = none ];
138 -> 66;

139 [label="SPLIT 2\nnew knowledge:\nT157 is ground\nT158 is ground\nT159 is ground\nT160 is ground\nreplacements:X247 -> T160", fontsize=14, style = filled, fillcolor = violet];
65 -> 139 [arrowhead = none ];
139 -> 67;

140 [label="INSTANCE with matching:\nT63 -> T157\nT64 -> T158\nT65 -> T159\nX85 -> X247", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 140 [arrowhead = none , style=dashed];
140 -> 29[style=dashed];

141 [label="INSTANCE with matching:\nT1 -> T160\nT2 -> T150", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 141 [arrowhead = none , style=dashed];
141 -> 1[style=dashed];

}

(2) Obligation:

Triples:

flattenD(cons(atom(X1), X2), .(X1, X3)) :- flattenD(X2, X3).
flattenD(cons(cons(X1, X2), X3), X4) :- flattenC(X1, X2, X3, X4).
flattenC(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) :- flattenD(X3, X4).
flattenC(atom(X1), cons(X2, X3), X4, .(X1, X5)) :- flattenC(X2, X3, X4, X5).
flattenC(cons(X1, X2), X3, X4, X5) :- flattenC(X1, X2, cons(X3, X4), X5).
countA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) :- countA(X3, X4).
countA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) :- flattenC(X2, X3, X4, X6).
countA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) :- ','(flattencB(X2, X3, X4, X6), countA(X6, X5)).
countA(cons(cons(X1, X2), X3), X4) :- flattenC(X1, X2, X3, X5).
countA(cons(cons(X1, X2), X3), X4) :- ','(flattencC(X1, X2, X3, X5), countA(X5, X4)).

Clauses:

countcA(atom(X1), s(0)).
countcA(cons(atom(X1), atom(X2)), s(s(0))).
countcA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) :- countcA(X3, X4).
countcA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) :- ','(flattencB(X2, X3, X4, X6), countcA(X6, X5)).
countcA(cons(cons(X1, X2), X3), X4) :- ','(flattencC(X1, X2, X3, X5), countcA(X5, X4)).
flattencD(atom(X1), .(X1, [])).
flattencD(cons(atom(X1), X2), .(X1, X3)) :- flattencD(X2, X3).
flattencD(cons(cons(X1, X2), X3), X4) :- flattencC(X1, X2, X3, X4).
flattencC(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) :- flattencD(X3, X4).
flattencC(atom(X1), cons(X2, X3), X4, .(X1, X5)) :- flattencC(X2, X3, X4, X5).
flattencC(cons(X1, X2), X3, X4, X5) :- flattencC(X1, X2, cons(X3, X4), X5).
flattencB(X1, X2, X3, X4) :- flattencC(X1, X2, X3, X4).

Afs:

countA(x1, x2)  =  countA(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:
countA_in: (b,f)
flattenC_in: (b,b,b,f)
flattenD_in: (b,f)
flattencB_in: (b,b,b,f)
flattencC_in: (b,b,b,f)
flattencD_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) → U6_GA(X1, X2, X3, X4, countA_in_ga(X3, X4))
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) → COUNTA_IN_GA(X3, X4)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → U7_GA(X1, X2, X3, X4, X5, flattenC_in_ggga(X2, X3, X4, X6))
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X6)
FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U3_GGGA(X1, X2, X3, X4, flattenD_in_ga(X3, X4))
FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → FLATTEND_IN_GA(X3, X4)
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → U1_GA(X1, X2, X3, flattenD_in_ga(X2, X3))
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → FLATTEND_IN_GA(X2, X3)
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → U2_GA(X1, X2, X3, X4, flattenC_in_ggga(X1, X2, X3, X4))
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X4)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U4_GGGA(X1, X2, X3, X4, X5, flattenC_in_ggga(X2, X3, X4, X5))
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X5)
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, flattenC_in_ggga(X1, X2, cons(X3, X4), X5))
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4), X5)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → U8_GA(X1, X2, X3, X4, X5, flattencB_in_ggga(X2, X3, X4, X6))
U8_GA(X1, X2, X3, X4, X5, flattencB_out_ggga(X2, X3, X4, X6)) → U9_GA(X1, X2, X3, X4, X5, countA_in_ga(X6, X5))
U8_GA(X1, X2, X3, X4, X5, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6, X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → U10_GA(X1, X2, X3, X4, flattenC_in_ggga(X1, X2, X3, X5))
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → U11_GA(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X5))
U11_GA(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X5)) → U12_GA(X1, X2, X3, X4, countA_in_ga(X5, X4))
U11_GA(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5, X4)

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3, X4) → U24_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U21_ggga(X1, X2, X3, X4, flattencD_in_ga(X3, X4))
flattencD_in_ga(atom(X1), .(X1, [])) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2), .(X1, X3)) → U19_ga(X1, X2, X3, flattencD_in_ga(X2, X3))
flattencD_in_ga(cons(cons(X1, X2), X3), X4) → U20_ga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X2, X3, X4, X5))
flattencC_in_ggga(cons(X1, X2), X3, X4, X5) → U23_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X1, X2, cons(X3, X4), X5))
U23_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, X3, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, X4, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The argument filtering Pi contains the following mapping:
countA_in_ga(x1, x2)  =  countA_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
flattenC_in_ggga(x1, x2, x3, x4)  =  flattenC_in_ggga(x1, x2, x3)
flattenD_in_ga(x1, x2)  =  flattenD_in_ga(x1)
flattencB_in_ggga(x1, x2, x3, x4)  =  flattencB_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5)  =  U24_ggga(x1, x2, x3, x5)
flattencC_in_ggga(x1, x2, x3, x4)  =  flattencC_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5)  =  U21_ggga(x1, x2, x3, x5)
flattencD_in_ga(x1, x2)  =  flattencD_in_ga(x1)
flattencD_out_ga(x1, x2)  =  flattencD_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
U23_ggga(x1, x2, x3, x4, x5, x6)  =  U23_ggga(x1, x2, x3, x4, x6)
flattencC_out_ggga(x1, x2, x3, x4)  =  flattencC_out_ggga(x1, x2, x3, x4)
flattencB_out_ggga(x1, x2, x3, x4)  =  flattencB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
COUNTA_IN_GA(x1, x2)  =  COUNTA_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x1, x2, x3, x4, x6)
FLATTENC_IN_GGGA(x1, x2, x3, x4)  =  FLATTENC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
FLATTEND_IN_GA(x1, x2)  =  FLATTEND_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(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, x5, x6)  =  U9_GA(x1, x2, x3, x4, x6)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x3, x5)

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:

COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) → U6_GA(X1, X2, X3, X4, countA_in_ga(X3, X4))
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) → COUNTA_IN_GA(X3, X4)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → U7_GA(X1, X2, X3, X4, X5, flattenC_in_ggga(X2, X3, X4, X6))
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X6)
FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U3_GGGA(X1, X2, X3, X4, flattenD_in_ga(X3, X4))
FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → FLATTEND_IN_GA(X3, X4)
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → U1_GA(X1, X2, X3, flattenD_in_ga(X2, X3))
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → FLATTEND_IN_GA(X2, X3)
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → U2_GA(X1, X2, X3, X4, flattenC_in_ggga(X1, X2, X3, X4))
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X4)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U4_GGGA(X1, X2, X3, X4, X5, flattenC_in_ggga(X2, X3, X4, X5))
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X5)
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, flattenC_in_ggga(X1, X2, cons(X3, X4), X5))
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4), X5)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → U8_GA(X1, X2, X3, X4, X5, flattencB_in_ggga(X2, X3, X4, X6))
U8_GA(X1, X2, X3, X4, X5, flattencB_out_ggga(X2, X3, X4, X6)) → U9_GA(X1, X2, X3, X4, X5, countA_in_ga(X6, X5))
U8_GA(X1, X2, X3, X4, X5, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6, X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → U10_GA(X1, X2, X3, X4, flattenC_in_ggga(X1, X2, X3, X5))
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → U11_GA(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X5))
U11_GA(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X5)) → U12_GA(X1, X2, X3, X4, countA_in_ga(X5, X4))
U11_GA(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5, X4)

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3, X4) → U24_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U21_ggga(X1, X2, X3, X4, flattencD_in_ga(X3, X4))
flattencD_in_ga(atom(X1), .(X1, [])) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2), .(X1, X3)) → U19_ga(X1, X2, X3, flattencD_in_ga(X2, X3))
flattencD_in_ga(cons(cons(X1, X2), X3), X4) → U20_ga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X2, X3, X4, X5))
flattencC_in_ggga(cons(X1, X2), X3, X4, X5) → U23_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X1, X2, cons(X3, X4), X5))
U23_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, X3, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, X4, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The argument filtering Pi contains the following mapping:
countA_in_ga(x1, x2)  =  countA_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
flattenC_in_ggga(x1, x2, x3, x4)  =  flattenC_in_ggga(x1, x2, x3)
flattenD_in_ga(x1, x2)  =  flattenD_in_ga(x1)
flattencB_in_ggga(x1, x2, x3, x4)  =  flattencB_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5)  =  U24_ggga(x1, x2, x3, x5)
flattencC_in_ggga(x1, x2, x3, x4)  =  flattencC_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5)  =  U21_ggga(x1, x2, x3, x5)
flattencD_in_ga(x1, x2)  =  flattencD_in_ga(x1)
flattencD_out_ga(x1, x2)  =  flattencD_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
U23_ggga(x1, x2, x3, x4, x5, x6)  =  U23_ggga(x1, x2, x3, x4, x6)
flattencC_out_ggga(x1, x2, x3, x4)  =  flattencC_out_ggga(x1, x2, x3, x4)
flattencB_out_ggga(x1, x2, x3, x4)  =  flattencB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
COUNTA_IN_GA(x1, x2)  =  COUNTA_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5, x6)  =  U7_GA(x1, x2, x3, x4, x6)
FLATTENC_IN_GGGA(x1, x2, x3, x4)  =  FLATTENC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
FLATTEND_IN_GA(x1, x2)  =  FLATTEND_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(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, x5, x6)  =  U9_GA(x1, x2, x3, x4, x6)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → FLATTEND_IN_GA(X3, X4)
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → FLATTEND_IN_GA(X2, X3)
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X4)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X5)
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4), X5)

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3, X4) → U24_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U21_ggga(X1, X2, X3, X4, flattencD_in_ga(X3, X4))
flattencD_in_ga(atom(X1), .(X1, [])) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2), .(X1, X3)) → U19_ga(X1, X2, X3, flattencD_in_ga(X2, X3))
flattencD_in_ga(cons(cons(X1, X2), X3), X4) → U20_ga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X2, X3, X4, X5))
flattencC_in_ggga(cons(X1, X2), X3, X4, X5) → U23_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X1, X2, cons(X3, X4), X5))
U23_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, X3, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, X4, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
flattencB_in_ggga(x1, x2, x3, x4)  =  flattencB_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5)  =  U24_ggga(x1, x2, x3, x5)
flattencC_in_ggga(x1, x2, x3, x4)  =  flattencC_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5)  =  U21_ggga(x1, x2, x3, x5)
flattencD_in_ga(x1, x2)  =  flattencD_in_ga(x1)
flattencD_out_ga(x1, x2)  =  flattencD_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
U23_ggga(x1, x2, x3, x4, x5, x6)  =  U23_ggga(x1, x2, x3, x4, x6)
flattencC_out_ggga(x1, x2, x3, x4)  =  flattencC_out_ggga(x1, x2, x3, x4)
flattencB_out_ggga(x1, x2, x3, x4)  =  flattencB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
FLATTENC_IN_GGGA(x1, x2, x3, x4)  =  FLATTENC_IN_GGGA(x1, x2, x3)
FLATTEND_IN_GA(x1, x2)  =  FLATTEND_IN_GA(x1)

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:

FLATTENC_IN_GGGA(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → FLATTEND_IN_GA(X3, X4)
FLATTEND_IN_GA(cons(atom(X1), X2), .(X1, X3)) → FLATTEND_IN_GA(X2, X3)
FLATTEND_IN_GA(cons(cons(X1, X2), X3), X4) → FLATTENC_IN_GGGA(X1, X2, X3, X4)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4, .(X1, X5)) → FLATTENC_IN_GGGA(X2, X3, X4, X5)
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4, X5) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4), X5)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
.(x1, x2)  =  .(x1, x2)
FLATTENC_IN_GGGA(x1, x2, x3, x4)  =  FLATTENC_IN_GGGA(x1, x2, x3)
FLATTEND_IN_GA(x1, x2)  =  FLATTEND_IN_GA(x1)

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:

FLATTENC_IN_GGGA(atom(X1), atom(X2), X3) → FLATTEND_IN_GA(X3)
FLATTEND_IN_GA(cons(atom(X1), X2)) → FLATTEND_IN_GA(X2)
FLATTEND_IN_GA(cons(cons(X1, X2), X3)) → FLATTENC_IN_GGGA(X1, X2, X3)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4) → FLATTENC_IN_GGGA(X2, X3, X4)
FLATTENC_IN_GGGA(cons(X1, X2), X3, X4) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4))

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

(12) 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:

FLATTENC_IN_GGGA(atom(X1), atom(X2), X3) → FLATTEND_IN_GA(X3)
FLATTEND_IN_GA(cons(atom(X1), X2)) → FLATTEND_IN_GA(X2)
FLATTEND_IN_GA(cons(cons(X1, X2), X3)) → FLATTENC_IN_GGGA(X1, X2, X3)
FLATTENC_IN_GGGA(atom(X1), cons(X2, X3), X4) → FLATTENC_IN_GGGA(X2, X3, X4)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTENC_IN_GGGA(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃   
POL(FLATTEND_IN_GA(x₁)) = 1 + 2·x₁   
POL(atom(x₁)) = 1 + x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   

(13) Obligation:

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

FLATTENC_IN_GGGA(cons(X1, X2), X3, X4) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4))

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• FLATTENC_IN_GGGA(cons(X1, X2), X3, X4) → FLATTENC_IN_GGGA(X1, X2, cons(X3, X4))
  The graph contains the following edges 1 > 1, 1 > 2

(15) YES

(16) Obligation:

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

COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), s(X5)) → U8_GA(X1, X2, X3, X4, X5, flattencB_in_ggga(X2, X3, X4, X6))
U8_GA(X1, X2, X3, X4, X5, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6, X5)
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), s(s(X4))) → COUNTA_IN_GA(X3, X4)
COUNTA_IN_GA(cons(cons(X1, X2), X3), X4) → U11_GA(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X5))
U11_GA(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5, X4)

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3, X4) → U24_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4))) → U21_ggga(X1, X2, X3, X4, flattencD_in_ga(X3, X4))
flattencD_in_ga(atom(X1), .(X1, [])) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2), .(X1, X3)) → U19_ga(X1, X2, X3, flattencD_in_ga(X2, X3))
flattencD_in_ga(cons(cons(X1, X2), X3), X4) → U20_ga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, X3, X4))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X2, X3, X4, X5))
flattencC_in_ggga(cons(X1, X2), X3, X4, X5) → U23_ggga(X1, X2, X3, X4, X5, flattencC_in_ggga(X1, X2, cons(X3, X4), X5))
U23_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, X5, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, X3, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, X4, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
flattencB_in_ggga(x1, x2, x3, x4)  =  flattencB_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5)  =  U24_ggga(x1, x2, x3, x5)
flattencC_in_ggga(x1, x2, x3, x4)  =  flattencC_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5)  =  U21_ggga(x1, x2, x3, x5)
flattencD_in_ga(x1, x2)  =  flattencD_in_ga(x1)
flattencD_out_ga(x1, x2)  =  flattencD_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x2, x4)
U20_ga(x1, x2, x3, x4, x5)  =  U20_ga(x1, x2, x3, x5)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
U23_ggga(x1, x2, x3, x4, x5, x6)  =  U23_ggga(x1, x2, x3, x4, x6)
flattencC_out_ggga(x1, x2, x3, x4)  =  flattencC_out_ggga(x1, x2, x3, x4)
flattencB_out_ggga(x1, x2, x3, x4)  =  flattencB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
COUNTA_IN_GA(x1, x2)  =  COUNTA_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x1, x2, x3, x4, x6)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)

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:

COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, flattencB_in_ggga(X2, X3, X4))
U8_GA(X1, X2, X3, X4, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6)
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → COUNTA_IN_GA(X3)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3) → U24_ggga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The set Q consists of the following terms:

flattencB_in_ggga(x0, x1, x2)
flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(19) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, flattencB_in_ggga(X2, X3, X4)) at position [4] we obtained the following new rules [LPAR04]:

COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, U24_ggga(X2, X3, X4, flattencC_in_ggga(X2, X3, X4)))

(20) Obligation:

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

U8_GA(X1, X2, X3, X4, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6)
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → COUNTA_IN_GA(X3)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, U24_ggga(X2, X3, X4, flattencC_in_ggga(X2, X3, X4)))

The TRS R consists of the following rules:

flattencB_in_ggga(X1, X2, X3) → U24_ggga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)

The set Q consists of the following terms:

flattencB_in_ggga(x0, x1, x2)
flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(21) 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.

(22) Obligation:

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

U8_GA(X1, X2, X3, X4, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6)
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → COUNTA_IN_GA(X3)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, U24_ggga(X2, X3, X4, flattencC_in_ggga(X2, X3, X4)))

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencB_in_ggga(x0, x1, x2)
flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(23) 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].

flattencB_in_ggga(x0, x1, x2)

(24) Obligation:

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

U8_GA(X1, X2, X3, X4, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6)
COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → COUNTA_IN_GA(X3)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, U24_ggga(X2, X3, X4, flattencC_in_ggga(X2, X3, X4)))

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(25) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

COUNTA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → COUNTA_IN_GA(X3)
COUNTA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → U8_GA(X1, X2, X3, X4, U24_ggga(X2, X3, X4, flattencC_in_ggga(X2, X3, X4)))

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

POL(.(x₁, x₂)) = 0   
POL(COUNTA_IN_GA(x₁)) = 1 + x₁   
POL(U11_GA(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U19_ga(x₁, x₂, x₃)) = 0   
POL(U20_ga(x₁, x₂, x₃, x₄)) = x₄   
POL(U21_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(U22_ggga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U23_ggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U24_ggga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U8_GA(x₁, x₂, x₃, x₄, x₅)) = x₄ + x₅   
POL([]) = 0   
POL(atom(x₁)) = 1   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(flattencB_out_ggga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(flattencC_in_ggga(x₁, x₂, x₃)) = 0   
POL(flattencC_out_ggga(x₁, x₂, x₃, x₄)) = x₄   
POL(flattencD_in_ga(x₁)) = 0   
POL(flattencD_out_ga(x₁, x₂)) = 0   

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

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)

(26) Obligation:

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

U8_GA(X1, X2, X3, X4, flattencB_out_ggga(X2, X3, X4, X6)) → COUNTA_IN_GA(X6)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Obligation:

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

U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U24_ggga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencB_out_ggga(X1, X2, X3, X4)
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(29) 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.

(30) Obligation:

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

U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)
U24_ggga(x0, x1, x2, x3)

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

(31) 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].

U24_ggga(x0, x1, x2, x3)

(32) Obligation:

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

U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)
COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)

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

(33) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

COUNTA_IN_GA(cons(cons(X1, X2), X3)) → U11_GA(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))

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

POL(.(x₁, x₂)) = 0   
POL(COUNTA_IN_GA(x₁)) = x₁   
POL(U11_GA(x₁, x₂, x₃, x₄)) = x₄   
POL(U19_ga(x₁, x₂, x₃)) = 1   
POL(U20_ga(x₁, x₂, x₃, x₄)) = 0   
POL(U21_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(U22_ggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U23_ggga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL([]) = 0   
POL(atom(x₁)) = 0   
POL(cons(x₁, x₂)) = 1 + x₁ + x₂   
POL(flattencC_in_ggga(x₁, x₂, x₃)) = 0   
POL(flattencC_out_ggga(x₁, x₂, x₃, x₄)) = x₄   
POL(flattencD_in_ga(x₁)) = x₁   
POL(flattencD_out_ga(x₁, x₂)) = 0   

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

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)

(34) Obligation:

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

U11_GA(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X5)) → COUNTA_IN_GA(X5)

The TRS R consists of the following rules:

flattencC_in_ggga(atom(X1), atom(X2), X3) → U21_ggga(X1, X2, X3, flattencD_in_ga(X3))
flattencC_in_ggga(atom(X1), cons(X2, X3), X4) → U22_ggga(X1, X2, X3, X4, flattencC_in_ggga(X2, X3, X4))
flattencC_in_ggga(cons(X1, X2), X3, X4) → U23_ggga(X1, X2, X3, X4, flattencC_in_ggga(X1, X2, cons(X3, X4)))
U23_ggga(X1, X2, X3, X4, flattencC_out_ggga(X1, X2, cons(X3, X4), X5)) → flattencC_out_ggga(cons(X1, X2), X3, X4, X5)
U22_ggga(X1, X2, X3, X4, flattencC_out_ggga(X2, X3, X4, X5)) → flattencC_out_ggga(atom(X1), cons(X2, X3), X4, .(X1, X5))
flattencD_in_ga(atom(X1)) → flattencD_out_ga(atom(X1), .(X1, []))
flattencD_in_ga(cons(atom(X1), X2)) → U19_ga(X1, X2, flattencD_in_ga(X2))
flattencD_in_ga(cons(cons(X1, X2), X3)) → U20_ga(X1, X2, X3, flattencC_in_ggga(X1, X2, X3))
U21_ggga(X1, X2, X3, flattencD_out_ga(X3, X4)) → flattencC_out_ggga(atom(X1), atom(X2), X3, .(X1, .(X2, X4)))
U20_ga(X1, X2, X3, flattencC_out_ggga(X1, X2, X3, X4)) → flattencD_out_ga(cons(cons(X1, X2), X3), X4)
U19_ga(X1, X2, flattencD_out_ga(X2, X3)) → flattencD_out_ga(cons(atom(X1), X2), .(X1, X3))

The set Q consists of the following terms:

flattencC_in_ggga(x0, x1, x2)
flattencD_in_ga(x0)
U23_ggga(x0, x1, x2, x3, x4)
U22_ggga(x0, x1, x2, x3, x4)
U20_ga(x0, x1, x2, x3)
U19_ga(x0, x1, x2)
U21_ggga(x0, x1, x2, x3)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(36) TRUE