Left Termination proof of /tmp/tmpDQ3DGn/flatten

Left Termination of the query pattern count_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 UsableRulesReductionPairsProof (⇔)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 Rewriting (⇔)

        ↳20 QDP

        ↳21 UsableRulesProof (⇔)

        ↳22 QDP

        ↳23 QReductionProof (⇔)

        ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 DependencyGraphProof (⇔)

        ↳28 QDP

        ↳29 UsableRulesProof (⇔)

        ↳30 QDP

        ↳31 QReductionProof (⇔)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔)

        ↳34 QDP

        ↳35 DependencyGraphProof (⇔)

        ↳36 TRUE

(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)).

Queries:

count(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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\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\ncount(T1, T2) !=? count(atom(X2), s(0))\ncount(T1, T2) !=? count(cons(atom(X14), X15), s(X16))", 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), X68), count(X68, T55))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: flatten(cons(cons(T51, T52), T53), X68)\n\nT51 is ground\nT52 is ground\nT53 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: count(T59, T55)\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 0 | flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 1 | flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 1 | flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: flatten(cons(cons(T51, T52), T53), X68), SCOPE: 3, CLAUSE: 2\n\nT51 is ground\nT52 is ground\nT53 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: flatten(cons(T72, cons(T73, T74)), X117)\n\nT72 is ground\nT73 is ground\nT74 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 0 | flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 1 | flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 2\n\nT72 is ground\nT73 is ground\nT74 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 1 | flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 2\n\nT72 is ground\nT73 is ground\nT74 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 1\n\nT72 is ground\nT73 is ground\nT74 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: flatten(cons(T72, cons(T73, T74)), X117), SCOPE: 4, CLAUSE: 2\n\nT72 is ground\nT73 is ground\nT74 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flatten(cons(T91, T92), X144)\n\nT91 is ground\nT92 is ground\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 0 | flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 1 | flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 2\n\nT91 is ground\nT92 is ground\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 1 | flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 2\n\nT91 is ground\nT92 is ground\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 1\n\nT91 is ground\nT92 is ground\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: flatten(cons(T91, T92), X144), SCOPE: 5, CLAUSE: 2\n\nT91 is ground\nT92 is ground\nX144 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: flatten(T104, X174)\n\nT104 is ground\nX174 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: flatten(T104, X174), SCOPE: 6, CLAUSE: 0 | flatten(T104, X174), SCOPE: 6, CLAUSE: 1 | flatten(T104, X174), SCOPE: 6, CLAUSE: 2\n\nT104 is ground\nX174 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: flatten(T104, X174), SCOPE: 6, CLAUSE: 0\n\nT104 is ground\nX174 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: flatten(T104, X174), SCOPE: 6, CLAUSE: 1 | flatten(T104, X174), SCOPE: 6, CLAUSE: 2\n\nT104 is ground\nX174 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(T104, X174), SCOPE: 6, CLAUSE: 1\n\nT104 is ground\nX174 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: flatten(T104, X174), SCOPE: 6, CLAUSE: 2\n\nT104 is ground\nX174 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: flatten(T121, X205)\n\nT121 is ground\nX205 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: flatten(cons(T130, cons(T131, T132)), X225)\n\nT130 is ground\nT131 is ground\nT132 is ground\nX225 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: flatten(cons(T142, cons(T143, T144)), X250)\n\nT142 is ground\nT143 is ground\nT144 is ground\nX250 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: flatten(cons(T156, cons(T157, cons(T158, T159))), X275)\n\nT156 is ground\nT157 is ground\nT158 is ground\nT159 is ground\nX275 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(T170, T171), T172), X300), count(X300, T174))\n\nT170 is ground\nT171 is ground\nT172 is ground\nX300 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(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 0 | ','(flatten(cons(cons(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 1 | ','(flatten(cons(cons(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 2\n\nT170 is ground\nT171 is ground\nT172 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(flatten(cons(cons(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 1 | ','(flatten(cons(cons(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 2\n\nT170 is ground\nT171 is ground\nT172 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(flatten(cons(cons(T170, T171), T172), X300), count(X300, T174)), SCOPE: 7, CLAUSE: 2\n\nT170 is ground\nT171 is ground\nT172 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(flatten(cons(T184, cons(T185, T186)), X329), count(X329, T174))\n\nT184 is ground\nT185 is ground\nT186 is ground\nX329 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: flatten(cons(T184, cons(T185, T186)), X329)\n\nT184 is ground\nT185 is ground\nT186 is ground\nX329 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: count(T190, T174)\n\nT190 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 substitution\nX2 -> 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 = lightblue];
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(X14), X15), s(X16)) :- count(X15, X16).\nand substitution\nX14 -> T8\nX15 -> T9\nT1 -> cons(atom(T8), T9)\nX16 -> 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 = lightblue];
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 = white];
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 = white];
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(X296, X297), X298), X299) :- ','(flatten(cons(cons(X296, X297), X298), X300), count(X300, X299)).\nand substitution\nX296 -> T170\nX297 -> T171\nX298 -> T172\nT1 -> cons(cons(T170, T171), T172)\nT2 -> T174\nX299 -> T174\nT173 -> T174", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 77 [arrowhead = none ];
77 -> 60;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X24), s(0)).\nand substitution\nX24 -> 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 = lightblue];
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(X38), X39), s(X40)) :- count(X39, X40).\nand substitution\nX38 -> T29\nX39 -> T30\nT9 -> cons(atom(T29), T30)\nX40 -> 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 = lightblue];
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(X64, X65), X66), X67) :- ','(flatten(cons(cons(X64, X65), X66), X68), count(X68, X67)).\nand substitution\nX64 -> T51\nX65 -> T52\nX66 -> T53\nT9 -> cons(cons(T51, T52), T53)\nT11 -> T55\nX67 -> T55\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 91 [arrowhead = none ];
91 -> 22;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT59 is ground\nreplacements:\nX68 -> T59", 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 -> T59\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 = white];
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 = white];
27 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="ONLY EVAL with clause\nflatten(cons(cons(X113, X114), X115), X116) :- flatten(cons(X113, cons(X114, X115)), X116).\nand substitution\nT51 -> T72\nX113 -> T72\nT52 -> T73\nX114 -> T73\nT53 -> T74\nX115 -> T74\nX68 -> X117\nX116 -> X117", fontsize=14, style = filled, fillcolor = white];
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 = white];
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(X141), X142), .(X141, X143)) :- flatten(X142, X143).\nand substitution\nX141 -> T90\nT72 -> atom(T90)\nT73 -> T91\nT74 -> T92\nX142 -> cons(T91, T92)\nX143 -> X144\nX117 -> .(T90, X144)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 105 [arrowhead = none ];
105 -> 34;

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

107 [label="EVAL with clause\nflatten(cons(cons(X271, X272), X273), X274) :- flatten(cons(X271, cons(X272, X273)), X274).\nand substitution\nX271 -> T156\nX272 -> T157\nT72 -> cons(T156, T157)\nT73 -> T158\nT74 -> T159\nX273 -> cons(T158, T159)\nX117 -> X275\nX274 -> X275", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 107 [arrowhead = none ];
107 -> 56;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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(X171), X172), .(X171, X173)) :- flatten(X172, X173).\nand substitution\nX171 -> T103\nT91 -> atom(T103)\nT92 -> T104\nX172 -> T104\nX173 -> X174\nX144 -> .(T103, X174)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 113 [arrowhead = none ];
113 -> 40;

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

115 [label="EVAL with clause\nflatten(cons(cons(X246, X247), X248), X249) :- flatten(cons(X246, cons(X247, X248)), X249).\nand substitution\nX246 -> T142\nX247 -> T143\nT91 -> cons(T142, T143)\nT92 -> T144\nX248 -> T144\nX144 -> X250\nX249 -> X250", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 115 [arrowhead = none ];
115 -> 54;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X184), .(X184, [])).\nand substitution\nX184 -> T111\nT104 -> atom(T111)\nX174 -> .(T111, [])", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X202), X203), .(X202, X204)) :- flatten(X203, X204).\nand substitution\nX202 -> T120\nX203 -> T121\nT104 -> cons(atom(T120), T121)\nX204 -> X205\nX174 -> .(T120, X205)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 125 [arrowhead = none ];
125 -> 50;

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

127 [label="EVAL with clause\nflatten(cons(cons(X221, X222), X223), X224) :- flatten(cons(X221, cons(X222, X223)), X224).\nand substitution\nX221 -> T130\nX222 -> T131\nX223 -> T132\nT104 -> cons(cons(T130, T131), T132)\nX174 -> X225\nX224 -> X225", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 127 [arrowhead = none ];
127 -> 52;

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

129 [label="INSTANCE with matching:\nT104 -> T121\nX174 -> X205", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 129 [arrowhead = none , style=dashed];
129 -> 40[style=dashed];

130 [label="INSTANCE with matching:\nT72 -> T130\nT73 -> T131\nT74 -> T132\nX117 -> X225", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 130 [arrowhead = none , style=dashed];
130 -> 29[style=dashed];

131 [label="INSTANCE with matching:\nT72 -> T142\nT73 -> T143\nT74 -> T144\nX117 -> X250", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 131 [arrowhead = none , style=dashed];
131 -> 29[style=dashed];

132 [label="INSTANCE with matching:\nT72 -> T156\nT73 -> T157\nT74 -> cons(T158, T159)\nX117 -> X275", 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 = white];
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 = white];
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 = white];
63 -> 136 [arrowhead = none ];
136 -> 64;

137 [label="ONLY EVAL with clause\nflatten(cons(cons(X325, X326), X327), X328) :- flatten(cons(X325, cons(X326, X327)), X328).\nand substitution\nT170 -> T184\nX325 -> T184\nT171 -> T185\nX326 -> T185\nT172 -> T186\nX327 -> T186\nX300 -> X329\nX328 -> X329", fontsize=14, style = filled, fillcolor = white];
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:\nT190 is ground\nreplacements:\nX329 -> T190", fontsize=14, style = filled, fillcolor = violet];
65 -> 139 [arrowhead = none ];
139 -> 67;

140 [label="INSTANCE with matching:\nT72 -> T184\nT73 -> T185\nT74 -> T186\nX117 -> X329", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 140 [arrowhead = none , style=dashed];
140 -> 29[style=dashed];

141 [label="INSTANCE with matching:\nT1 -> T190\nT2 -> T174", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 141 [arrowhead = none , style=dashed];
141 -> 1[style=dashed];

}

(2) Obligation:

Triples:

flatten40(cons(atom(T120), T121), .(T120, X205)) :- flatten40(T121, X205).
flatten40(cons(cons(T130, T131), T132), X225) :- flatten29(T130, T131, T132, X225).
flatten29(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) :- flatten40(T104, X174).
flatten29(atom(T90), cons(T142, T143), T144, .(T90, X250)) :- flatten29(T142, T143, T144, X250).
flatten29(cons(T156, T157), T158, T159, X275) :- flatten29(T156, T157, cons(T158, T159), X275).
count1(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) :- count1(T30, T32).
count1(cons(atom(T8), cons(cons(T72, T73), T74)), s(T55)) :- flatten29(T72, T73, T74, X117).
count1(cons(atom(T8), cons(cons(T51, T52), T53)), s(T55)) :- ','(flattenc24(T51, T52, T53, T59), count1(T59, T55)).
count1(cons(cons(T184, T185), T186), T174) :- flatten29(T184, T185, T186, X329).
count1(cons(cons(T184, T185), T186), T174) :- ','(flattenc29(T184, T185, T186, T190), count1(T190, T174)).

Clauses:

countc1(atom(T4), s(0)).
countc1(cons(atom(T8), atom(T16)), s(s(0))).
countc1(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) :- countc1(T30, T32).
countc1(cons(atom(T8), cons(cons(T51, T52), T53)), s(T55)) :- ','(flattenc24(T51, T52, T53, T59), countc1(T59, T55)).
countc1(cons(cons(T184, T185), T186), T174) :- ','(flattenc29(T184, T185, T186, T190), countc1(T190, T174)).
flattenc40(atom(T111), .(T111, [])).
flattenc40(cons(atom(T120), T121), .(T120, X205)) :- flattenc40(T121, X205).
flattenc40(cons(cons(T130, T131), T132), X225) :- flattenc29(T130, T131, T132, X225).
flattenc29(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) :- flattenc40(T104, X174).
flattenc29(atom(T90), cons(T142, T143), T144, .(T90, X250)) :- flattenc29(T142, T143, T144, X250).
flattenc29(cons(T156, T157), T158, T159, X275) :- flattenc29(T156, T157, cons(T158, T159), X275).
flattenc24(T72, T73, T74, X117) :- flattenc29(T72, T73, T74, X117).

Afs:

count1(x1, x2) = count1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
count1_in: (b,f)
flatten29_in: (b,b,b,f)
flatten40_in: (b,f)
flattenc24_in: (b,b,b,f)
flattenc29_in: (b,b,b,f)
flattenc40_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) → U6_GA(T8, T29, T30, T32, count1_in_ga(T30, T32))
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) → COUNT1_IN_GA(T30, T32)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T72, T73), T74)), s(T55)) → U7_GA(T8, T72, T73, T74, T55, flatten29_in_ggga(T72, T73, T74, X117))
COUNT1_IN_GA(cons(atom(T8), cons(cons(T72, T73), T74)), s(T55)) → FLATTEN29_IN_GGGA(T72, T73, T74, X117)
FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U3_GGGA(T90, T103, T104, X174, flatten40_in_ga(T104, X174))
FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → FLATTEN40_IN_GA(T104, X174)
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → U1_GA(T120, T121, X205, flatten40_in_ga(T121, X205))
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → FLATTEN40_IN_GA(T121, X205)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → U2_GA(T130, T131, T132, X225, flatten29_in_ggga(T130, T131, T132, X225))
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → FLATTEN29_IN_GGGA(T130, T131, T132, X225)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U4_GGGA(T90, T142, T143, T144, X250, flatten29_in_ggga(T142, T143, T144, X250))
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → FLATTEN29_IN_GGGA(T142, T143, T144, X250)
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → U5_GGGA(T156, T157, T158, T159, X275, flatten29_in_ggga(T156, T157, cons(T158, T159), X275))
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159), X275)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), s(T55)) → U8_GA(T8, T51, T52, T53, T55, flattenc24_in_ggga(T51, T52, T53, T59))
U8_GA(T8, T51, T52, T53, T55, flattenc24_out_ggga(T51, T52, T53, T59)) → U9_GA(T8, T51, T52, T53, T55, count1_in_ga(T59, T55))
U8_GA(T8, T51, T52, T53, T55, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59, T55)
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → U10_GA(T184, T185, T186, T174, flatten29_in_ggga(T184, T185, T186, X329))
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → FLATTEN29_IN_GGGA(T184, T185, T186, X329)
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → U11_GA(T184, T185, T186, T174, flattenc29_in_ggga(T184, T185, T186, T190))
U11_GA(T184, T185, T186, T174, flattenc29_out_ggga(T184, T185, T186, T190)) → U12_GA(T184, T185, T186, T174, count1_in_ga(T190, T174))
U11_GA(T184, T185, T186, T174, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190, T174)

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74, X117) → U24_ggga(T72, T73, T74, X117, flattenc29_in_ggga(T72, T73, T74, X117))
flattenc29_in_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U21_ggga(T90, T103, T104, X174, flattenc40_in_ga(T104, X174))
flattenc40_in_ga(atom(T111), .(T111, [])) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121), .(T120, X205)) → U19_ga(T120, T121, X205, flattenc40_in_ga(T121, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132), X225) → U20_ga(T130, T131, T132, X225, flattenc29_in_ggga(T130, T131, T132, X225))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U22_ggga(T90, T142, T143, T144, X250, flattenc29_in_ggga(T142, T143, T144, X250))
flattenc29_in_ggga(cons(T156, T157), T158, T159, X275) → U23_ggga(T156, T157, T158, T159, X275, flattenc29_in_ggga(T156, T157, cons(T158, T159), X275))
U23_ggga(T156, T157, T158, T159, X275, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, X250, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, X225, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, X205, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, X174, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, X117, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

The argument filtering Pi contains the following mapping:
count1_in_ga(x1, x2) = count1_in_ga(x1)
cons(x1, x2) = cons(x1, x2)
atom(x1) = atom(x1)
flatten29_in_ggga(x1, x2, x3, x4) = flatten29_in_ggga(x1, x2, x3)
flatten40_in_ga(x1, x2) = flatten40_in_ga(x1)
flattenc24_in_ggga(x1, x2, x3, x4) = flattenc24_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5) = U24_ggga(x1, x2, x3, x5)
flattenc29_in_ggga(x1, x2, x3, x4) = flattenc29_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5) = U21_ggga(x1, x2, x3, x5)
flattenc40_in_ga(x1, x2) = flattenc40_in_ga(x1)
flattenc40_out_ga(x1, x2) = flattenc40_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)
flattenc29_out_ggga(x1, x2, x3, x4) = flattenc29_out_ggga(x1, x2, x3, x4)
flattenc24_out_ggga(x1, x2, x3, x4) = flattenc24_out_ggga(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
COUNT1_IN_GA(x1, x2) = COUNT1_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)
FLATTEN29_IN_GGGA(x1, x2, x3, x4) = FLATTEN29_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x1, x2, x3, x5)
FLATTEN40_IN_GA(x1, x2) = FLATTEN40_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:

COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) → U6_GA(T8, T29, T30, T32, count1_in_ga(T30, T32))
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) → COUNT1_IN_GA(T30, T32)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T72, T73), T74)), s(T55)) → U7_GA(T8, T72, T73, T74, T55, flatten29_in_ggga(T72, T73, T74, X117))
COUNT1_IN_GA(cons(atom(T8), cons(cons(T72, T73), T74)), s(T55)) → FLATTEN29_IN_GGGA(T72, T73, T74, X117)
FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U3_GGGA(T90, T103, T104, X174, flatten40_in_ga(T104, X174))
FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → FLATTEN40_IN_GA(T104, X174)
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → U1_GA(T120, T121, X205, flatten40_in_ga(T121, X205))
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → FLATTEN40_IN_GA(T121, X205)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → U2_GA(T130, T131, T132, X225, flatten29_in_ggga(T130, T131, T132, X225))
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → FLATTEN29_IN_GGGA(T130, T131, T132, X225)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U4_GGGA(T90, T142, T143, T144, X250, flatten29_in_ggga(T142, T143, T144, X250))
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → FLATTEN29_IN_GGGA(T142, T143, T144, X250)
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → U5_GGGA(T156, T157, T158, T159, X275, flatten29_in_ggga(T156, T157, cons(T158, T159), X275))
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159), X275)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), s(T55)) → U8_GA(T8, T51, T52, T53, T55, flattenc24_in_ggga(T51, T52, T53, T59))
U8_GA(T8, T51, T52, T53, T55, flattenc24_out_ggga(T51, T52, T53, T59)) → U9_GA(T8, T51, T52, T53, T55, count1_in_ga(T59, T55))
U8_GA(T8, T51, T52, T53, T55, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59, T55)
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → U10_GA(T184, T185, T186, T174, flatten29_in_ggga(T184, T185, T186, X329))
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → FLATTEN29_IN_GGGA(T184, T185, T186, X329)
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → U11_GA(T184, T185, T186, T174, flattenc29_in_ggga(T184, T185, T186, T190))
U11_GA(T184, T185, T186, T174, flattenc29_out_ggga(T184, T185, T186, T190)) → U12_GA(T184, T185, T186, T174, count1_in_ga(T190, T174))
U11_GA(T184, T185, T186, T174, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190, T174)

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74, X117) → U24_ggga(T72, T73, T74, X117, flattenc29_in_ggga(T72, T73, T74, X117))
flattenc29_in_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U21_ggga(T90, T103, T104, X174, flattenc40_in_ga(T104, X174))
flattenc40_in_ga(atom(T111), .(T111, [])) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121), .(T120, X205)) → U19_ga(T120, T121, X205, flattenc40_in_ga(T121, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132), X225) → U20_ga(T130, T131, T132, X225, flattenc29_in_ggga(T130, T131, T132, X225))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U22_ggga(T90, T142, T143, T144, X250, flattenc29_in_ggga(T142, T143, T144, X250))
flattenc29_in_ggga(cons(T156, T157), T158, T159, X275) → U23_ggga(T156, T157, T158, T159, X275, flattenc29_in_ggga(T156, T157, cons(T158, T159), X275))
U23_ggga(T156, T157, T158, T159, X275, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, X250, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, X225, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, X205, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, X174, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, X117, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

(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:

FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → FLATTEN40_IN_GA(T104, X174)
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → FLATTEN40_IN_GA(T121, X205)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → FLATTEN29_IN_GGGA(T130, T131, T132, X225)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → FLATTEN29_IN_GGGA(T142, T143, T144, X250)
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159), X275)

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74, X117) → U24_ggga(T72, T73, T74, X117, flattenc29_in_ggga(T72, T73, T74, X117))
flattenc29_in_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U21_ggga(T90, T103, T104, X174, flattenc40_in_ga(T104, X174))
flattenc40_in_ga(atom(T111), .(T111, [])) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121), .(T120, X205)) → U19_ga(T120, T121, X205, flattenc40_in_ga(T121, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132), X225) → U20_ga(T130, T131, T132, X225, flattenc29_in_ggga(T130, T131, T132, X225))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U22_ggga(T90, T142, T143, T144, X250, flattenc29_in_ggga(T142, T143, T144, X250))
flattenc29_in_ggga(cons(T156, T157), T158, T159, X275) → U23_ggga(T156, T157, T158, T159, X275, flattenc29_in_ggga(T156, T157, cons(T158, T159), X275))
U23_ggga(T156, T157, T158, T159, X275, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, X250, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, X225, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, X205, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, X174, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, X117, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x1, x2)
atom(x1) = atom(x1)
flattenc24_in_ggga(x1, x2, x3, x4) = flattenc24_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5) = U24_ggga(x1, x2, x3, x5)
flattenc29_in_ggga(x1, x2, x3, x4) = flattenc29_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5) = U21_ggga(x1, x2, x3, x5)
flattenc40_in_ga(x1, x2) = flattenc40_in_ga(x1)
flattenc40_out_ga(x1, x2) = flattenc40_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)
flattenc29_out_ggga(x1, x2, x3, x4) = flattenc29_out_ggga(x1, x2, x3, x4)
flattenc24_out_ggga(x1, x2, x3, x4) = flattenc24_out_ggga(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
FLATTEN29_IN_GGGA(x1, x2, x3, x4) = FLATTEN29_IN_GGGA(x1, x2, x3)
FLATTEN40_IN_GA(x1, x2) = FLATTEN40_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:

FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → FLATTEN40_IN_GA(T104, X174)
FLATTEN40_IN_GA(cons(atom(T120), T121), .(T120, X205)) → FLATTEN40_IN_GA(T121, X205)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132), X225) → FLATTEN29_IN_GGGA(T130, T131, T132, X225)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144, .(T90, X250)) → FLATTEN29_IN_GGGA(T142, T143, T144, X250)
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159, X275) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159), X275)

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)
FLATTEN29_IN_GGGA(x1, x2, x3, x4) = FLATTEN29_IN_GGGA(x1, x2, x3)
FLATTEN40_IN_GA(x1, x2) = FLATTEN40_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:

FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104) → FLATTEN40_IN_GA(T104)
FLATTEN40_IN_GA(cons(atom(T120), T121)) → FLATTEN40_IN_GA(T121)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132)) → FLATTEN29_IN_GGGA(T130, T131, T132)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144) → FLATTEN29_IN_GGGA(T142, T143, T144)
FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159))

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:

FLATTEN29_IN_GGGA(atom(T90), atom(T103), T104) → FLATTEN40_IN_GA(T104)
FLATTEN40_IN_GA(cons(atom(T120), T121)) → FLATTEN40_IN_GA(T121)
FLATTEN40_IN_GA(cons(cons(T130, T131), T132)) → FLATTEN29_IN_GGGA(T130, T131, T132)
FLATTEN29_IN_GGGA(atom(T90), cons(T142, T143), T144) → FLATTEN29_IN_GGGA(T142, T143, T144)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN29_IN_GGGA(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(FLATTEN40_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:

FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159))

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:

FLATTEN29_IN_GGGA(cons(T156, T157), T158, T159) → FLATTEN29_IN_GGGA(T156, T157, cons(T158, T159))
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:

COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), s(T55)) → U8_GA(T8, T51, T52, T53, T55, flattenc24_in_ggga(T51, T52, T53, T59))
U8_GA(T8, T51, T52, T53, T55, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59, T55)
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), s(s(T32))) → COUNT1_IN_GA(T30, T32)
COUNT1_IN_GA(cons(cons(T184, T185), T186), T174) → U11_GA(T184, T185, T186, T174, flattenc29_in_ggga(T184, T185, T186, T190))
U11_GA(T184, T185, T186, T174, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190, T174)

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74, X117) → U24_ggga(T72, T73, T74, X117, flattenc29_in_ggga(T72, T73, T74, X117))
flattenc29_in_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174))) → U21_ggga(T90, T103, T104, X174, flattenc40_in_ga(T104, X174))
flattenc40_in_ga(atom(T111), .(T111, [])) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121), .(T120, X205)) → U19_ga(T120, T121, X205, flattenc40_in_ga(T121, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132), X225) → U20_ga(T130, T131, T132, X225, flattenc29_in_ggga(T130, T131, T132, X225))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250)) → U22_ggga(T90, T142, T143, T144, X250, flattenc29_in_ggga(T142, T143, T144, X250))
flattenc29_in_ggga(cons(T156, T157), T158, T159, X275) → U23_ggga(T156, T157, T158, T159, X275, flattenc29_in_ggga(T156, T157, cons(T158, T159), X275))
U23_ggga(T156, T157, T158, T159, X275, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, X250, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, X225, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, X205, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, X174, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, X117, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x1, x2)
atom(x1) = atom(x1)
flattenc24_in_ggga(x1, x2, x3, x4) = flattenc24_in_ggga(x1, x2, x3)
U24_ggga(x1, x2, x3, x4, x5) = U24_ggga(x1, x2, x3, x5)
flattenc29_in_ggga(x1, x2, x3, x4) = flattenc29_in_ggga(x1, x2, x3)
U21_ggga(x1, x2, x3, x4, x5) = U21_ggga(x1, x2, x3, x5)
flattenc40_in_ga(x1, x2) = flattenc40_in_ga(x1)
flattenc40_out_ga(x1, x2) = flattenc40_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)
flattenc29_out_ggga(x1, x2, x3, x4) = flattenc29_out_ggga(x1, x2, x3, x4)
flattenc24_out_ggga(x1, x2, x3, x4) = flattenc24_out_ggga(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
COUNT1_IN_GA(x1, x2) = COUNT1_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:

COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, flattenc24_in_ggga(T51, T52, T53))
U8_GA(T8, T51, T52, T53, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59)
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → COUNT1_IN_GA(T30)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))
U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74) → U24_ggga(T72, T73, T74, flattenc29_in_ggga(T72, T73, T74))
flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

The set Q consists of the following terms:

flattenc24_in_ggga(x0, x1, x2)
flattenc29_in_ggga(x0, x1, x2)
flattenc40_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 COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, flattenc24_in_ggga(T51, T52, T53)) at position [4] we obtained the following new rules [LPAR04]:

COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, U24_ggga(T51, T52, T53, flattenc29_in_ggga(T51, T52, T53)))

(20) Obligation:

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

U8_GA(T8, T51, T52, T53, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59)
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → COUNT1_IN_GA(T30)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))
U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, U24_ggga(T51, T52, T53, flattenc29_in_ggga(T51, T52, T53)))

The TRS R consists of the following rules:

flattenc24_in_ggga(T72, T73, T74) → U24_ggga(T72, T73, T74, flattenc29_in_ggga(T72, T73, T74))
flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)

The set Q consists of the following terms:

flattenc24_in_ggga(x0, x1, x2)
flattenc29_in_ggga(x0, x1, x2)
flattenc40_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(T8, T51, T52, T53, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59)
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → COUNT1_IN_GA(T30)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))
U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, U24_ggga(T51, T52, T53, flattenc29_in_ggga(T51, T52, T53)))

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc24_in_ggga(x0, x1, x2)
flattenc29_in_ggga(x0, x1, x2)
flattenc40_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].

flattenc24_in_ggga(x0, x1, x2)

(24) Obligation:

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

U8_GA(T8, T51, T52, T53, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59)
COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → COUNT1_IN_GA(T30)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))
U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, U24_ggga(T51, T52, T53, flattenc29_in_ggga(T51, T52, T53)))

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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].

The following pairs can be oriented strictly and are deleted.

COUNT1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → COUNT1_IN_GA(T30)
COUNT1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → U8_GA(T8, T51, T52, T53, U24_ggga(T51, T52, T53, flattenc29_in_ggga(T51, T52, T53)))

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

POL(.(x₁, x₂)) = 0
POL(COUNT1_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(flattenc24_out_ggga(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(flattenc29_in_ggga(x₁, x₂, x₃)) = 0
POL(flattenc29_out_ggga(x₁, x₂, x₃, x₄)) = x₄
POL(flattenc40_in_ga(x₁)) = 0
POL(flattenc40_out_ga(x₁, x₂)) = 0

The following usable rules [FROCOS05] were oriented:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)

(26) Obligation:

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

U8_GA(T8, T51, T52, T53, flattenc24_out_ggga(T51, T52, T53, T59)) → COUNT1_IN_GA(T59)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))
U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U24_ggga(T72, T73, T74, flattenc29_out_ggga(T72, T73, T74, X117)) → flattenc24_out_ggga(T72, T73, T74, X117)
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)
COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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].

The following pairs can be oriented strictly and are deleted.

COUNT1_IN_GA(cons(cons(T184, T185), T186)) → U11_GA(T184, T185, T186, flattenc29_in_ggga(T184, T185, T186))

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

POL(U11_GA(x₁, x₂, x₃, x₄)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 1 ]

· x₄

POL(flattenc29_out_ggga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 1 1 /

· x₄

POL(COUNT1_IN_GA(x₁)) = 0 +
[ 1, 0 ]

· x₁

POL(cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(flattenc29_in_ggga(x₁, x₂, x₃)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 1 /

· x₃

POL(atom(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U21_ggga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(flattenc40_in_ga(x₁)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(U22_ggga(x₁, x₂, x₃, x₄, x₅)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄ +
/ 0 0 \

\ 0 0 /

· x₅

POL(U23_ggga(x₁, x₂, x₃, x₄, x₅)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄ +
/ 0 0 \

\ 0 1 /

· x₅

POL(U19_ga(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(flattenc40_out_ga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(.(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL([]) =
/ 0 \

\ 0 /

POL(U20_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

The following usable rules [FROCOS05] were oriented:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)

(34) Obligation:

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

U11_GA(T184, T185, T186, flattenc29_out_ggga(T184, T185, T186, T190)) → COUNT1_IN_GA(T190)

The TRS R consists of the following rules:

flattenc29_in_ggga(atom(T90), atom(T103), T104) → U21_ggga(T90, T103, T104, flattenc40_in_ga(T104))
flattenc29_in_ggga(atom(T90), cons(T142, T143), T144) → U22_ggga(T90, T142, T143, T144, flattenc29_in_ggga(T142, T143, T144))
flattenc29_in_ggga(cons(T156, T157), T158, T159) → U23_ggga(T156, T157, T158, T159, flattenc29_in_ggga(T156, T157, cons(T158, T159)))
U23_ggga(T156, T157, T158, T159, flattenc29_out_ggga(T156, T157, cons(T158, T159), X275)) → flattenc29_out_ggga(cons(T156, T157), T158, T159, X275)
U22_ggga(T90, T142, T143, T144, flattenc29_out_ggga(T142, T143, T144, X250)) → flattenc29_out_ggga(atom(T90), cons(T142, T143), T144, .(T90, X250))
flattenc40_in_ga(atom(T111)) → flattenc40_out_ga(atom(T111), .(T111, []))
flattenc40_in_ga(cons(atom(T120), T121)) → U19_ga(T120, T121, flattenc40_in_ga(T121))
flattenc40_in_ga(cons(cons(T130, T131), T132)) → U20_ga(T130, T131, T132, flattenc29_in_ggga(T130, T131, T132))
U21_ggga(T90, T103, T104, flattenc40_out_ga(T104, X174)) → flattenc29_out_ggga(atom(T90), atom(T103), T104, .(T90, .(T103, X174)))
U20_ga(T130, T131, T132, flattenc29_out_ggga(T130, T131, T132, X225)) → flattenc40_out_ga(cons(cons(T130, T131), T132), X225)
U19_ga(T120, T121, flattenc40_out_ga(T121, X205)) → flattenc40_out_ga(cons(atom(T120), T121), .(T120, X205))

The set Q consists of the following terms:

flattenc29_in_ggga(x0, x1, x2)
flattenc40_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.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) Rewriting (EQUIVALENT transformation)

(20) Obligation:

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) QReductionProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) QReductionProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) DependencyGraphProof (EQUIVALENT transformation)

(36) TRUE