Left Termination of the query pattern flatten_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

flatten(atom(X), Y) :- ','(!, eq(Y, .(X, []))).
flatten(L, Y) :- ','(head(L, atom(H)), ','(!, ','(eq(Y, .(H, Z)), ','(tail(L, T), flatten(T, Z))))).
flatten(L, X) :- ','(head(L, cons(U, V)), ','(tail(L, W), flatten(cons(U, cons(V, W)), X))).
head(nil, X1).
head(cons(H, X2), H).
tail(nil, nil).
tail(cons(X3, T), T).
eq(X, X).

Queries:

flatten(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: flatten(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: flatten(T1, T2), SCOPE: 1, CLAUSE: 0 | flatten(T1, T2), SCOPE: 1, CLAUSE: 1 | flatten(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T7, .(T5, []))) | flatten(atom(T5), T2), SCOPE: 1, CLAUSE: 1 | flatten(atom(T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: flatten(T1, T2), SCOPE: 1, CLAUSE: 1 | flatten(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nflatten(T1, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T7, .(T5, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T7, .(T5, [])), SCOPE: 2, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T16, atom(X18)), ','(!_1, ','(eq(T18, .(X18, X19)), ','(tail(T16, X20), flatten(X20, X19))))) | flatten(T16, T2), SCOPE: 1, CLAUSE: 2\n\nT16 is ground\nX18 is free; \nX19 is free; \nX20 is free; \nflatten(T16, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T16, atom(X18)), ','(!_1, ','(eq(T18, .(X18, X19)), ','(tail(T16, X20), flatten(X20, X19))))), SCOPE: 3, CLAUSE: 3 | ','(head(T16, atom(X18)), ','(!_1, ','(eq(T18, .(X18, X19)), ','(tail(T16, X20), flatten(X20, X19))))), SCOPE: 3, CLAUSE: 4 | ?_3 | flatten(T16, T2), SCOPE: 1, CLAUSE: 2\n\nT16 is ground\nX18 is free; \nX19 is free; \nX20 is free; \nflatten(T16, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_1, ','(eq(T18, .(X26, X19)), ','(tail(nil, X20), flatten(X20, X19)))) | ','(head(nil, atom(X18)), ','(!_1, ','(eq(T18, .(X18, X19)), ','(tail(nil, X20), flatten(X20, X19))))), SCOPE: 3, CLAUSE: 4 | ?_3 | flatten(nil, T2), SCOPE: 1, CLAUSE: 2\n\nX18 is free\nX19 is free; \nX20 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T16, atom(X18)), ','(!_1, ','(eq(T18, .(X18, X19)), ','(tail(T16, X20), flatten(X20, X19))))), SCOPE: 3, CLAUSE: 4 | ?_3 | flatten(T16, T2), SCOPE: 1, CLAUSE: 2\n\nT16 is ground\nX18 is free; \nX19 is free; \nX20 is free; \nflatten(T16, T2) !=? flatten(atom(X6), X7)\nhead(T16, atom(X18)) !=? head(nil, X25)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(eq(T18, .(X26, X19)), ','(tail(nil, X20), flatten(X20, X19)))\n\nX19 is free\nX20 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(eq(T18, .(X26, X19)), ','(tail(nil, X20), flatten(X20, X19))), SCOPE: 4, CLAUSE: 7\n\nX19 is free\nX20 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(nil, X20), flatten(X20, T28))\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(tail(nil, X20), flatten(X20, T28)), SCOPE: 5, CLAUSE: 5 | ','(tail(nil, X20), flatten(X20, T28)), SCOPE: 5, CLAUSE: 6\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(tail(nil, X20), flatten(X20, T28)), SCOPE: 5, CLAUSE: 5\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(tail(nil, X20), flatten(X20, T28)), SCOPE: 5, CLAUSE: 6\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: flatten(nil, T28)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: flatten(nil, T28), SCOPE: 6, CLAUSE: 0 | flatten(nil, T28), SCOPE: 6, CLAUSE: 1 | flatten(nil, T28), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: flatten(nil, T28), SCOPE: 6, CLAUSE: 1 | flatten(nil, T28), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(head(nil, atom(X71)), ','(!_6, ','(eq(T37, .(X71, X72)), ','(tail(nil, X73), flatten(X73, X72))))) | flatten(nil, T28), SCOPE: 6, CLAUSE: 2\n\nX71 is free\nX72 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(nil, atom(X71)), ','(!_6, ','(eq(T37, .(X71, X72)), ','(tail(nil, X73), flatten(X73, X72))))), SCOPE: 7, CLAUSE: 3 | ','(head(nil, atom(X71)), ','(!_6, ','(eq(T37, .(X71, X72)), ','(tail(nil, X73), flatten(X73, X72))))), SCOPE: 7, CLAUSE: 4 | ?_7 | flatten(nil, T28), SCOPE: 6, CLAUSE: 2\n\nX71 is free\nX72 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_6, ','(eq(T37, .(X79, X72)), ','(tail(nil, X73), flatten(X73, X72)))) | ','(head(nil, atom(X71)), ','(!_6, ','(eq(T37, .(X71, X72)), ','(tail(nil, X73), flatten(X73, X72))))), SCOPE: 7, CLAUSE: 4 | ?_7 | flatten(nil, T28), SCOPE: 6, CLAUSE: 2\n\nX71 is free\nX72 is free; \nX73 is free; \nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(eq(T37, .(X79, X72)), ','(tail(nil, X73), flatten(X73, X72)))\n\nX72 is free\nX73 is free; \nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(eq(T37, .(X79, X72)), ','(tail(nil, X73), flatten(X73, X72))), SCOPE: 8, CLAUSE: 7\n\nX72 is free\nX73 is free; \nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(nil, X73), flatten(X73, T47))\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(nil, X73), flatten(X73, T47)), SCOPE: 9, CLAUSE: 5 | ','(tail(nil, X73), flatten(X73, T47)), SCOPE: 9, CLAUSE: 6\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(nil, X73), flatten(X73, T47)), SCOPE: 9, CLAUSE: 5\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail(nil, X73), flatten(X73, T47)), SCOPE: 9, CLAUSE: 6\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flatten(nil, T47)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(!_1, ','(eq(T18, .(T60, X19)), ','(tail(cons(atom(T60), T59), X20), flatten(X20, X19)))) | ?_3 | flatten(cons(atom(T60), T59), T2), SCOPE: 1, CLAUSE: 2\n\nT60 is ground\nT59 is ground\nX19 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ?_3 | flatten(T16, T2), SCOPE: 1, CLAUSE: 2\n\nT16 is ground\nflatten(T16, T2) !=? flatten(atom(X6), X7)\nhead(T16, atom(X18)) !=? head(nil, X25)\nhead(T16, atom(X18)) !=? head(cons(X120, X121), X120)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(eq(T18, .(T60, X19)), ','(tail(cons(atom(T60), T59), X20), flatten(X20, X19)))\n\nT60 is ground\nT59 is ground\nX19 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(eq(T18, .(T60, X19)), ','(tail(cons(atom(T60), T59), X20), flatten(X20, X19))), SCOPE: 10, CLAUSE: 7\n\nT60 is ground\nT59 is ground\nX19 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(cons(atom(T68), T59), X20), flatten(X20, T70))\n\nT59 is ground\nT68 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail(cons(atom(T68), T59), X20), flatten(X20, T70)), SCOPE: 11, CLAUSE: 5 | ','(tail(cons(atom(T68), T59), X20), flatten(X20, T70)), SCOPE: 11, CLAUSE: 6\n\nT59 is ground\nT68 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(cons(atom(T68), T59), X20), flatten(X20, T70)), SCOPE: 11, CLAUSE: 6\n\nT59 is ground\nT68 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: flatten(T79, T70)\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: flatten(T16, T2), SCOPE: 1, CLAUSE: 2\n\nT16 is ground\nflatten(T16, T2) !=? flatten(atom(X6), X7)\nhead(T16, atom(X18)) !=? head(nil, X25)\nhead(T16, atom(X18)) !=? head(cons(X120, X121), X120)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(head(T88, cons(X164, X165)), ','(tail(T88, X166), flatten(cons(X164, cons(X165, X166)), T90)))\n\nT88 is ground\nX164 is free; \nX165 is free; \nX166 is free; \nflatten(T88, T2) !=? flatten(atom(X6), X7)\nhead(T88, atom(X18)) !=? head(nil, X25)\nhead(T88, atom(X18)) !=? head(cons(X120, X121), X120)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(head(T88, cons(X164, X165)), ','(tail(T88, X166), flatten(cons(X164, cons(X165, X166)), T90))), SCOPE: 12, CLAUSE: 3 | ','(head(T88, cons(X164, X165)), ','(tail(T88, X166), flatten(cons(X164, cons(X165, X166)), T90))), SCOPE: 12, CLAUSE: 4\n\nT88 is ground\nX164 is free; \nX165 is free; \nX166 is free; \nflatten(T88, T2) !=? flatten(atom(X6), X7)\nhead(T88, atom(X18)) !=? head(nil, X25)\nhead(T88, atom(X18)) !=? head(cons(X120, X121), X120)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(head(T88, cons(X164, X165)), ','(tail(T88, X166), flatten(cons(X164, cons(X165, X166)), T90))), SCOPE: 12, CLAUSE: 4\n\nT88 is ground\nX164 is free; \nX165 is free; \nX166 is free; \nflatten(T88, T2) !=? flatten(atom(X6), X7)\nhead(T88, atom(X18)) !=? head(nil, X25)\nhead(T88, atom(X18)) !=? head(cons(X120, X121), X120)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(tail(cons(cons(T102, T103), T101), X166), flatten(cons(T102, cons(T103, X166)), T90))\n\nT102 is ground\nT103 is ground\nT101 is ground\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(tail(cons(cons(T102, T103), T101), X166), flatten(cons(T102, cons(T103, X166)), T90)), SCOPE: 13, CLAUSE: 5 | ','(tail(cons(cons(T102, T103), T101), X166), flatten(cons(T102, cons(T103, X166)), T90)), SCOPE: 13, CLAUSE: 6\n\nT102 is ground\nT103 is ground\nT101 is ground\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(tail(cons(cons(T102, T103), T101), X166), flatten(cons(T102, cons(T103, X166)), T90)), SCOPE: 13, CLAUSE: 6\n\nT102 is ground\nT103 is ground\nT101 is ground\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: flatten(cons(T117, cons(T118, T119)), T90)\n\nT117 is ground\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 55 [arrowhead = none ];
55 -> 2;

56 [label="EVAL with clause\nflatten(atom(X6), X7) :- ','(!_1, eq(X7, .(X6, []))).\nand substitution\nX6 -> T5\nT1 -> atom(T5)\nT2 -> T7\nX7 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 56 [arrowhead = none ];
56 -> 3;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 57 [arrowhead = none ];
57 -> 4;

58 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 58 [arrowhead = none ];
58 -> 5;

59 [label="ONLY EVAL with clause\nflatten(X16, X17) :- ','(head(X16, atom(X18)), ','(!_1, ','(eq(X17, .(X18, X19)), ','(tail(X16, X20), flatten(X20, X19))))).\nand substitution\nT1 -> T16\nX16 -> T16\nT2 -> T18\nX17 -> T18\nT17 -> T18", fontsize=14, style = filled, fillcolor = white];
4 -> 59 [arrowhead = none ];
59 -> 10;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 60 [arrowhead = none ];
60 -> 6;

61 [label="EVAL with clause\neq(X12, X12).\nand substitution\nT7 -> .(T13, [])\nX12 -> .(T13, [])\nT5 -> T13\nT12 -> .(T13, [])", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 62 [arrowhead = none ];
62 -> 8;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 64 [arrowhead = none ];
64 -> 11;

65 [label="EVAL with clause\nhead(nil, X25).\nand substitution\nT16 -> nil\nX18 -> X26\nX25 -> atom(X26)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 65 [arrowhead = none ];
65 -> 12;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 66 [arrowhead = none ];
66 -> 13;

67 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="EVAL with clause\nhead(cons(X120, X121), X120).\nand substitution\nX120 -> atom(T60)\nX121 -> T59\nT16 -> cons(atom(T60), T59)\nX18 -> T60\nT58 -> atom(T60)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 68 [arrowhead = none ];
68 -> 37;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 69 [arrowhead = none ];
69 -> 38;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 70 [arrowhead = none ];
70 -> 15;

71 [label="EVAL with clause\neq(X34, X34).\nand substitution\nT18 -> .(T26, T28)\nX34 -> .(T26, T28)\nX26 -> T26\nX19 -> T28\nT25 -> .(T26, T28)\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 71 [arrowhead = none ];
71 -> 16;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 73 [arrowhead = none ];
73 -> 18;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 74 [arrowhead = none ];
74 -> 19;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 75 [arrowhead = none ];
75 -> 20;

76 [label="ONLY EVAL with clause\ntail(nil, nil).\nand substitution\nX20 -> nil", fontsize=14, style = filled, fillcolor = white];
19 -> 76 [arrowhead = none ];
76 -> 21;

77 [label="BACKTRACK\nfor clause: tail(cons(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 77 [arrowhead = none ];
77 -> 36;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 78 [arrowhead = none ];
78 -> 22;

79 [label="BACKTRACK\nfor clause: flatten(atom(X), Y) :- ','(!, eq(Y, .(X, [])))because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 79 [arrowhead = none ];
79 -> 23;

80 [label="ONLY EVAL with clause\nflatten(X69, X70) :- ','(head(X69, atom(X71)), ','(!_6, ','(eq(X70, .(X71, X72)), ','(tail(X69, X73), flatten(X73, X72))))).\nand substitution\nX69 -> nil\nT28 -> T37\nX70 -> T37\nT36 -> T37", fontsize=14, style = filled, fillcolor = white];
23 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 81 [arrowhead = none ];
81 -> 25;

82 [label="ONLY EVAL with clause\nhead(nil, X78).\nand substitution\nX71 -> X79\nX78 -> atom(X79)", fontsize=14, style = filled, fillcolor = white];
25 -> 82 [arrowhead = none ];
82 -> 26;

83 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 83 [arrowhead = none ];
83 -> 27;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 84 [arrowhead = none ];
84 -> 28;

85 [label="EVAL with clause\neq(X86, X86).\nand substitution\nT37 -> .(T45, T47)\nX86 -> .(T45, T47)\nX79 -> T45\nX72 -> T47\nT44 -> .(T45, T47)\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 85 [arrowhead = none ];
85 -> 29;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 86 [arrowhead = none ];
86 -> 30;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 87 [arrowhead = none ];
87 -> 31;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 88 [arrowhead = none ];
88 -> 32;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 89 [arrowhead = none ];
89 -> 33;

90 [label="ONLY EVAL with clause\ntail(nil, nil).\nand substitution\nX73 -> nil", fontsize=14, style = filled, fillcolor = white];
32 -> 90 [arrowhead = none ];
90 -> 34;

91 [label="BACKTRACK\nfor clause: tail(cons(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
33 -> 91 [arrowhead = none ];
91 -> 35;

92 [label="INSTANCE with matching:\nT28 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 92 [arrowhead = none , style=dashed];
92 -> 21[style=dashed];

93 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 94 [arrowhead = none ];
94 -> 46;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="EVAL with clause\neq(X130, X130).\nand substitution\nT18 -> .(T68, T70)\nX130 -> .(T68, T70)\nT60 -> T68\nX19 -> T70\nT67 -> .(T68, T70)\nT69 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 98 [arrowhead = none ];
98 -> 43;

99 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 99 [arrowhead = none ];
99 -> 44;

100 [label="ONLY EVAL with clause\ntail(cons(X141, X142), X142).\nand substitution\nT68 -> T78\nX141 -> atom(T78)\nT59 -> T79\nX142 -> T79\nX20 -> T79", fontsize=14, style = filled, fillcolor = white];
44 -> 100 [arrowhead = none ];
100 -> 45;

101 [label="INSTANCE with matching:\nT1 -> T79\nT2 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 101 [arrowhead = none , style=dashed];
101 -> 1[style=dashed];

102 [label="ONLY EVAL with clause\nflatten(X162, X163) :- ','(head(X162, cons(X164, X165)), ','(tail(X162, X166), flatten(cons(X164, cons(X165, X166)), X163))).\nand substitution\nT16 -> T88\nX162 -> T88\nT2 -> T90\nX163 -> T90\nT89 -> T90", fontsize=14, style = filled, fillcolor = white];
46 -> 102 [arrowhead = none ];
102 -> 47;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 103 [arrowhead = none ];
103 -> 48;

104 [label="BACKTRACK\nfor clause: head(nil, X1)\nwith clash: (head(T88, atom(X18)), head(nil, X25))", fontsize=14, style = filled, fillcolor = white];
48 -> 104 [arrowhead = none ];
104 -> 49;

105 [label="EVAL with clause\nhead(cons(X184, X185), X184).\nand substitution\nX184 -> cons(T102, T103)\nX185 -> T101\nT88 -> cons(cons(T102, T103), T101)\nX164 -> T102\nX165 -> T103\nT100 -> cons(T102, T103)", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 106 [arrowhead = none ];
106 -> 51;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 107 [arrowhead = none ];
107 -> 52;

108 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 108 [arrowhead = none ];
108 -> 53;

109 [label="ONLY EVAL with clause\ntail(cons(X207, X208), X208).\nand substitution\nT102 -> T117\nT103 -> T118\nX207 -> cons(T117, T118)\nT101 -> T119\nX208 -> T119\nX166 -> T119", fontsize=14, style = filled, fillcolor = white];
53 -> 109 [arrowhead = none ];
109 -> 54;

110 [label="INSTANCE with matching:\nT1 -> cons(T117, cons(T118, T119))\nT2 -> T90", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 110 [arrowhead = none , style=dashed];
110 -> 1[style=dashed];

}

(2) Obligation:

Triples:

flatten21(.(T45, T47)) :- flatten21(T47).
flatten1(nil, .(T26, T28)) :- flatten21(T28).
flatten1(cons(atom(T78), T79), .(T78, T70)) :- flatten1(T79, T70).
flatten1(cons(cons(T117, T118), T119), T90) :- flatten1(cons(T117, cons(T118, T119)), T90).

Clauses:

flattenc21(.(T45, T47)) :- flattenc21(T47).
flattenc1(atom(T13), .(T13, [])).
flattenc1(nil, .(T26, T28)) :- flattenc21(T28).
flattenc1(cons(atom(T78), T79), .(T78, T70)) :- flattenc1(T79, T70).
flattenc1(cons(cons(T117, T118), T119), T90) :- flattenc1(cons(T117, cons(T118, T119)), T90).

Afs:

flatten1(x1, x2)  =  flatten1(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:
flatten1_in: (b,f)
flatten21_in: (f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FLATTEN1_IN_GA(nil, .(T26, T28)) → U2_GA(T26, T28, flatten21_in_a(T28))
FLATTEN1_IN_GA(nil, .(T26, T28)) → FLATTEN21_IN_A(T28)
FLATTEN21_IN_A(.(T45, T47)) → U1_A(T45, T47, flatten21_in_a(T47))
FLATTEN21_IN_A(.(T45, T47)) → FLATTEN21_IN_A(T47)
FLATTEN1_IN_GA(cons(atom(T78), T79), .(T78, T70)) → U3_GA(T78, T79, T70, flatten1_in_ga(T79, T70))
FLATTEN1_IN_GA(cons(atom(T78), T79), .(T78, T70)) → FLATTEN1_IN_GA(T79, T70)
FLATTEN1_IN_GA(cons(cons(T117, T118), T119), T90) → U4_GA(T117, T118, T119, T90, flatten1_in_ga(cons(T117, cons(T118, T119)), T90))
FLATTEN1_IN_GA(cons(cons(T117, T118), T119), T90) → FLATTEN1_IN_GA(cons(T117, cons(T118, T119)), T90)

R is empty.
The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
nil  =  nil
flatten21_in_a(x1)  =  flatten21_in_a
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
FLATTEN21_IN_A(x1)  =  FLATTEN21_IN_A
U1_A(x1, x2, x3)  =  U1_A(x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_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:

FLATTEN1_IN_GA(nil, .(T26, T28)) → U2_GA(T26, T28, flatten21_in_a(T28))
FLATTEN1_IN_GA(nil, .(T26, T28)) → FLATTEN21_IN_A(T28)
FLATTEN21_IN_A(.(T45, T47)) → U1_A(T45, T47, flatten21_in_a(T47))
FLATTEN21_IN_A(.(T45, T47)) → FLATTEN21_IN_A(T47)
FLATTEN1_IN_GA(cons(atom(T78), T79), .(T78, T70)) → U3_GA(T78, T79, T70, flatten1_in_ga(T79, T70))
FLATTEN1_IN_GA(cons(atom(T78), T79), .(T78, T70)) → FLATTEN1_IN_GA(T79, T70)
FLATTEN1_IN_GA(cons(cons(T117, T118), T119), T90) → U4_GA(T117, T118, T119, T90, flatten1_in_ga(cons(T117, cons(T118, T119)), T90))
FLATTEN1_IN_GA(cons(cons(T117, T118), T119), T90) → FLATTEN1_IN_GA(cons(T117, cons(T118, T119)), T90)

R is empty.
The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
nil  =  nil
flatten21_in_a(x1)  =  flatten21_in_a
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
FLATTEN21_IN_A(x1)  =  FLATTEN21_IN_A
U1_A(x1, x2, x3)  =  U1_A(x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_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 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

FLATTEN21_IN_A(.(T45, T47)) → FLATTEN21_IN_A(T47)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
FLATTEN21_IN_A(x1)  =  FLATTEN21_IN_A

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

(8) PiDPToQDPProof (SOUND transformation)

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

(9) Obligation:

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

FLATTEN21_IN_A → FLATTEN21_IN_A

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

(10) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = FLATTEN21_IN_A evaluates to t =FLATTEN21_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLATTEN21_IN_A to FLATTEN21_IN_A.

(11) NO

(12) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T117, T118), T119), T90) → FLATTEN1_IN_GA(cons(T117, cons(T118, T119)), T90)
FLATTEN1_IN_GA(cons(atom(T78), T79), .(T78, T70)) → FLATTEN1_IN_GA(T79, T70)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T117, T118), T119)) → FLATTEN1_IN_GA(cons(T117, cons(T118, T119)))
FLATTEN1_IN_GA(cons(atom(T78), T79)) → FLATTEN1_IN_GA(T79)

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

(15) 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:

FLATTEN1_IN_GA(cons(cons(T117, T118), T119)) → FLATTEN1_IN_GA(cons(T117, cons(T118, T119)))
FLATTEN1_IN_GA(cons(atom(T78), T79)) → FLATTEN1_IN_GA(T79)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN1_IN_GA(x₁)) = 2·x₁   
POL(atom(x₁)) = x₁   
POL(cons(x₁, x₂)) = 2 + 2·x₁ + x₂   

(16) Obligation:

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

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) YES