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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T5, .(T3, []))) | flatten(atom(T3), T2), SCOPE: 1, CLAUSE: 1 | flatten(atom(T3), T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT3 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 | ?_1\n\nT1 is ground\nX6 is free; \nX7 is free; \nflatten(T1, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T5, .(T3, []))\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T5, .(T3, [])), SCOPE: 2, CLAUSE: 7\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T8, atom(X14)), ','(!_1, ','(eq(T10, .(X14, X15)), ','(tail(T8, X16), flatten(X16, X15))))) | flatten(T8, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT8 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX15 is free; \nX16 is free; \nflatten(T8, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T8, atom(X14)), ','(!_1, ','(eq(T10, .(X14, X15)), ','(tail(T8, X16), flatten(X16, X15))))), SCOPE: 3, CLAUSE: 3 | ','(head(T8, atom(X14)), ','(!_1, ','(eq(T10, .(X14, X15)), ','(tail(T8, X16), flatten(X16, X15))))), SCOPE: 3, CLAUSE: 4 | flatten(T8, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT8 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX15 is free; \nX16 is free; \nflatten(T8, T2) !=? flatten(atom(X6), X7)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_1, ','(eq(T10, .(X19, X15)), ','(tail(nil, X16), flatten(X16, X15)))) | ','(head(nil, atom(X14)), ','(!_1, ','(eq(T10, .(X14, X15)), ','(tail(nil, X16), flatten(X16, X15))))), SCOPE: 3, CLAUSE: 4 | flatten(nil, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nX6 is free\nX7 is free; \nX14 is free; \nX15 is free; \nX16 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T8, atom(X14)), ','(!_1, ','(eq(T10, .(X14, X15)), ','(tail(T8, X16), flatten(X16, X15))))), SCOPE: 3, CLAUSE: 4 | flatten(T8, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT8 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX15 is free; \nX16 is free; \nX18 is free; \nflatten(T8, T2) !=? flatten(atom(X6), X7)\nhead(T8, atom(X14)) !=? head(nil, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(eq(T10, .(X19, X15)), ','(tail(nil, X16), flatten(X16, X15)))\n\nX15 is free\nX16 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(eq(T10, .(X19, X15)), ','(tail(nil, X16), flatten(X16, X15))), SCOPE: 4, CLAUSE: 7\n\nX15 is free\nX16 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(nil, X16), flatten(X16, T14))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX15 is free\nX19 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(tail(nil, X16), flatten(X16, T14)), SCOPE: 5, CLAUSE: 5 | ','(tail(nil, X16), flatten(X16, T14)), SCOPE: 5, CLAUSE: 6\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: flatten(nil, T14) | ','(tail(nil, X16), flatten(X16, T14)), SCOPE: 5, CLAUSE: 6\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: flatten(nil, T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(nil, X16), flatten(X16, T14)), SCOPE: 5, CLAUSE: 6\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: flatten(nil, T14), SCOPE: 6, CLAUSE: 0 | flatten(nil, T14), SCOPE: 6, CLAUSE: 1 | flatten(nil, T14), SCOPE: 6, CLAUSE: 2 | ?_6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: flatten(nil, T14), SCOPE: 6, CLAUSE: 1 | flatten(nil, T14), SCOPE: 6, CLAUSE: 2 | ?_6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(head(nil, atom(X28)), ','(!_6, ','(eq(T16, .(X28, X29)), ','(tail(nil, X30), flatten(X30, X29))))) | flatten(nil, T14), SCOPE: 6, CLAUSE: 2 | ?_6\n\nX28 is free\nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(nil, atom(X28)), ','(!_6, ','(eq(T16, .(X28, X29)), ','(tail(nil, X30), flatten(X30, X29))))), SCOPE: 7, CLAUSE: 3 | ','(head(nil, atom(X28)), ','(!_6, ','(eq(T16, .(X28, X29)), ','(tail(nil, X30), flatten(X30, X29))))), SCOPE: 7, CLAUSE: 4 | flatten(nil, T14), SCOPE: 6, CLAUSE: 2 | ?_6\n\nX28 is free\nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_6, ','(eq(T16, .(X33, X29)), ','(tail(nil, X30), flatten(X30, X29)))) | ','(head(nil, atom(X28)), ','(!_6, ','(eq(T16, .(X28, X29)), ','(tail(nil, X30), flatten(X30, X29))))), SCOPE: 7, CLAUSE: 4 | flatten(nil, T14), SCOPE: 6, CLAUSE: 2 | ?_6\n\nX28 is free\nX29 is free; \nX30 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(eq(T16, .(X33, X29)), ','(tail(nil, X30), flatten(X30, X29)))\n\nX29 is free\nX30 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(eq(T16, .(X33, X29)), ','(tail(nil, X30), flatten(X30, X29))), SCOPE: 8, CLAUSE: 7\n\nX29 is free\nX30 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(nil, X30), flatten(X30, T20))\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\nX29 is free\nX33 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(nil, X30), flatten(X30, T20)), SCOPE: 9, CLAUSE: 5 | ','(tail(nil, X30), flatten(X30, T20)), SCOPE: 9, CLAUSE: 6\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flatten(nil, T20) | ','(tail(nil, X30), flatten(X30, T20)), SCOPE: 9, CLAUSE: 6\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(!_1, ','(eq(T10, .(T23, X15)), ','(tail(cons(atom(T23), T22), X16), flatten(X16, X15)))) | flatten(cons(atom(T23), T22), T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT23 is ground\nT22 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX15 is free; \nX16 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: flatten(T8, T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nflatten(T8, T2) !=? flatten(atom(X6), X7)\nhead(T8, atom(X14)) !=? head(nil, X18)\nhead(T8, atom(X14)) !=? head(cons(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(eq(T10, .(T23, X15)), ','(tail(cons(atom(T23), T22), X16), flatten(X16, X15)))\n\nT23 is ground\nT22 is ground\nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(eq(T10, .(T23, X15)), ','(tail(cons(atom(T23), T22), X16), flatten(X16, X15))), SCOPE: 10, CLAUSE: 7\n\nT23 is ground\nT22 is ground\nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(tail(cons(atom(T25), T22), X16), flatten(X16, T27))\n\nT22 is ground\nT25 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\nX15 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(cons(atom(T25), T22), X16), flatten(X16, T27)), SCOPE: 11, CLAUSE: 5 | ','(tail(cons(atom(T25), T22), X16), flatten(X16, T27)), SCOPE: 11, CLAUSE: 6\n\nT22 is ground\nT25 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(cons(atom(T25), T22), X16), flatten(X16, T27)), SCOPE: 11, CLAUSE: 6\n\nT22 is ground\nT25 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: flatten(T29, T27)\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(head(T30, cons(X52, X53)), ','(tail(T30, X54), flatten(cons(X52, cons(X53, X54)), T32)))\n\nT30 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nX52 is free; \nX53 is free; \nX54 is free; \nflatten(T30, T2) !=? flatten(atom(X6), X7)\nhead(T30, atom(X14)) !=? head(nil, X18)\nhead(T30, atom(X14)) !=? head(cons(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(head(T30, cons(X52, X53)), ','(tail(T30, X54), flatten(cons(X52, cons(X53, X54)), T32))), SCOPE: 12, CLAUSE: 3 | ','(head(T30, cons(X52, X53)), ','(tail(T30, X54), flatten(cons(X52, cons(X53, X54)), T32))), SCOPE: 12, CLAUSE: 4\n\nT30 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nX52 is free; \nX53 is free; \nX54 is free; \nflatten(T30, T2) !=? flatten(atom(X6), X7)\nhead(T30, atom(X14)) !=? head(nil, X18)\nhead(T30, atom(X14)) !=? head(cons(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(head(T30, cons(X52, X53)), ','(tail(T30, X54), flatten(cons(X52, cons(X53, X54)), T32))), SCOPE: 12, CLAUSE: 4\n\nT30 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nX52 is free; \nX53 is free; \nX54 is free; \nX56 is free; \nflatten(T30, T2) !=? flatten(atom(X6), X7)\nhead(T30, atom(X14)) !=? head(nil, X18)\nhead(T30, atom(X14)) !=? head(cons(X40, X41), X40)\nhead(T30, cons(X52, X53)) !=? head(nil, X56)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail(cons(cons(T35, T36), T34), X54), flatten(cons(T35, cons(T36, X54)), T32))\n\nT35 is ground\nT36 is ground\nT34 is ground\nX6 is free; \nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nX52 is free; \nX53 is free; \nX54 is free; \nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\nX6 is free\nX7 is free; \nX14 is free; \nX18 is free; \nX40 is free; \nX41 is free; \nX52 is free; \nX53 is free; \nX56 is free; \nX59 is free; \nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(tail(cons(cons(T35, T36), T34), X54), flatten(cons(T35, cons(T36, X54)), T32)), SCOPE: 13, CLAUSE: 5 | ','(tail(cons(cons(T35, T36), T34), X54), flatten(cons(T35, cons(T36, X54)), T32)), SCOPE: 13, CLAUSE: 6\n\nT35 is ground\nT36 is ground\nT34 is ground\nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(tail(cons(cons(T35, T36), T34), X54), flatten(cons(T35, cons(T36, X54)), T32)), SCOPE: 13, CLAUSE: 6\n\nT35 is ground\nT36 is ground\nT34 is ground\nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: flatten(cons(T37, cons(T38, T39)), T32)\n\nT37 is ground\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 2;

52 [label="EVAL with clause\nflatten(atom(X6), X7) :- ','(!, eq(X7, .(X6, []))).\nand substitution\nX6 -> T3\nT1 -> atom(T3)\nT2 -> T5\nX7 -> T5\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="EVAL with clause\nflatten(X12, X13) :- ','(head(X12, atom(X14)), ','(!, ','(eq(X13, .(X14, X15)), ','(tail(X12, X16), flatten(X16, X15))))).\nand substitution\nT1 -> T8\nX12 -> T8\nT2 -> T10\nX13 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 10;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="EVAL with clause\neq(X9, X9).\nand substitution\nT5 -> .(T7, [])\nX9 -> .(T7, [])\nT3 -> T7\nT6 -> .(T7, [])", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 59 [arrowhead = none ];
59 -> 9;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 60 [arrowhead = none ];
60 -> 11;

61 [label="EVAL with clause\nhead(nil, X18).\nand substitution\nT8 -> nil\nX14 -> X19\nX18 -> atom(X19)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 61 [arrowhead = none ];
61 -> 12;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="EVAL with clause\nhead(cons(X40, X41), X40).\nand substitution\nX40 -> atom(T23)\nX41 -> T22\nT8 -> cons(atom(T23), T22)\nX14 -> T23\nT21 -> atom(T23)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 64 [arrowhead = none ];
64 -> 34;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 65 [arrowhead = none ];
65 -> 35;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 66 [arrowhead = none ];
66 -> 15;

67 [label="EVAL with clause\neq(X21, X21).\nand substitution\nT10 -> .(T12, T14)\nX21 -> .(T12, T14)\nX19 -> T12\nX15 -> T14\nT11 -> .(T12, T14)\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 67 [arrowhead = none ];
67 -> 16;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 68 [arrowhead = none ];
68 -> 17;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 69 [arrowhead = none ];
69 -> 18;

70 [label="EVAL with clause\ntail(nil, nil).\nand substitution\nX16 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 70 [arrowhead = none ];
70 -> 19;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 71 [arrowhead = none ];
71 -> 20;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 72 [arrowhead = none ];
72 -> 21;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 73 [arrowhead = none ];
73 -> 22;

74 [label="BACKTRACK\nfor clause: tail(cons(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 74 [arrowhead = none ];
74 -> 33;

75 [label="BACKTRACK\nfor clause: flatten(atom(X), Y) :- ','(!, eq(Y, .(X, [])))because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 75 [arrowhead = none ];
75 -> 23;

76 [label="EVAL with clause\nflatten(X26, X27) :- ','(head(X26, atom(X28)), ','(!, ','(eq(X27, .(X28, X29)), ','(tail(X26, X30), flatten(X30, X29))))).\nand substitution\nX26 -> nil\nT14 -> T16\nX27 -> T16\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 76 [arrowhead = none ];
76 -> 24;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 77 [arrowhead = none ];
77 -> 25;

78 [label="EVAL with clause\nhead(nil, X32).\nand substitution\nX28 -> X33\nX32 -> atom(X33)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 78 [arrowhead = none ];
78 -> 26;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 80 [arrowhead = none ];
80 -> 28;

81 [label="EVAL with clause\neq(X35, X35).\nand substitution\nT16 -> .(T18, T20)\nX35 -> .(T18, T20)\nX33 -> T18\nX29 -> T20\nT17 -> .(T18, T20)\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 81 [arrowhead = none ];
81 -> 29;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 82 [arrowhead = none ];
82 -> 30;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 83 [arrowhead = none ];
83 -> 31;

84 [label="EVAL with clause\ntail(nil, nil).\nand substitution\nX30 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="INSTANCE with matching:\nT14 -> T20\nX16 -> X30", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 85 [arrowhead = none , style=dashed];
85 -> 19[style=dashed];

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 86 [arrowhead = none ];
86 -> 36;

87 [label="EVAL with clause\nflatten(X50, X51) :- ','(head(X50, cons(X52, X53)), ','(tail(X50, X54), flatten(cons(X52, cons(X53, X54)), X51))).\nand substitution\nT8 -> T30\nX50 -> T30\nT2 -> T32\nX51 -> T32\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 87 [arrowhead = none ];
87 -> 43;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 88 [arrowhead = none ];
88 -> 37;

89 [label="EVAL with clause\neq(X43, X43).\nand substitution\nT10 -> .(T25, T27)\nX43 -> .(T25, T27)\nT23 -> T25\nX15 -> T27\nT24 -> .(T25, T27)\nT26 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 89 [arrowhead = none ];
89 -> 38;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 90 [arrowhead = none ];
90 -> 39;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="EVAL with clause\ntail(cons(X46, X47), X47).\nand substitution\nT25 -> T28\nX46 -> atom(T28)\nT22 -> T29\nX47 -> T29\nX16 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="INSTANCE with matching:\nT1 -> T29\nT2 -> T27", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 94 [arrowhead = none , style=dashed];
94 -> 1[style=dashed];

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="BACKTRACK\nfor clause: head(nil, X1)\nwith clash: (head(T30, atom(X14)), head(nil, X18))", fontsize=14, style = filled, fillcolor = white];
44 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="EVAL with clause\nhead(cons(X59, X60), X59).\nand substitution\nX59 -> cons(T35, T36)\nX60 -> T34\nT30 -> cons(cons(T35, T36), T34)\nX52 -> T35\nX53 -> T36\nT33 -> cons(T35, T36)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 99 [arrowhead = none ];
99 -> 48;

100 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 100 [arrowhead = none ];
100 -> 49;

101 [label="EVAL with clause\ntail(cons(X63, X64), X64).\nand substitution\nT35 -> T37\nT36 -> T38\nX63 -> cons(T37, T38)\nT34 -> T39\nX64 -> T39\nX54 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="INSTANCE with matching:\nT1 -> cons(T37, cons(T38, T39))\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 102 [arrowhead = none , style=dashed];
102 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

flatten19(.(T18, T20), X16) :- flatten19(T20, X30).
flatten1(atom(T7), .(T7, [])).
flatten1(nil, .(T12, .(T18, T20))) :- flatten19(T20, X30).
flatten1(cons(atom(T28), T29), .(T28, T27)) :- flatten1(T29, T27).
flatten1(cons(cons(T37, T38), T39), T32) :- flatten1(cons(T37, cons(T38, T39)), T32).

Queries:

flatten1(g,a).

(3) PrologToPiTRSProof (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)
flatten19_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → U2_GA(T12, T18, T20, flatten19_in_aa(T20, X30))
FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → FLATTEN19_IN_AA(T20, X30)
FLATTEN19_IN_AA(.(T18, T20), X16) → U1_AA(T18, T20, X16, flatten19_in_aa(T20, X30))
FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → U3_GA(T28, T29, T27, flatten1_in_ga(T29, T27))
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → U4_GA(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
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

(6) Obligation:

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

FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → U2_GA(T12, T18, T20, flatten19_in_aa(T20, X30))
FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → FLATTEN19_IN_AA(T20, X30)
FLATTEN19_IN_AA(.(T18, T20), X16) → U1_AA(T18, T20, X16, flatten19_in_aa(T20, X30))
FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → U3_GA(T28, T29, T27, flatten1_in_ga(T29, T27))
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → U4_GA(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

FLATTEN19_IN_AA → FLATTEN19_IN_AA

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

(14) 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 = FLATTEN19_IN_AA evaluates to t =FLATTEN19_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLATTEN19_IN_AA to FLATTEN19_IN_AA.

(15) FALSE

(16) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x1, x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39)) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)))
FLATTEN1_IN_GA(cons(atom(T28), T29)) → FLATTEN1_IN_GA(T29)

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

(21) 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(T37, T38), T39)) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)))
FLATTEN1_IN_GA(cons(atom(T28), T29)) → FLATTEN1_IN_GA(T29)

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   

(22) Obligation:

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) PrologToPiTRSProof (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)
flatten19_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(26) Obligation:

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

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)

(27) DependencyPairsProof (EQUIVALENT transformation)

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

FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → U2_GA(T12, T18, T20, flatten19_in_aa(T20, X30))
FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → FLATTEN19_IN_AA(T20, X30)
FLATTEN19_IN_AA(.(T18, T20), X16) → U1_AA(T18, T20, X16, flatten19_in_aa(T20, X30))
FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → U3_GA(T28, T29, T27, flatten1_in_ga(T29, T27))
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → U4_GA(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(28) Obligation:

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

FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → U2_GA(T12, T18, T20, flatten19_in_aa(T20, X30))
FLATTEN1_IN_GA(nil, .(T12, .(T18, T20))) → FLATTEN19_IN_AA(T20, X30)
FLATTEN19_IN_AA(.(T18, T20), X16) → U1_AA(T18, T20, X16, flatten19_in_aa(T20, X30))
FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → U3_GA(T28, T29, T27, flatten1_in_ga(T29, T27))
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → U4_GA(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(30) Complex Obligation (AND)

(31) Obligation:

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

FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA

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

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

FLATTEN19_IN_AA(.(T18, T20), X16) → FLATTEN19_IN_AA(T20, X30)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
FLATTEN19_IN_AA(x1, x2)  =  FLATTEN19_IN_AA

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

(34) PiDPToQDPProof (SOUND transformation)

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

(35) Obligation:

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

FLATTEN19_IN_AA → FLATTEN19_IN_AA

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

(36) 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 = FLATTEN19_IN_AA evaluates to t =FLATTEN19_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLATTEN19_IN_AA to FLATTEN19_IN_AA.

(37) FALSE

(38) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)

The TRS R consists of the following rules:

flatten1_in_ga(atom(T7), .(T7, [])) → flatten1_out_ga(atom(T7), .(T7, []))
flatten1_in_ga(nil, .(T12, .(T18, T20))) → U2_ga(T12, T18, T20, flatten19_in_aa(T20, X30))
flatten19_in_aa(.(T18, T20), X16) → U1_aa(T18, T20, X16, flatten19_in_aa(T20, X30))
U1_aa(T18, T20, X16, flatten19_out_aa(T20, X30)) → flatten19_out_aa(.(T18, T20), X16)
U2_ga(T12, T18, T20, flatten19_out_aa(T20, X30)) → flatten1_out_ga(nil, .(T12, .(T18, T20)))
flatten1_in_ga(cons(atom(T28), T29), .(T28, T27)) → U3_ga(T28, T29, T27, flatten1_in_ga(T29, T27))
flatten1_in_ga(cons(cons(T37, T38), T39), T32) → U4_ga(T37, T38, T39, T32, flatten1_in_ga(cons(T37, cons(T38, T39)), T32))
U4_ga(T37, T38, T39, T32, flatten1_out_ga(cons(T37, cons(T38, T39)), T32)) → flatten1_out_ga(cons(cons(T37, T38), T39), T32)
U3_ga(T28, T29, T27, flatten1_out_ga(T29, T27)) → flatten1_out_ga(cons(atom(T28), T29), .(T28, T27))

The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
atom(x1)  =  atom(x1)
flatten1_out_ga(x1, x2)  =  flatten1_out_ga(x2)
nil  =  nil
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
flatten19_in_aa(x1, x2)  =  flatten19_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
flatten19_out_aa(x1, x2)  =  flatten19_out_aa(x1)
.(x1, x2)  =  .(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)

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

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39), T32) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)), T32)
FLATTEN1_IN_GA(cons(atom(T28), T29), .(T28, T27)) → FLATTEN1_IN_GA(T29, T27)

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

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

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

FLATTEN1_IN_GA(cons(cons(T37, T38), T39)) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)))
FLATTEN1_IN_GA(cons(atom(T28), T29)) → FLATTEN1_IN_GA(T29)

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

(43) 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(T37, T38), T39)) → FLATTEN1_IN_GA(cons(T37, cons(T38, T39)))
FLATTEN1_IN_GA(cons(atom(T28), T29)) → FLATTEN1_IN_GA(T29)

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   

(44) Obligation:

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

(45) PisEmptyProof (EQUIVALENT transformation)

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

(46) TRUE