Runtime Complexity proof of /tmp/tmpjzHhN1/fold1.pl

Runtime Complexity of the query pattern
fold(g,g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(1)).

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 72 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 61 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, c).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Query: fold(g,g,a)

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, eq(T8, T10)) | fold(T8, [], T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX8 is free\nX9 is free\nfold(T1, T2, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(T8, T10)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T8, T10), SCOPE: 2, CLAUSE: 7\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25))))\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 3 | ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 4\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 4\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25)))\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 6\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 6\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(myop(T22, T39, X30), fold(X30, T40, T25))\n\nT22 is ground\nT39 is ground\nT40 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(myop(T22, T39, X30), fold(X30, T40, T25)), SCOPE: 5, CLAUSE: 2\n\nT22 is ground\nT39 is ground\nT40 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: fold(c, T40, T25)\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: fold(c, T40, T25), SCOPE: 6, CLAUSE: 0 | fold(c, T40, T25), SCOPE: 6, CLAUSE: 1\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(!_6, eq(c, T46)) | fold(c, [], T25), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: fold(c, T40, T25), SCOPE: 6, CLAUSE: 1\n\nT40 is ground\nX63 is free\nX64 is free\nfold(c, T40, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(c, T46)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: eq(c, T46), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(c, X83, X85), fold(X85, X84, T55))))\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(c, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(c, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 3 | ','(head(T53, X83), ','(tail(T53, X84), ','(myop(c, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 4\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(c, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(c, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 4\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(c, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(T61, T62), X84), ','(myop(c, T61, X85), fold(X85, X84, T55)))\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(tail(.(T61, T62), X84), ','(myop(c, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T61, T62), X84), ','(myop(c, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 6\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(T61, T62), X84), ','(myop(c, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 6\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(myop(c, T69, X85), fold(X85, T70, T55))\n\nT69 is ground\nT70 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(myop(c, T69, X85), fold(X85, T70, T55)), SCOPE: 10, CLAUSE: 2\n\nT69 is ground\nT70 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 77 [arrowhead = none ];
77 -> 4;

78 [label="EVAL with clause\nfold(X8, [], X9) :- ','(!_1, eq(X8, X9)).\nand substitutionT1 -> T8,\nX8 -> T8,\nT2 -> [],\nT3 -> T10,\nX9 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 79 [arrowhead = none ];
79 -> 8;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 80 [arrowhead = none ];
80 -> 9;

81 [label="ONLY EVAL with clause\nfold(X25, X26, X27) :- ','(head(X26, X28), ','(tail(X26, X29), ','(myop(X25, X28, X30), fold(X30, X29, X27)))).\nand substitutionT1 -> T22,\nX25 -> T22,\nT2 -> T23,\nX26 -> T23,\nT3 -> T25,\nX27 -> T25,\nT24 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 81 [arrowhead = none ];
81 -> 20;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 82 [arrowhead = none ];
82 -> 12;

83 [label="EVAL with clause\neq(X12, X12).\nand substitutionT8 -> T13,\nX12 -> T13,\nT10 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 83 [arrowhead = none ];
83 -> 14;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 85 [arrowhead = none ];
85 -> 18;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T22, T23, T3), fold(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 87 [arrowhead = none ];
87 -> 24;

88 [label="EVAL with clause\nhead(.(X40, X41), X40).\nand substitutionX40 -> T31,\nX41 -> T32,\nT23 -> .(T31, T32),\nX28 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 88 [arrowhead = none ];
88 -> 29;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 89 [arrowhead = none ];
89 -> 30;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 90 [arrowhead = none ];
90 -> 31;

91 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 91 [arrowhead = none ];
91 -> 33;

92 [label="ONLY EVAL with clause\ntail(.(X52, X53), X53).\nand substitutionT31 -> T39,\nX52 -> T39,\nT32 -> T40,\nX53 -> T40,\nX29 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 92 [arrowhead = none ];
92 -> 37;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 93 [arrowhead = none ];
93 -> 38;

94 [label="EVAL with clause\nmyop(a, b, c).\nand substitutionT22 -> a,\nT39 -> b,\nX30 -> c", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="EVAL with clause\nfold(X63, [], X64) :- ','(!_6, eq(X63, X64)).\nand substitutionX63 -> c,\nT40 -> [],\nT25 -> T46,\nX64 -> T46,\nT45 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 97 [arrowhead = none ];
97 -> 44;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 99 [arrowhead = none ];
99 -> 48;

100 [label="ONLY EVAL with clause\nfold(X80, X81, X82) :- ','(head(X81, X83), ','(tail(X81, X84), ','(myop(X80, X83, X85), fold(X85, X84, X82)))).\nand substitutionX80 -> c,\nT40 -> T53,\nX81 -> T53,\nT25 -> T55,\nX82 -> T55,\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 100 [arrowhead = none ];
100 -> 58;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="EVAL with clause\neq(X67, X67).\nand substitutionX67 -> c,\nT46 -> c", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 102 [arrowhead = none ];
102 -> 53;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 103 [arrowhead = none ];
103 -> 55;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 104 [arrowhead = none ];
104 -> 56;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 105 [arrowhead = none ];
105 -> 60;

106 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(c, T53, T25), fold(X63, [], X64))", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 106 [arrowhead = none ];
106 -> 62;

107 [label="EVAL with clause\nhead(.(X95, X96), X95).\nand substitutionX95 -> T61,\nX96 -> T62,\nT53 -> .(T61, T62),\nX83 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 107 [arrowhead = none ];
107 -> 65;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 108 [arrowhead = none ];
108 -> 66;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 109 [arrowhead = none ];
109 -> 69;

110 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 110 [arrowhead = none ];
110 -> 70;

111 [label="ONLY EVAL with clause\ntail(.(X107, X108), X108).\nand substitutionT61 -> T69,\nX107 -> T69,\nT62 -> T70,\nX108 -> T70,\nX84 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 111 [arrowhead = none ];
111 -> 72;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 112 [arrowhead = none ];
112 -> 75;

113 [label="BACKTRACK\nfor clause: myop(a, b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 113 [arrowhead = none ];
113 -> 76;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, [])) → f2_out1(c)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f2_in

Defined Pair Symbols:none

Compound Symbols:none

(3) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(4) BOUNDS(O(1), O(1))

(5) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, eq(T6, T8)) | fold(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX6 is free\nX7 is free\nfold(T1, T2, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(T6, T8)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(T6, T8), SCOPE: 2, CLAUSE: 7\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18))))\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 3 | ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18)))\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(myop(T15, T31, X19), fold(X19, T32, T18))\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(myop(T15, T31, X19), fold(X19, T32, T18)), SCOPE: 5, CLAUSE: 2\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: fold(c, T32, T18)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: fold(c, T32, T18), SCOPE: 6, CLAUSE: 0 | fold(c, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(!_6, eq(c, T38)) | fold(c, [], T18), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: fold(c, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground\nX49 is free\nX50 is free\nfold(c, T32, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: eq(c, T38)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: eq(c, T38), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(c, X69, X71), fold(X71, X70, T47))))\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(c, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(c, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 3 | ','(head(T45, X69), ','(tail(T45, X70), ','(myop(c, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(c, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(c, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(c, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T53, T54), X70), ','(myop(c, T53, X71), fold(X71, X70, T47)))\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(.(T53, T54), X70), ','(myop(c, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T53, T54), X70), ','(myop(c, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(tail(.(T53, T54), X70), ','(myop(c, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(myop(c, T61, X71), fold(X71, T62, T47))\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(myop(c, T61, X71), fold(X71, T62, T47)), SCOPE: 10, CLAUSE: 2\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 75 [arrowhead = none ];
75 -> 3;

76 [label="EVAL with clause\nfold(X6, [], X7) :- ','(!_1, eq(X6, X7)).\nand substitutionT1 -> T6,\nX6 -> T6,\nT2 -> [],\nT3 -> T8,\nX7 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 76 [arrowhead = none ];
76 -> 5;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 78 [arrowhead = none ];
78 -> 10;

79 [label="ONLY EVAL with clause\nfold(X14, X15, X16) :- ','(head(X15, X17), ','(tail(X15, X18), ','(myop(X14, X17, X19), fold(X19, X18, X16)))).\nand substitutionT1 -> T15,\nX14 -> T15,\nT2 -> T16,\nX15 -> T16,\nT3 -> T18,\nX16 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 79 [arrowhead = none ];
79 -> 19;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 80 [arrowhead = none ];
80 -> 11;

81 [label="EVAL with clause\neq(X10, X10).\nand substitutionT6 -> T11,\nX10 -> T11,\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 82 [arrowhead = none ];
82 -> 15;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 83 [arrowhead = none ];
83 -> 16;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T15, T16, T3), fold(X6, [], X7))", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="EVAL with clause\nhead(.(X26, X27), X26).\nand substitutionX26 -> T23,\nX27 -> T24,\nT16 -> .(T23, T24),\nX17 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 87 [arrowhead = none ];
87 -> 26;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 88 [arrowhead = none ];
88 -> 27;

89 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 89 [arrowhead = none ];
89 -> 28;

90 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitutionT23 -> T31,\nX38 -> T31,\nT24 -> T32,\nX39 -> T32,\nX18 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 90 [arrowhead = none ];
90 -> 32;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 91 [arrowhead = none ];
91 -> 34;

92 [label="EVAL with clause\nmyop(a, b, c).\nand substitutionT15 -> a,\nT31 -> b,\nX19 -> c", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 92 [arrowhead = none ];
92 -> 35;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 93 [arrowhead = none ];
93 -> 36;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 94 [arrowhead = none ];
94 -> 42;

95 [label="EVAL with clause\nfold(X49, [], X50) :- ','(!_6, eq(X49, X50)).\nand substitutionX49 -> c,\nT32 -> [],\nT18 -> T38,\nX50 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 95 [arrowhead = none ];
95 -> 43;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="ONLY EVAL with clause\nfold(X66, X67, X68) :- ','(head(X67, X69), ','(tail(X67, X70), ','(myop(X66, X69, X71), fold(X71, X70, X68)))).\nand substitutionX66 -> c,\nT32 -> T45,\nX67 -> T45,\nT18 -> T47,\nX68 -> T47,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 98 [arrowhead = none ];
98 -> 57;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 99 [arrowhead = none ];
99 -> 49;

100 [label="EVAL with clause\neq(X53, X53).\nand substitutionX53 -> c,\nT38 -> c", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 100 [arrowhead = none ];
100 -> 51;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 101 [arrowhead = none ];
101 -> 52;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 102 [arrowhead = none ];
102 -> 54;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 103 [arrowhead = none ];
103 -> 59;

104 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(c, T45, T18), fold(X49, [], X50))", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 104 [arrowhead = none ];
104 -> 61;

105 [label="EVAL with clause\nhead(.(X81, X82), X81).\nand substitutionX81 -> T53,\nX82 -> T54,\nT45 -> .(T53, T54),\nX69 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 105 [arrowhead = none ];
105 -> 63;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 106 [arrowhead = none ];
106 -> 64;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 107 [arrowhead = none ];
107 -> 67;

108 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 108 [arrowhead = none ];
108 -> 68;

109 [label="ONLY EVAL with clause\ntail(.(X93, X94), X94).\nand substitutionT53 -> T61,\nX93 -> T61,\nT54 -> T62,\nX94 -> T62,\nX70 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 109 [arrowhead = none ];
109 -> 71;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 110 [arrowhead = none ];
110 -> 73;

111 [label="BACKTRACK\nfor clause: myop(a, b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 111 [arrowhead = none ];
111 -> 74;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, [])) → f1_out1(c)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f1_in

Defined Pair Symbols:none

Compound Symbols:none