Runtime Complexity proof of /tmp/tmpZW3CXb/short1.pl

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

0 Prolog

↳1 BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 68 ms)

↳4 CdtProblem

    ↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳8 CdtProblem

    ↳9 CdtKnowledgeProof (⇔, 1 ms)

    ↳10 BOUNDS(O(1), O(1))

↳11 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 63 ms)

↳12 CdtProblem

    ↳13 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳14 CdtProblem

    ↳15 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳16 CdtProblem

(0) Obligation:

Clauses:

p(X, Y, Z) :- ','(=(X, Y), ','(=(Z, 1), !)).
p(X, Y, Z) :- ','(=(Z, 1), ','(=(Y, X), p(X, Y, Z))).
=(X, X).

Query: p(a,a,a)

(1) BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

p(X, Y, Z) :- ','(user_defined_=(X, Y), ','(user_defined_=(Z, 1), !)).
p(X, Y, Z) :- ','(user_defined_=(Z, 1), ','(user_defined_=(Y, X), p(X, Y, Z))).
user_defined_=(X, X).

Query: p(a,a,a)

(3) 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: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(user_defined_=(T16, T17), ','(user_defined_=(T18, 1), !_1)) | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(user_defined_=(T16, T17), ','(user_defined_=(T18, 1), !_1)), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(user_defined_=(T22, 1), !_1) | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(user_defined_=(T22, 1), !_1), SCOPE: 3, CLAUSE: 2 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: !_1 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(user_defined_=(T41, 1), ','(user_defined_=(T42, T43), p(T43, T42, T41)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(user_defined_=(T41, 1), ','(user_defined_=(T42, T43), p(T43, T42, T41))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(user_defined_=(T50, T51), p(T51, T50, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(user_defined_=(T50, T51), p(T51, T50, 1)), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: p(T57, T57, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(T57, T57, 1), SCOPE: 6, CLAUSE: 0 | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(user_defined_=(T63, T63), ','(user_defined_=(1, 1), !_6)) | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(user_defined_=(T63, T63), ','(user_defined_=(1, 1), !_6)), SCOPE: 7, CLAUSE: 2 | ?_7 | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(user_defined_=(1, 1), !_6) | ?_7 | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(user_defined_=(1, 1), !_6), SCOPE: 8, CLAUSE: 2 | ?_8 | ?_7 | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: !_6 | ?_8 | ?_7 | p(T57, T57, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(user_defined_=(T82, 1), ','(user_defined_=(T83, T84), p(T84, T83, T82)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(user_defined_=(T82, 1), ','(user_defined_=(T83, T84), p(T84, T83, T82))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(user_defined_=(T91, T92), p(T92, T91, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(user_defined_=(T91, T92), p(T92, T91, 1)), SCOPE: 10, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(T98, T98, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 73 [arrowhead = none ];
73 -> 4;

74 [label="ONLY EVAL with clause\np(X7, X8, X9) :- ','(user_defined_=(X7, X8), ','(user_defined_=(X9, 1), !_1)).\nand substitutionT1 -> T16,\nX7 -> T16,\nT2 -> T17,\nX8 -> T17,\nT3 -> T18,\nX9 -> T18,\nT13 -> T16,\nT14 -> T17,\nT15 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 75 [arrowhead = none ];
75 -> 11;

76 [label="EVAL with clause\nuser_defined_=(X12, X12).\nand substitutionT16 -> T21,\nX12 -> T21,\nT17 -> T21,\nT18 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 76 [arrowhead = none ];
76 -> 15;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 77 [arrowhead = none ];
77 -> 16;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 78 [arrowhead = none ];
78 -> 18;

79 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 79 [arrowhead = none ];
79 -> 58;

80 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT22 -> 1,\nX15 -> 1,\nT25 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 82 [arrowhead = none ];
82 -> 23;

83 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 83 [arrowhead = none ];
83 -> 25;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 84 [arrowhead = none ];
84 -> 24;

85 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 85 [arrowhead = none ];
85 -> 26;

86 [label="ONLY EVAL with clause\np(X25, X26, X27) :- ','(user_defined_=(X27, 1), ','(user_defined_=(X26, X25), p(X25, X26, X27))).\nand substitutionT1 -> T43,\nX25 -> T43,\nT2 -> T42,\nX26 -> T42,\nT3 -> T41,\nX27 -> T41,\nT40 -> T41,\nT39 -> T42,\nT38 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 86 [arrowhead = none ];
86 -> 29;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 87 [arrowhead = none ];
87 -> 30;

88 [label="EVAL with clause\nuser_defined_=(X33, X33).\nand substitutionT41 -> 1,\nX33 -> 1,\nT49 -> 1,\nT42 -> T50,\nT43 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 88 [arrowhead = none ];
88 -> 34;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 90 [arrowhead = none ];
90 -> 37;

91 [label="EVAL with clause\nuser_defined_=(X39, X39).\nand substitutionT50 -> T57,\nX39 -> T57,\nT51 -> T57,\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 91 [arrowhead = none ];
91 -> 39;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 92 [arrowhead = none ];
92 -> 40;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="ONLY EVAL with clause\np(X49, X50, X51) :- ','(user_defined_=(X49, X50), ','(user_defined_=(X51, 1), !_6)).\nand substitutionT57 -> T63,\nX49 -> T63,\nX50 -> T63,\nX51 -> 1,\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 94 [arrowhead = none ];
94 -> 44;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 95 [arrowhead = none ];
95 -> 46;

96 [label="ONLY EVAL with clause\nuser_defined_=(X54, X54).\nand substitutionT63 -> T66,\nX54 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 96 [arrowhead = none ];
96 -> 49;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 97 [arrowhead = none ];
97 -> 50;

98 [label="ONLY EVAL with clause\nuser_defined_=(X57, X57).\nand substitutionX57 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 98 [arrowhead = none ];
98 -> 55;

99 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 99 [arrowhead = none ];
99 -> 56;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 100 [arrowhead = none ];
100 -> 57;

101 [label="ONLY EVAL with clause\np(X67, X68, X69) :- ','(user_defined_=(X69, 1), ','(user_defined_=(X68, X67), p(X67, X68, X69))).\nand substitutionT1 -> T84,\nX67 -> T84,\nT2 -> T83,\nX68 -> T83,\nT3 -> T82,\nX69 -> T82,\nT81 -> T82,\nT80 -> T83,\nT79 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 101 [arrowhead = none ];
101 -> 60;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 102 [arrowhead = none ];
102 -> 62;

103 [label="EVAL with clause\nuser_defined_=(X75, X75).\nand substitutionT82 -> 1,\nX75 -> 1,\nT90 -> 1,\nT83 -> T91,\nT84 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 103 [arrowhead = none ];
103 -> 65;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 104 [arrowhead = none ];
104 -> 66;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 105 [arrowhead = none ];
105 -> 68;

106 [label="EVAL with clause\nuser_defined_=(X81, X81).\nand substitutionT91 -> T98,\nX81 -> T98,\nT92 -> T98,\nT97 -> T98", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 106 [arrowhead = none ];
106 -> 71;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 107 [arrowhead = none ];
107 -> 72;

108 [label="INSTANCE with matching:\nT57 -> T98", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 108 [arrowhead = none , style=dashed];
108 -> 39[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(1)
f2_in → U1(f39_in)
f2_in → U2(f39_in)
U1(f39_out1) → f2_out1(1)
U2(f39_out1) → f2_out1(1)
f39_in → f39_out1

Tuples:

F2_IN → c1(U1'(f39_in), F39_IN)
F2_IN → c2(U2'(f39_in), F39_IN)

S tuples:

F2_IN → c1(U1'(f39_in), F39_IN)
F2_IN → c2(U2'(f39_in), F39_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f39_in

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(1)
f2_in → U1(f39_in)
f2_in → U2(f39_in)
U1(f39_out1) → f2_out1(1)
U2(f39_out1) → f2_out1(1)
f39_in → f39_out1

Tuples:

F2_IN → c(U1'(f39_in))
F2_IN → c(F39_IN)
F2_IN → c(U2'(f39_in))

S tuples:

F2_IN → c(U1'(f39_in))
F2_IN → c(F39_IN)
F2_IN → c(U2'(f39_in))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f39_in

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(1)
f2_in → U1(f39_in)
f2_in → U2(f39_in)
U1(f39_out1) → f2_out1(1)
U2(f39_out1) → f2_out1(1)
f39_in → f39_out1

Tuples:

F2_IN → c

S tuples:

F2_IN → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f39_in

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN → c
F2_IN → c
F2_IN → c
F2_IN → c

Now S is empty

(10) BOUNDS(O(1), O(1))

(11) 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: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(user_defined_=(T10, T11), ','(user_defined_=(T12, 1), !_1)) | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(user_defined_=(T10, T11), ','(user_defined_=(T12, 1), !_1)), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(user_defined_=(T16, 1), !_1) | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(user_defined_=(T16, 1), !_1), SCOPE: 3, CLAUSE: 2 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: !_1 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", 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];
19 [label="19: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(user_defined_=(T35, 1), ','(user_defined_=(T36, T37), p(T37, T36, T35)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(user_defined_=(T35, 1), ','(user_defined_=(T36, T37), p(T37, T36, T35))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(user_defined_=(T44, T45), p(T45, T44, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(user_defined_=(T44, T45), p(T45, T44, 1)), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(T51, T51, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: p(T51, T51, 1), SCOPE: 6, CLAUSE: 0 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(user_defined_=(T57, T57), ','(user_defined_=(1, 1), !_6)) | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(user_defined_=(T57, T57), ','(user_defined_=(1, 1), !_6)), SCOPE: 7, CLAUSE: 2 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(user_defined_=(1, 1), !_6) | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(user_defined_=(1, 1), !_6), SCOPE: 8, CLAUSE: 2 | ?_8 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: !_6 | ?_8 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(user_defined_=(T76, 1), ','(user_defined_=(T77, T78), p(T78, T77, T76)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(user_defined_=(T76, 1), ','(user_defined_=(T77, T78), p(T78, T77, T76))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(user_defined_=(T85, T86), p(T86, T85, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(user_defined_=(T85, T86), p(T86, T85, 1)), SCOPE: 10, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: p(T92, T92, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 71 [arrowhead = none ];
71 -> 3;

72 [label="ONLY EVAL with clause\np(X4, X5, X6) :- ','(user_defined_=(X4, X5), ','(user_defined_=(X6, 1), !_1)).\nand substitutionT1 -> T10,\nX4 -> T10,\nT2 -> T11,\nX5 -> T11,\nT3 -> T12,\nX6 -> T12,\nT7 -> T10,\nT8 -> T11,\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 72 [arrowhead = none ];
72 -> 5;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="EVAL with clause\nuser_defined_=(X9, X9).\nand substitutionT10 -> T15,\nX9 -> T15,\nT11 -> T15,\nT12 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 77 [arrowhead = none ];
77 -> 54;

78 [label="EVAL with clause\nuser_defined_=(X12, X12).\nand substitutionT16 -> 1,\nX12 -> 1,\nT19 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 80 [arrowhead = none ];
80 -> 14;

81 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 81 [arrowhead = none ];
81 -> 19;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 82 [arrowhead = none ];
82 -> 17;

83 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="ONLY EVAL with clause\np(X22, X23, X24) :- ','(user_defined_=(X24, 1), ','(user_defined_=(X23, X22), p(X22, X23, X24))).\nand substitutionT1 -> T37,\nX22 -> T37,\nT2 -> T36,\nX23 -> T36,\nT3 -> T35,\nX24 -> T35,\nT34 -> T35,\nT33 -> T36,\nT32 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 84 [arrowhead = none ];
84 -> 27;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 85 [arrowhead = none ];
85 -> 28;

86 [label="EVAL with clause\nuser_defined_=(X30, X30).\nand substitutionT35 -> 1,\nX30 -> 1,\nT43 -> 1,\nT36 -> T44,\nT37 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 86 [arrowhead = none ];
86 -> 31;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 87 [arrowhead = none ];
87 -> 32;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 88 [arrowhead = none ];
88 -> 33;

89 [label="EVAL with clause\nuser_defined_=(X36, X36).\nand substitutionT44 -> T51,\nX36 -> T51,\nT45 -> T51,\nT50 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 89 [arrowhead = none ];
89 -> 36;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 91 [arrowhead = none ];
91 -> 41;

92 [label="ONLY EVAL with clause\np(X46, X47, X48) :- ','(user_defined_=(X46, X47), ','(user_defined_=(X48, 1), !_6)).\nand substitutionT51 -> T57,\nX46 -> T57,\nX47 -> T57,\nX48 -> 1,\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 92 [arrowhead = none ];
92 -> 43;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 93 [arrowhead = none ];
93 -> 45;

94 [label="ONLY EVAL with clause\nuser_defined_=(X51, X51).\nand substitutionT57 -> T60,\nX51 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 94 [arrowhead = none ];
94 -> 47;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 95 [arrowhead = none ];
95 -> 48;

96 [label="ONLY EVAL with clause\nuser_defined_=(X54, X54).\nand substitutionX54 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 96 [arrowhead = none ];
96 -> 51;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 97 [arrowhead = none ];
97 -> 52;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 98 [arrowhead = none ];
98 -> 53;

99 [label="ONLY EVAL with clause\np(X64, X65, X66) :- ','(user_defined_=(X66, 1), ','(user_defined_=(X65, X64), p(X64, X65, X66))).\nand substitutionT1 -> T78,\nX64 -> T78,\nT2 -> T77,\nX65 -> T77,\nT3 -> T76,\nX66 -> T76,\nT75 -> T76,\nT74 -> T77,\nT73 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 99 [arrowhead = none ];
99 -> 59;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 100 [arrowhead = none ];
100 -> 61;

101 [label="EVAL with clause\nuser_defined_=(X72, X72).\nand substitutionT76 -> 1,\nX72 -> 1,\nT84 -> 1,\nT77 -> T85,\nT78 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 101 [arrowhead = none ];
101 -> 63;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 102 [arrowhead = none ];
102 -> 64;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 103 [arrowhead = none ];
103 -> 67;

104 [label="EVAL with clause\nuser_defined_=(X78, X78).\nand substitutionT85 -> T92,\nX78 -> T92,\nT86 -> T92,\nT91 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 104 [arrowhead = none ];
104 -> 69;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 105 [arrowhead = none ];
105 -> 70;

106 [label="INSTANCE with matching:\nT51 -> T92", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 106 [arrowhead = none , style=dashed];
106 -> 36[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1(1)
f1_in → U1(f36_in)
f1_in → U2(f36_in)
U1(f36_out1) → f1_out1(1)
U2(f36_out1) → f1_out1(1)
f36_in → f36_out1

Tuples:

F1_IN → c1(U1'(f36_in), F36_IN)
F1_IN → c2(U2'(f36_in), F36_IN)

S tuples:

F1_IN → c1(U1'(f36_in), F36_IN)
F1_IN → c2(U2'(f36_in), F36_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f36_in

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

(13) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1(1)
f1_in → U1(f36_in)
f1_in → U2(f36_in)
U1(f36_out1) → f1_out1(1)
U2(f36_out1) → f1_out1(1)
f36_in → f36_out1

Tuples:

F1_IN → c(U1'(f36_in))
F1_IN → c(F36_IN)
F1_IN → c(U2'(f36_in))

S tuples:

F1_IN → c(U1'(f36_in))
F1_IN → c(F36_IN)
F1_IN → c(U2'(f36_in))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f36_in

Defined Pair Symbols:

F1_IN

Compound Symbols:

c

(15) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1(1)
f1_in → U1(f36_in)
f1_in → U2(f36_in)
U1(f36_out1) → f1_out1(1)
U2(f36_out1) → f1_out1(1)
f36_in → f36_out1

Tuples:

F1_IN → c

S tuples:

F1_IN → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f36_in

Defined Pair Symbols:

F1_IN

Compound Symbols:

c