Runtime Complexity proof of /tmp/tmp0BMzlr/select1.pl

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

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 162 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 106 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 169 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 110 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

select(X1, [], X2) :- ','(!, failure(a)).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).

Query: select(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: select(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, failure(a)) | select(T8, [], T3), SCOPE: 1, CLAUSE: 1 | select(T8, [], T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nselect(T1, T2, T3) !=? select(X10, [], X11)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: failure(a), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T18, T17), tail(T18, T20)) | select(T17, T18, T3), SCOPE: 1, CLAUSE: 2\n\nT17 is ground\nT18 is ground\nX10 is free\nX11 is free\nselect(T17, T18, T3) !=? select(X10, [], X11)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T18, T17), tail(T18, T20)), SCOPE: 3, CLAUSE: 3 | ','(head(T18, T17), tail(T18, T20)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T17, T18, T3), SCOPE: 1, CLAUSE: 2\n\nT17 is ground\nT18 is ground\nX10 is free\nX11 is free\nselect(T17, T18, T3) !=? select(X10, [], X11)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T18, T17), tail(T18, T20)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T17, T18, T3), SCOPE: 1, CLAUSE: 2\n\nT17 is ground\nT18 is ground\nX10 is free\nX11 is free\nselect(T17, T18, T3) !=? select(X10, [], X11)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: tail(.(T26, T27), T20) | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ?_3 | select(T17, T18, T3), SCOPE: 1, CLAUSE: 2\n\nT17 is ground\nT18 is ground\nX10 is free\nX11 is free\nX26 is free\nX27 is free\nselect(T17, T18, T3) !=? select(X10, [], X11)\nhead(T18, T17) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: tail(.(T26, T27), T20), SCOPE: 4, CLAUSE: 5 | tail(.(T26, T27), T20), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: tail(.(T26, T27), T20), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true | ?_4 | ?_3 | select(T32, .(T32, T33), T3), SCOPE: 1, CLAUSE: 2\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ?_4 | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX32 is free\nX33 is free\ntail(.(T26, T27), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ?_4 | ?_3 | select(T32, .(T32, T33), T3), SCOPE: 1, CLAUSE: 2\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_3 | select(T32, .(T32, T33), T3), SCOPE: 1, CLAUSE: 2\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: select(T32, .(T32, T33), T3), SCOPE: 1, CLAUSE: 2\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(head(.(T42, T43), T46), ','(tail(.(T42, T43), X47), select(T42, X47, T47)))\n\nT42 is ground\nT43 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(head(.(T42, T43), T46), ','(tail(.(T42, T43), X47), select(T42, X47, T47))), SCOPE: 5, CLAUSE: 3 | ','(head(.(T42, T43), T46), ','(tail(.(T42, T43), X47), select(T42, X47, T47))), SCOPE: 5, CLAUSE: 4\n\nT42 is ground\nT43 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(.(T42, T43), T46), ','(tail(.(T42, T43), X47), select(T42, X47, T47))), SCOPE: 5, CLAUSE: 4\n\nT42 is ground\nT43 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(tail(.(T54, T55), X47), select(T54, X47, T56))\n\nT54 is ground\nT55 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(tail(.(T54, T55), X47), select(T54, X47, T56)), SCOPE: 6, CLAUSE: 5 | ','(tail(.(T54, T55), X47), select(T54, X47, T56)), SCOPE: 6, CLAUSE: 6\n\nT54 is ground\nT55 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail(.(T54, T55), X47), select(T54, X47, T56)), SCOPE: 6, CLAUSE: 6\n\nT54 is ground\nT55 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: select(T65, T66, T56)\n\nT65 is ground\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX32 is free\nX33 is free\ntail(.(T26, T27), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX32 is free\nX33 is free\ntail(.(T26, T27), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(head(.(T77, T78), T81), ','(tail(.(T77, T78), X82), select(T77, X82, T82)))\n\nT77 is ground\nT78 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T77, T78), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(head(.(T77, T78), T81), ','(tail(.(T77, T78), X82), select(T77, X82, T82))), SCOPE: 7, CLAUSE: 3 | ','(head(.(T77, T78), T81), ','(tail(.(T77, T78), X82), select(T77, X82, T82))), SCOPE: 7, CLAUSE: 4\n\nT77 is ground\nT78 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T77, T78), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(head(.(T77, T78), T81), ','(tail(.(T77, T78), X82), select(T77, X82, T82))), SCOPE: 7, CLAUSE: 4\n\nT77 is ground\nT78 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T77, T78), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(tail(.(T89, T90), X82), select(T89, X82, T91))\n\nT89 is ground\nT90 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T89, T90), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(tail(.(T89, T90), X82), select(T89, X82, T91)), SCOPE: 8, CLAUSE: 5 | ','(tail(.(T89, T90), X82), select(T89, X82, T91)), SCOPE: 8, CLAUSE: 6\n\nT89 is ground\nT90 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T89, T90), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(tail(.(T89, T90), X82), select(T89, X82, T91)), SCOPE: 8, CLAUSE: 6\n\nT89 is ground\nT90 is ground\nX32 is free\nX33 is free\nX82 is free\ntail(.(T89, T90), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: select(T100, T101, T91)\n\nT100 is ground\nT101 is ground\nX32 is free\nX33 is free\ntail(.(T100, T101), T20) !=? tail(.(X32, X33), X33)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: select(T17, T18, T3), SCOPE: 1, CLAUSE: 2\n\nT17 is ground\nT18 is ground\nX10 is free\nX11 is free\nX26 is free\nX27 is free\nselect(T17, T18, T3) !=? select(X10, [], X11)\nhead(T18, T17) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(head(T113, T116), ','(tail(T113, X117), select(T112, X117, T117)))\n\nT112 is ground\nT113 is ground\nX10 is free\nX11 is free\nX26 is free\nX27 is free\nX117 is free\nselect(T112, T113, T3) !=? select(X10, [], X11)\nhead(T113, T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(head(T113, T116), ','(tail(T113, X117), select(T112, X117, T117))), SCOPE: 9, CLAUSE: 3 | ','(head(T113, T116), ','(tail(T113, X117), select(T112, X117, T117))), SCOPE: 9, CLAUSE: 4\n\nT112 is ground\nT113 is ground\nX10 is free\nX11 is free\nX26 is free\nX27 is free\nX117 is free\nselect(T112, T113, T3) !=? select(X10, [], X11)\nhead(T113, T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(head(T113, T116), ','(tail(T113, X117), select(T112, X117, T117))), SCOPE: 9, CLAUSE: 4\n\nT112 is ground\nT113 is ground\nX10 is free\nX11 is free\nX26 is free\nX27 is free\nX117 is free\nselect(T112, T113, T3) !=? select(X10, [], X11)\nhead(T113, T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(tail(.(T125, T126), X117), select(T112, X117, T127))\n\nT112 is ground\nT125 is ground\nT126 is ground\nX26 is free\nX27 is free\nX117 is free\nhead(.(T125, T126), T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(tail(.(T125, T126), X117), select(T112, X117, T127)), SCOPE: 10, CLAUSE: 5 | ','(tail(.(T125, T126), X117), select(T112, X117, T127)), SCOPE: 10, CLAUSE: 6\n\nT112 is ground\nT125 is ground\nT126 is ground\nX26 is free\nX27 is free\nX117 is free\nhead(.(T125, T126), T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(tail(.(T125, T126), X117), select(T112, X117, T127)), SCOPE: 10, CLAUSE: 6\n\nT112 is ground\nT125 is ground\nT126 is ground\nX26 is free\nX27 is free\nX117 is free\nhead(.(T125, T126), T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: select(T112, T137, T127)\n\nT112 is ground\nT136 is ground\nT137 is ground\nX26 is free\nX27 is free\nhead(.(T136, T137), T112) !=? head(.(X26, X27), X26)", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 99 [arrowhead = none ];
99 -> 4;

100 [label="EVAL with clause\nselect(X10, [], X11) :- ','(!_1, failure(a)).\nand substitutionT1 -> T8,\nX10 -> T8,\nT2 -> [],\nT3 -> T9,\nX11 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 100 [arrowhead = none ];
100 -> 7;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 101 [arrowhead = none ];
101 -> 9;

102 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 102 [arrowhead = none ];
102 -> 10;

103 [label="ONLY EVAL with clause\nselect(X18, X19, X20) :- ','(head(X19, X18), tail(X19, X20)).\nand substitutionT1 -> T17,\nX18 -> T17,\nT2 -> T18,\nX19 -> T18,\nT3 -> T20,\nX20 -> T20,\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 103 [arrowhead = none ];
103 -> 18;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 104 [arrowhead = none ];
104 -> 13;

105 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 105 [arrowhead = none ];
105 -> 14;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 106 [arrowhead = none ];
106 -> 19;

107 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T17, T18, T3), select(X10, [], X11))", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 107 [arrowhead = none ];
107 -> 20;

108 [label="EVAL with clause\nhead(.(X26, X27), X26).\nand substitutionX26 -> T26,\nX27 -> T27,\nT18 -> .(T26, T27),\nT17 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 108 [arrowhead = none ];
108 -> 30;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 109 [arrowhead = none ];
109 -> 31;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 110 [arrowhead = none ];
110 -> 32;

111 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 111 [arrowhead = none ];
111 -> 82;

112 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="EVAL with clause\ntail(.(X32, X33), X33).\nand substitutionT26 -> T32,\nX32 -> T32,\nT27 -> T33,\nX33 -> T33,\nT20 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 113 [arrowhead = none ];
113 -> 36;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 116 [arrowhead = none ];
116 -> 59;

117 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 118 [arrowhead = none ];
118 -> 42;

119 [label="EVAL with clause\nselect(X43, X44, .(X45, X46)) :- ','(head(X44, X45), ','(tail(X44, X47), select(X43, X47, X46))).\nand substitutionT32 -> T42,\nX43 -> T42,\nT33 -> T43,\nX44 -> .(T42, T43),\nX45 -> T46,\nX46 -> T47,\nT3 -> .(T46, T47),\nT44 -> T46,\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 119 [arrowhead = none ];
119 -> 47;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 121 [arrowhead = none ];
121 -> 49;

122 [label="BACKTRACK\nfor clause: head([], X3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 122 [arrowhead = none ];
122 -> 50;

123 [label="EVAL with clause\nhead(.(X55, X56), X55).\nand substitutionT42 -> T54,\nX55 -> T54,\nT43 -> T55,\nX56 -> T55,\nT46 -> T54,\nT47 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 123 [arrowhead = none ];
123 -> 52;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 124 [arrowhead = none ];
124 -> 53;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 125 [arrowhead = none ];
125 -> 54;

126 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 126 [arrowhead = none ];
126 -> 55;

127 [label="ONLY EVAL with clause\ntail(.(X65, X66), X66).\nand substitutionT54 -> T65,\nX65 -> T65,\nT55 -> T66,\nX66 -> T66,\nX47 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 127 [arrowhead = none ];
127 -> 58;

128 [label="INSTANCE with matching:\nT1 -> T65\nT2 -> T66\nT3 -> T56", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 128 [arrowhead = none , style=dashed];
128 -> 2[style=dashed];

129 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 129 [arrowhead = none ];
129 -> 60;

130 [label="EVAL with clause\nselect(X78, X79, .(X80, X81)) :- ','(head(X79, X80), ','(tail(X79, X82), select(X78, X82, X81))).\nand substitutionT26 -> T77,\nX78 -> T77,\nT27 -> T78,\nX79 -> .(T77, T78),\nX80 -> T81,\nX81 -> T82,\nT3 -> .(T81, T82),\nT79 -> T81,\nT80 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 130 [arrowhead = none ];
130 -> 65;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 131 [arrowhead = none ];
131 -> 66;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 132 [arrowhead = none ];
132 -> 69;

133 [label="BACKTRACK\nfor clause: head([], X3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 133 [arrowhead = none ];
133 -> 71;

134 [label="EVAL with clause\nhead(.(X90, X91), X90).\nand substitutionT77 -> T89,\nX90 -> T89,\nT78 -> T90,\nX91 -> T90,\nT81 -> T89,\nT82 -> T91", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 134 [arrowhead = none ];
134 -> 73;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 135 [arrowhead = none ];
135 -> 74;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 136 [arrowhead = none ];
136 -> 76;

137 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 137 [arrowhead = none ];
137 -> 77;

138 [label="ONLY EVAL with clause\ntail(.(X100, X101), X101).\nand substitutionT89 -> T100,\nX100 -> T100,\nT90 -> T101,\nX101 -> T101,\nX82 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 138 [arrowhead = none ];
138 -> 79;

139 [label="INSTANCE with matching:\nT1 -> T100\nT2 -> T101\nT3 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 139 [arrowhead = none , style=dashed];
139 -> 2[style=dashed];

140 [label="EVAL with clause\nselect(X113, X114, .(X115, X116)) :- ','(head(X114, X115), ','(tail(X114, X117), select(X113, X117, X116))).\nand substitutionT17 -> T112,\nX113 -> T112,\nT18 -> T113,\nX114 -> T113,\nX115 -> T116,\nX116 -> T117,\nT3 -> .(T116, T117),\nT114 -> T116,\nT115 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 140 [arrowhead = none ];
140 -> 86;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 141 [arrowhead = none ];
141 -> 88;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 142 [arrowhead = none ];
142 -> 91;

143 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T112, T113, T3), select(X10, [], X11))", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 143 [arrowhead = none ];
143 -> 92;

144 [label="EVAL with clause\nhead(.(X125, X126), X125).\nand substitutionX125 -> T125,\nX126 -> T126,\nT113 -> .(T125, T126),\nT116 -> T125,\nT117 -> T127", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 144 [arrowhead = none ];
144 -> 94;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 145 [arrowhead = none ];
145 -> 95;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 146 [arrowhead = none ];
146 -> 96;

147 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 147 [arrowhead = none ];
147 -> 97;

148 [label="ONLY EVAL with clause\ntail(.(X135, X136), X136).\nand substitutionT125 -> T136,\nX135 -> T136,\nT126 -> T137,\nX136 -> T137,\nX117 -> T137", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 148 [arrowhead = none ];
148 -> 98;

149 [label="INSTANCE with matching:\nT1 -> T112\nT2 -> T137\nT3 -> T127", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 149 [arrowhead = none , style=dashed];
149 -> 2[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, .(z0, z1)) → f2_out1
f2_in(z0, .(z0, z1)) → U1(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z0, z1)) → U2(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z1, z2)) → U3(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1, z0, .(z0, z1)) → f2_out1
U2(f2_out1, z0, .(z0, z1)) → f2_out1
U3(f2_out1, z0, .(z1, z2)) → f2_out1

Tuples:

F2_IN(z0, .(z0, z1)) → c1(U1'(f2_in(z0, z1), z0, .(z0, z1)), F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(U2'(f2_in(z0, z1), z0, .(z0, z1)), F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(U3'(f2_in(z0, z2), z0, .(z1, z2)), F2_IN(z0, z2))

S tuples:

F2_IN(z0, .(z0, z1)) → c1(U1'(f2_in(z0, z1), z0, .(z0, z1)), F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(U2'(f2_in(z0, z1), z0, .(z0, z1)), F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(U3'(f2_in(z0, z2), z0, .(z1, z2)), F2_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2, c3

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

Removed 3 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, .(z0, z1)) → f2_out1
f2_in(z0, .(z0, z1)) → U1(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z0, z1)) → U2(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z1, z2)) → U3(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1, z0, .(z0, z1)) → f2_out1
U2(f2_out1, z0, .(z0, z1)) → f2_out1
U3(f2_out1, z0, .(z1, z2)) → f2_out1

Tuples:

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

S tuples:

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2, c3

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [2] + x₂
POL(F2_IN(x₁, x₂)) = [2]x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, .(z0, z1)) → f2_out1
f2_in(z0, .(z0, z1)) → U1(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z0, z1)) → U2(f2_in(z0, z1), z0, .(z0, z1))
f2_in(z0, .(z1, z2)) → U3(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1, z0, .(z0, z1)) → f2_out1
U2(f2_out1, z0, .(z0, z1)) → f2_out1
U3(f2_out1, z0, .(z1, z2)) → f2_out1

Tuples:

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

S tuples:none
K tuples:

F2_IN(z0, .(z0, z1)) → c1(F2_IN(z0, z1))
F2_IN(z0, .(z0, z1)) → c2(F2_IN(z0, z1))
F2_IN(z0, .(z1, z2)) → c3(F2_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, U2, U3

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2, c3

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))

(9) 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: select(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, failure(a)) | select(T6, [], T3), SCOPE: 1, CLAUSE: 1 | select(T6, [], T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX8 is free\nX9 is free\nselect(T1, T2, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: failure(a), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(head(T12, T11), tail(T12, T14)) | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 3 | ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: tail(.(T20, T21), T14) | ?_3 | select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nX21 is free\nX22 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)\nhead(T12, T11) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: tail(.(T20, T21), T14), SCOPE: 4, CLAUSE: 5 | tail(.(T20, T21), T14), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: tail(.(T20, T21), T14), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | ?_4 | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_4 | ?_3 | select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground\nX27 is free\nX28 is free\ntail(.(T20, T21), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_4 | ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ?_3 | select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: select(T26, .(T26, T27), T3), SCOPE: 1, CLAUSE: 2\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(.(T36, T37), T40), ','(tail(.(T36, T37), X42), select(T36, X42, T41)))\n\nT36 is ground\nT37 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(head(.(T36, T37), T40), ','(tail(.(T36, T37), X42), select(T36, X42, T41))), SCOPE: 5, CLAUSE: 3 | ','(head(.(T36, T37), T40), ','(tail(.(T36, T37), X42), select(T36, X42, T41))), SCOPE: 5, CLAUSE: 4\n\nT36 is ground\nT37 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(head(.(T36, T37), T40), ','(tail(.(T36, T37), X42), select(T36, X42, T41))), SCOPE: 5, CLAUSE: 4\n\nT36 is ground\nT37 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail(.(T48, T49), X42), select(T48, X42, T50))\n\nT48 is ground\nT49 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tail(.(T48, T49), X42), select(T48, X42, T50)), SCOPE: 6, CLAUSE: 5 | ','(tail(.(T48, T49), X42), select(T48, X42, T50)), SCOPE: 6, CLAUSE: 6\n\nT48 is ground\nT49 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail(.(T48, T49), X42), select(T48, X42, T50)), SCOPE: 6, CLAUSE: 6\n\nT48 is ground\nT49 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: select(T59, T60, T50)\n\nT59 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ?_3 | select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground\nX27 is free\nX28 is free\ntail(.(T20, T21), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: select(T20, .(T20, T21), T3), SCOPE: 1, CLAUSE: 2\n\nT20 is ground\nT21 is ground\nX27 is free\nX28 is free\ntail(.(T20, T21), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(.(T71, T72), T75), ','(tail(.(T71, T72), X77), select(T71, X77, T76)))\n\nT71 is ground\nT72 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T71, T72), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(head(.(T71, T72), T75), ','(tail(.(T71, T72), X77), select(T71, X77, T76))), SCOPE: 7, CLAUSE: 3 | ','(head(.(T71, T72), T75), ','(tail(.(T71, T72), X77), select(T71, X77, T76))), SCOPE: 7, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T71, T72), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(head(.(T71, T72), T75), ','(tail(.(T71, T72), X77), select(T71, X77, T76))), SCOPE: 7, CLAUSE: 4\n\nT71 is ground\nT72 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T71, T72), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(.(T83, T84), X77), select(T83, X77, T85))\n\nT83 is ground\nT84 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T83, T84), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(T83, T84), X77), select(T83, X77, T85)), SCOPE: 8, CLAUSE: 5 | ','(tail(.(T83, T84), X77), select(T83, X77, T85)), SCOPE: 8, CLAUSE: 6\n\nT83 is ground\nT84 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T83, T84), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(tail(.(T83, T84), X77), select(T83, X77, T85)), SCOPE: 8, CLAUSE: 6\n\nT83 is ground\nT84 is ground\nX27 is free\nX28 is free\nX77 is free\ntail(.(T83, T84), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: select(T94, T95, T85)\n\nT94 is ground\nT95 is ground\nX27 is free\nX28 is free\ntail(.(T94, T95), T14) !=? tail(.(X27, X28), X28)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nX21 is free\nX22 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)\nhead(T12, T11) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head(T107, T110), ','(tail(T107, X112), select(T106, X112, T111)))\n\nT106 is ground\nT107 is ground\nX8 is free\nX9 is free\nX21 is free\nX22 is free\nX112 is free\nselect(T106, T107, T3) !=? select(X8, [], X9)\nhead(T107, T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(head(T107, T110), ','(tail(T107, X112), select(T106, X112, T111))), SCOPE: 9, CLAUSE: 3 | ','(head(T107, T110), ','(tail(T107, X112), select(T106, X112, T111))), SCOPE: 9, CLAUSE: 4\n\nT106 is ground\nT107 is ground\nX8 is free\nX9 is free\nX21 is free\nX22 is free\nX112 is free\nselect(T106, T107, T3) !=? select(X8, [], X9)\nhead(T107, T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(head(T107, T110), ','(tail(T107, X112), select(T106, X112, T111))), SCOPE: 9, CLAUSE: 4\n\nT106 is ground\nT107 is ground\nX8 is free\nX9 is free\nX21 is free\nX22 is free\nX112 is free\nselect(T106, T107, T3) !=? select(X8, [], X9)\nhead(T107, T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail(.(T119, T120), X112), select(T106, X112, T121))\n\nT106 is ground\nT119 is ground\nT120 is ground\nX21 is free\nX22 is free\nX112 is free\nhead(.(T119, T120), T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(tail(.(T119, T120), X112), select(T106, X112, T121)), SCOPE: 10, CLAUSE: 5 | ','(tail(.(T119, T120), X112), select(T106, X112, T121)), SCOPE: 10, CLAUSE: 6\n\nT106 is ground\nT119 is ground\nT120 is ground\nX21 is free\nX22 is free\nX112 is free\nhead(.(T119, T120), T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(tail(.(T119, T120), X112), select(T106, X112, T121)), SCOPE: 10, CLAUSE: 6\n\nT106 is ground\nT119 is ground\nT120 is ground\nX21 is free\nX22 is free\nX112 is free\nhead(.(T119, T120), T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: select(T106, T131, T121)\n\nT106 is ground\nT130 is ground\nT131 is ground\nX21 is free\nX22 is free\nhead(.(T130, T131), T106) !=? head(.(X21, X22), X21)", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 94 [arrowhead = none ];
94 -> 3;

95 [label="EVAL with clause\nselect(X8, [], X9) :- ','(!_1, failure(a)).\nand substitutionT1 -> T6,\nX8 -> T6,\nT2 -> [],\nT3 -> T7,\nX9 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 95 [arrowhead = none ];
95 -> 5;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 96 [arrowhead = none ];
96 -> 6;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 97 [arrowhead = none ];
97 -> 8;

98 [label="ONLY EVAL with clause\nselect(X13, X14, X15) :- ','(head(X14, X13), tail(X14, X15)).\nand substitutionT1 -> T11,\nX13 -> T11,\nT2 -> T12,\nX14 -> T12,\nT3 -> T14,\nX15 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 98 [arrowhead = none ];
98 -> 15;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 100 [arrowhead = none ];
100 -> 12;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 101 [arrowhead = none ];
101 -> 16;

102 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T11, T12, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 102 [arrowhead = none ];
102 -> 17;

103 [label="EVAL with clause\nhead(.(X21, X22), X21).\nand substitutionX21 -> T20,\nX22 -> T21,\nT12 -> .(T20, T21),\nT11 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 103 [arrowhead = none ];
103 -> 21;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 104 [arrowhead = none ];
104 -> 22;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 105 [arrowhead = none ];
105 -> 23;

106 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 106 [arrowhead = none ];
106 -> 78;

107 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 107 [arrowhead = none ];
107 -> 24;

108 [label="EVAL with clause\ntail(.(X27, X28), X28).\nand substitutionT20 -> T26,\nX27 -> T26,\nT21 -> T27,\nX28 -> T27,\nT14 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 108 [arrowhead = none ];
108 -> 25;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 109 [arrowhead = none ];
109 -> 26;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 110 [arrowhead = none ];
110 -> 27;

111 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 111 [arrowhead = none ];
111 -> 56;

112 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 112 [arrowhead = none ];
112 -> 28;

113 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 113 [arrowhead = none ];
113 -> 29;

114 [label="EVAL with clause\nselect(X38, X39, .(X40, X41)) :- ','(head(X39, X40), ','(tail(X39, X42), select(X38, X42, X41))).\nand substitutionT26 -> T36,\nX38 -> T36,\nT27 -> T37,\nX39 -> .(T36, T37),\nX40 -> T40,\nX41 -> T41,\nT3 -> .(T40, T41),\nT38 -> T40,\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 114 [arrowhead = none ];
114 -> 34;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 115 [arrowhead = none ];
115 -> 35;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 116 [arrowhead = none ];
116 -> 37;

117 [label="BACKTRACK\nfor clause: head([], X3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 117 [arrowhead = none ];
117 -> 39;

118 [label="EVAL with clause\nhead(.(X50, X51), X50).\nand substitutionT36 -> T48,\nX50 -> T48,\nT37 -> T49,\nX51 -> T49,\nT40 -> T48,\nT41 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 118 [arrowhead = none ];
118 -> 43;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="ONLY EVAL with clause\ntail(.(X60, X61), X61).\nand substitutionT48 -> T59,\nX60 -> T59,\nT49 -> T60,\nX61 -> T60,\nX42 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 122 [arrowhead = none ];
122 -> 51;

123 [label="INSTANCE with matching:\nT1 -> T59\nT2 -> T60\nT3 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 123 [arrowhead = none , style=dashed];
123 -> 1[style=dashed];

124 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 124 [arrowhead = none ];
124 -> 57;

125 [label="EVAL with clause\nselect(X73, X74, .(X75, X76)) :- ','(head(X74, X75), ','(tail(X74, X77), select(X73, X77, X76))).\nand substitutionT20 -> T71,\nX73 -> T71,\nT21 -> T72,\nX74 -> .(T71, T72),\nX75 -> T75,\nX76 -> T76,\nT3 -> .(T75, T76),\nT73 -> T75,\nT74 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 125 [arrowhead = none ];
125 -> 61;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 126 [arrowhead = none ];
126 -> 62;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 127 [arrowhead = none ];
127 -> 63;

128 [label="BACKTRACK\nfor clause: head([], X3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 128 [arrowhead = none ];
128 -> 64;

129 [label="EVAL with clause\nhead(.(X85, X86), X85).\nand substitutionT71 -> T83,\nX85 -> T83,\nT72 -> T84,\nX86 -> T84,\nT75 -> T83,\nT76 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 129 [arrowhead = none ];
129 -> 67;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 130 [arrowhead = none ];
130 -> 68;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 131 [arrowhead = none ];
131 -> 70;

132 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 132 [arrowhead = none ];
132 -> 72;

133 [label="ONLY EVAL with clause\ntail(.(X95, X96), X96).\nand substitutionT83 -> T94,\nX95 -> T94,\nT84 -> T95,\nX96 -> T95,\nX77 -> T95", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 133 [arrowhead = none ];
133 -> 75;

134 [label="INSTANCE with matching:\nT1 -> T94\nT2 -> T95\nT3 -> T85", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 134 [arrowhead = none , style=dashed];
134 -> 1[style=dashed];

135 [label="EVAL with clause\nselect(X108, X109, .(X110, X111)) :- ','(head(X109, X110), ','(tail(X109, X112), select(X108, X112, X111))).\nand substitutionT11 -> T106,\nX108 -> T106,\nT12 -> T107,\nX109 -> T107,\nX110 -> T110,\nX111 -> T111,\nT3 -> .(T110, T111),\nT108 -> T110,\nT109 -> T111", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 135 [arrowhead = none ];
135 -> 80;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 136 [arrowhead = none ];
136 -> 81;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 137 [arrowhead = none ];
137 -> 83;

138 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T106, T107, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 138 [arrowhead = none ];
138 -> 84;

139 [label="EVAL with clause\nhead(.(X120, X121), X120).\nand substitutionX120 -> T119,\nX121 -> T120,\nT107 -> .(T119, T120),\nT110 -> T119,\nT111 -> T121", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 139 [arrowhead = none ];
139 -> 85;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 140 [arrowhead = none ];
140 -> 87;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 141 [arrowhead = none ];
141 -> 89;

142 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 142 [arrowhead = none ];
142 -> 90;

143 [label="ONLY EVAL with clause\ntail(.(X130, X131), X131).\nand substitutionT119 -> T130,\nX130 -> T130,\nT120 -> T131,\nX131 -> T131,\nX112 -> T131", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 143 [arrowhead = none ];
143 -> 93;

144 [label="INSTANCE with matching:\nT1 -> T106\nT2 -> T131\nT3 -> T121", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, .(z0, z1)) → f1_out1
f1_in(z0, .(z0, z1)) → U1(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z0, z1)) → U2(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z1, z2)) → U3(f1_in(z0, z2), z0, .(z1, z2))
U1(f1_out1, z0, .(z0, z1)) → f1_out1
U2(f1_out1, z0, .(z0, z1)) → f1_out1
U3(f1_out1, z0, .(z1, z2)) → f1_out1

Tuples:

F1_IN(z0, .(z0, z1)) → c1(U1'(f1_in(z0, z1), z0, .(z0, z1)), F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(U2'(f1_in(z0, z1), z0, .(z0, z1)), F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(U3'(f1_in(z0, z2), z0, .(z1, z2)), F1_IN(z0, z2))

S tuples:

F1_IN(z0, .(z0, z1)) → c1(U1'(f1_in(z0, z1), z0, .(z0, z1)), F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(U2'(f1_in(z0, z1), z0, .(z0, z1)), F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(U3'(f1_in(z0, z2), z0, .(z1, z2)), F1_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2, c3

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

Removed 3 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, .(z0, z1)) → f1_out1
f1_in(z0, .(z0, z1)) → U1(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z0, z1)) → U2(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z1, z2)) → U3(f1_in(z0, z2), z0, .(z1, z2))
U1(f1_out1, z0, .(z0, z1)) → f1_out1
U2(f1_out1, z0, .(z0, z1)) → f1_out1
U3(f1_out1, z0, .(z1, z2)) → f1_out1

Tuples:

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

S tuples:

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2, c3

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

We considered the (Usable) Rules:none
And the Tuples:

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [2] + x₂
POL(F1_IN(x₁, x₂)) = [2]x₂
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, .(z0, z1)) → f1_out1
f1_in(z0, .(z0, z1)) → U1(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z0, z1)) → U2(f1_in(z0, z1), z0, .(z0, z1))
f1_in(z0, .(z1, z2)) → U3(f1_in(z0, z2), z0, .(z1, z2))
U1(f1_out1, z0, .(z0, z1)) → f1_out1
U2(f1_out1, z0, .(z0, z1)) → f1_out1
U3(f1_out1, z0, .(z1, z2)) → f1_out1

Tuples:

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

S tuples:none
K tuples:

F1_IN(z0, .(z0, z1)) → c1(F1_IN(z0, z1))
F1_IN(z0, .(z0, z1)) → c2(F1_IN(z0, z1))
F1_IN(z0, .(z1, z2)) → c3(F1_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2, c3