Runtime Complexity of the query pattern
overlap(g,g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member(X, Xs), member(X, Ys)).
member(X1, []) :- ','(!, failure(a)).
member(X, Y) :- head(Y, X).
member(X, Y) :- ','(tail(Y, T), member(X, T)).
head([], X2).
head(.(H, X3), H).
tail([], []).
tail(.(X4, T), T).
failure(b).

Query: overlap(g,g)

(1) 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: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: overlap(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(member(X9, T5), member(X9, T6))\n\nT5 is ground\nT6 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 1 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(','(!_2, failure(a)), member(X15, T6)) | ','(member(X9, []), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, []), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX9 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 2 | ','(member(X9, T5), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX9 is free\nX14 is free\nmember(X9, T5) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(failure(a), member(X15, T6))\n\nT6 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(failure(a), member(X15, T6)), SCOPE: 3, CLAUSE: 8\n\nT6 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(head(T9, X28), member(X28, T6)) | ','(member(X9, T9), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT9 is ground\nX9 is free\nX14 is free\nX28 is free\nmember(X9, T9) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(head(T9, X28), member(X28, T6)), SCOPE: 4, CLAUSE: 4 | ','(head(T9, X28), member(X28, T6)), SCOPE: 4, CLAUSE: 5 | ?_4 | ','(member(X9, T9), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT9 is ground\nX9 is free\nX14 is free\nX28 is free\nmember(X9, T9) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(head(T9, X28), member(X28, T6)), SCOPE: 4, CLAUSE: 5 | ?_4 | ','(member(X9, T9), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT9 is ground\nX9 is free\nX14 is free\nX28 is free\nmember(X9, T9) !=? member(X14, [])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: member(T14, T6) | ?_4 | ','(member(X9, .(T14, T15)), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT14 is ground\nT15 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ?_4 | ','(member(X9, T9), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT9 is ground\nX9 is free\nX14 is free\nX28 is free\nX37 is free\nX38 is free\nmember(X9, T9) !=? member(X14, [])\nhead(T9, X28) !=? head(.(X37, X38), X37)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: member(T14, T6)\n\nT6 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_4 | ','(member(X9, .(T14, T15)), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT14 is ground\nT15 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: member(T14, T6), SCOPE: 5, CLAUSE: 1 | member(T14, T6), SCOPE: 5, CLAUSE: 2 | member(T14, T6), SCOPE: 5, CLAUSE: 3\n\nT6 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(!_5, failure(a)) | member(T20, []), SCOPE: 5, CLAUSE: 2 | member(T20, []), SCOPE: 5, CLAUSE: 3\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: member(T14, T6), SCOPE: 5, CLAUSE: 2 | member(T14, T6), SCOPE: 5, CLAUSE: 3\n\nT6 is ground\nT14 is ground\nX43 is free\nmember(T14, T6) !=? member(X43, [])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: failure(a), SCOPE: 6, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: head(T26, T25) | member(T25, T26), SCOPE: 5, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX43 is free\nmember(T25, T26) !=? member(X43, [])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: head(T26, T25), SCOPE: 7, CLAUSE: 4 | head(T26, T25), SCOPE: 7, CLAUSE: 5 | ?_7 | member(T25, T26), SCOPE: 5, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX43 is free\nmember(T25, T26) !=? member(X43, [])", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: head(T26, T25), SCOPE: 7, CLAUSE: 5 | ?_7 | member(T25, T26), SCOPE: 5, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX43 is free\nmember(T25, T26) !=? member(X43, [])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true | ?_7 | member(T32, .(T32, T33)), SCOPE: 5, CLAUSE: 3\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ?_7 | member(T25, T26), SCOPE: 5, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX43 is free\nX55 is free\nX56 is free\nmember(T25, T26) !=? member(X43, [])\nhead(T26, T25) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ?_7 | member(T32, .(T32, T33)), SCOPE: 5, CLAUSE: 3\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: member(T32, .(T32, T33)), SCOPE: 5, CLAUSE: 3\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(tail(.(T39, T40), X65), member(T39, X65))\n\nT39 is ground\nT40 is ground\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(tail(.(T39, T40), X65), member(T39, X65)), SCOPE: 8, CLAUSE: 6 | ','(tail(.(T39, T40), X65), member(T39, X65)), SCOPE: 8, CLAUSE: 7\n\nT39 is ground\nT40 is ground\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(tail(.(T39, T40), X65), member(T39, X65)), SCOPE: 8, CLAUSE: 7\n\nT39 is ground\nT40 is ground\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: member(T47, T48)\n\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: member(T25, T26), SCOPE: 5, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX43 is free\nX55 is free\nX56 is free\nmember(T25, T26) !=? member(X43, [])\nhead(T26, T25) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail(T56, X83), member(T55, X83))\n\nT55 is ground\nT56 is ground\nX43 is free\nX55 is free\nX56 is free\nX83 is free\nmember(T55, T56) !=? member(X43, [])\nhead(T56, T55) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(T56, X83), member(T55, X83)), SCOPE: 9, CLAUSE: 6 | ','(tail(T56, X83), member(T55, X83)), SCOPE: 9, CLAUSE: 7\n\nT55 is ground\nT56 is ground\nX43 is free\nX55 is free\nX56 is free\nX83 is free\nmember(T55, T56) !=? member(X43, [])\nhead(T56, T55) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(T56, X83), member(T55, X83)), SCOPE: 9, CLAUSE: 7\n\nT55 is ground\nT56 is ground\nX43 is free\nX55 is free\nX56 is free\nX83 is free\nmember(T55, T56) !=? member(X43, [])\nhead(T56, T55) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: member(T55, T63)\n\nT55 is ground\nT62 is ground\nT63 is ground\nX55 is free\nX56 is free\nhead(.(T62, T63), T55) !=? head(.(X55, X56), X55)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(member(X9, .(T14, T15)), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT14 is ground\nT15 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(','(tail(.(T69, T70), X103), member(X104, X103)), member(X104, T6))\n\nT6 is ground\nT69 is ground\nT70 is ground\nX104 is free\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(','(tail(.(T69, T70), X103), member(X104, X103)), member(X104, T6)), SCOPE: 10, CLAUSE: 6 | ','(','(tail(.(T69, T70), X103), member(X104, X103)), member(X104, T6)), SCOPE: 10, CLAUSE: 7\n\nT6 is ground\nT69 is ground\nT70 is ground\nX104 is free\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(','(tail(.(T69, T70), X103), member(X104, X103)), member(X104, T6)), SCOPE: 10, CLAUSE: 7\n\nT6 is ground\nT69 is ground\nT70 is ground\nX104 is free\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(member(X104, T76), member(X104, T6))\n\nT6 is ground\nT76 is ground\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(member(X9, T9), member(X9, T6)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT9 is ground\nX9 is free\nX14 is free\nX28 is free\nX37 is free\nX38 is free\nmember(X9, T9) !=? member(X14, [])\nhead(T9, X28) !=? head(.(X37, X38), X37)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(','(tail(T79, X128), member(X129, X128)), member(X129, T6))\n\nT6 is ground\nT79 is ground\nX9 is free\nX14 is free\nX28 is free\nX37 is free\nX38 is free\nX129 is free\nX128 is free\nmember(X9, T79) !=? member(X14, [])\nhead(T79, X28) !=? head(.(X37, X38), X37)", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(','(tail(T79, X128), member(X129, X128)), member(X129, T6)), SCOPE: 11, CLAUSE: 6 | ','(','(tail(T79, X128), member(X129, X128)), member(X129, T6)), SCOPE: 11, CLAUSE: 7\n\nT6 is ground\nT79 is ground\nX9 is free\nX14 is free\nX28 is free\nX37 is free\nX38 is free\nX129 is free\nX128 is free\nmember(X9, T79) !=? member(X14, [])\nhead(T79, X28) !=? head(.(X37, X38), X37)", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(','(tail(T79, X128), member(X129, X128)), member(X129, T6)), SCOPE: 11, CLAUSE: 7\n\nT6 is ground\nT79 is ground\nX9 is free\nX14 is free\nX28 is free\nX37 is free\nX38 is free\nX129 is free\nX128 is free\nmember(X9, T79) !=? member(X14, [])\nhead(T79, X28) !=? head(.(X37, X38), X37)", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 95 [arrowhead = none ];
95 -> 3;

96 [label="ONLY EVAL with clause\noverlap(X7, X8) :- ','(member(X9, X7), member(X9, X8)).\nand substitutionT1 -> T5,\nX7 -> T5,\nT2 -> T6,\nX8 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 97 [arrowhead = none ];
97 -> 6;

98 [label="EVAL with clause\nmember(X14, []) :- ','(!_2, failure(a)).\nand substitutionX9 -> X15,\nX14 -> X15,\nT5 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 98 [arrowhead = none ];
98 -> 8;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 99 [arrowhead = none ];
99 -> 9;

100 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 100 [arrowhead = none ];
100 -> 10;

101 [label="ONLY EVAL with clause\nmember(X26, X27) :- head(X27, X26).\nand substitutionX9 -> X28,\nX26 -> X28,\nT5 -> T9,\nX27 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 101 [arrowhead = none ];
101 -> 14;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 102 [arrowhead = none ];
102 -> 11;

103 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 103 [arrowhead = none ];
103 -> 12;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 104 [arrowhead = none ];
104 -> 16;

105 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(X9, T9), member(X14, []))", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 105 [arrowhead = none ];
105 -> 17;

106 [label="EVAL with clause\nhead(.(X37, X38), X37).\nand substitutionX37 -> T14,\nX38 -> T15,\nT9 -> .(T14, T15),\nX28 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 106 [arrowhead = none ];
106 -> 19;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 107 [arrowhead = none ];
107 -> 23;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 108 [arrowhead = none ];
108 -> 26;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 109 [arrowhead = none ];
109 -> 27;

110 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 110 [arrowhead = none ];
110 -> 86;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 111 [arrowhead = none ];
111 -> 30;

112 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 112 [arrowhead = none ];
112 -> 73;

113 [label="EVAL with clause\nmember(X43, []) :- ','(!_5, failure(a)).\nand substitutionT14 -> T20,\nX43 -> T20,\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 113 [arrowhead = none ];
113 -> 32;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 114 [arrowhead = none ];
114 -> 33;

115 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 115 [arrowhead = none ];
115 -> 34;

116 [label="ONLY EVAL with clause\nmember(X48, X49) :- head(X49, X48).\nand substitutionT14 -> T25,\nX48 -> T25,\nT6 -> T26,\nX49 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 116 [arrowhead = none ];
116 -> 41;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 117 [arrowhead = none ];
117 -> 35;

118 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 118 [arrowhead = none ];
118 -> 36;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 119 [arrowhead = none ];
119 -> 42;

120 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(T25, T26), member(X43, []))", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 120 [arrowhead = none ];
120 -> 43;

121 [label="EVAL with clause\nhead(.(X55, X56), X55).\nand substitutionX55 -> T32,\nX56 -> T33,\nT26 -> .(T32, T33),\nT25 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 121 [arrowhead = none ];
121 -> 44;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 122 [arrowhead = none ];
122 -> 46;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 123 [arrowhead = none ];
123 -> 47;

124 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 124 [arrowhead = none ];
124 -> 57;

125 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 125 [arrowhead = none ];
125 -> 48;

126 [label="ONLY EVAL with clause\nmember(X63, X64) :- ','(tail(X64, X65), member(X63, X65)).\nand substitutionT32 -> T39,\nX63 -> T39,\nT33 -> T40,\nX64 -> .(T39, T40)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 126 [arrowhead = none ];
126 -> 51;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 127 [arrowhead = none ];
127 -> 53;

128 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 128 [arrowhead = none ];
128 -> 54;

129 [label="ONLY EVAL with clause\ntail(.(X72, X73), X73).\nand substitutionT39 -> T47,\nX72 -> T47,\nT40 -> T48,\nX73 -> T48,\nX65 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 129 [arrowhead = none ];
129 -> 56;

130 [label="INSTANCE with matching:\nT14 -> T47\nT6 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 130 [arrowhead = none , style=dashed];
130 -> 26[style=dashed];

131 [label="ONLY EVAL with clause\nmember(X81, X82) :- ','(tail(X82, X83), member(X81, X83)).\nand substitutionT25 -> T55,\nX81 -> T55,\nT26 -> T56,\nX82 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 131 [arrowhead = none ];
131 -> 60;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 132 [arrowhead = none ];
132 -> 62;

133 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(T55, T56), member(X43, []))", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 133 [arrowhead = none ];
133 -> 64;

134 [label="EVAL with clause\ntail(.(X89, X90), X90).\nand substitutionX89 -> T62,\nX90 -> T63,\nT56 -> .(T62, T63),\nX83 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 134 [arrowhead = none ];
134 -> 66;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 135 [arrowhead = none ];
135 -> 67;

136 [label="INSTANCE with matching:\nT14 -> T55\nT6 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 136 [arrowhead = none , style=dashed];
136 -> 26[style=dashed];

137 [label="ONLY EVAL with clause\nmember(X101, X102) :- ','(tail(X102, X103), member(X101, X103)).\nand substitutionX9 -> X104,\nX101 -> X104,\nT14 -> T69,\nT15 -> T70,\nX102 -> .(T69, T70)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 137 [arrowhead = none ];
137 -> 77;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 138 [arrowhead = none ];
138 -> 78;

139 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 139 [arrowhead = none ];
139 -> 79;

140 [label="ONLY EVAL with clause\ntail(.(X113, X114), X114).\nand substitutionT69 -> T75,\nX113 -> T75,\nT70 -> T76,\nX114 -> T76,\nX103 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 140 [arrowhead = none ];
140 -> 84;

141 [label="INSTANCE with matching:\nX9 -> X104\nT5 -> T76", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 141 [arrowhead = none , style=dashed];
141 -> 5[style=dashed];

142 [label="ONLY EVAL with clause\nmember(X126, X127) :- ','(tail(X127, X128), member(X126, X128)).\nand substitutionX9 -> X129,\nX126 -> X129,\nT9 -> T79,\nX127 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 142 [arrowhead = none ];
142 -> 90;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 143 [arrowhead = none ];
143 -> 91;

144 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(X9, T79), member(X14, []))", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 144 [arrowhead = none ];
144 -> 93;

145 [label="BACKTRACK\nfor clause: tail(.(X4, T), T)\nwith clash: (head(T79, X28), head(.(X37, X38), X37))", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 145 [arrowhead = none ];
145 -> 94;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(U2'(f19_in(z0, z2, z1), .(z0, z1), z2), F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(U3'(f26_in(z0, z1), z0, .(z0, z1)), F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(U4'(f26_in(z0, z2), z0, .(z1, z2)), F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(U6'(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2), F26_IN(z0, z1), F27_IN(z0, z2, z1))

S tuples:

F1_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(U2'(f19_in(z0, z2, z1), .(z0, z1), z2), F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(U3'(f26_in(z0, z1), z0, .(z0, z1)), F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(U4'(f26_in(z0, z2), z0, .(z1, z2)), F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(U6'(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2), F26_IN(z0, z1), F27_IN(z0, z2, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c, c2, c6, c7, c10, c12

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F5_IN(.(z0, z1), z2) → c2(U2'(f19_in(z0, z2, z1), .(z0, z1), z2), F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(U3'(f26_in(z0, z1), z0, .(z0, z1)), F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(U4'(f26_in(z0, z2), z0, .(z1, z2)), F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(U6'(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2), F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F5_IN(z0, z1))

S tuples:

F5_IN(.(z0, z1), z2) → c2(U2'(f19_in(z0, z2, z1), .(z0, z1), z2), F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(U3'(f26_in(z0, z1), z0, .(z0, z1)), F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(U4'(f26_in(z0, z2), z0, .(z1, z2)), F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(U6'(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2), F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F5_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F5_IN, F26_IN, F27_IN, F19_IN, F1_IN

Compound Symbols:

c2, c6, c7, c10, c12, c1

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

Removed 6 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(9) 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.

F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F19_IN(x₁, x₂, x₃)) = x₁ + [2]x₃   
POL(F1_IN(x₁, x₂)) = [3]x₁ + [3]x₂   
POL(F26_IN(x₁, x₂)) = 0   
POL(F27_IN(x₁, x₂, x₃)) = x₁ + [2]x₂   
POL(F5_IN(x₁, x₂)) = [2]x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F19_IN(x₁, x₂, x₃)) = x₂ + x₂·x₃   
POL(F1_IN(x₁, x₂)) = x₁·x₂   
POL(F26_IN(x₁, x₂)) = x₂   
POL(F27_IN(x₁, x₂, x₃)) = x₂·x₃   
POL(F5_IN(x₁, x₂)) = x₁·x₂   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f1_out1
f5_in(.(z0, z1), z2) → U2(f19_in(z0, z2, z1), .(z0, z1), z2)
U2(f19_out1, .(z0, z1), z2) → f5_out1
U2(f19_out3, .(z0, z1), z2) → f5_out1
f26_in(z0, .(z0, z1)) → f26_out1
f26_in(z0, .(z0, z1)) → U3(f26_in(z0, z1), z0, .(z0, z1))
f26_in(z0, .(z1, z2)) → U4(f26_in(z0, z2), z0, .(z1, z2))
U3(f26_out1, z0, .(z0, z1)) → f26_out1
U4(f26_out1, z0, .(z1, z2)) → f26_out1
f27_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1, z0, z1, z2) → f27_out2
f19_in(z0, z1, z2) → U6(f26_in(z0, z1), f27_in(z0, z2, z1), z0, z1, z2)
U6(f26_out1, z0, z1, z2, z3) → f19_out1
U6(z0, f27_out2, z1, z2, z3) → f19_out3

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:none
K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F19_IN(z0, z2, z1))
F19_IN(z0, z1, z2) → c12(F26_IN(z0, z1), F27_IN(z0, z2, z1))
F27_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F26_IN(z0, .(z0, z1)) → c6(F26_IN(z0, z1))
F26_IN(z0, .(z1, z2)) → c7(F26_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f26_in, U3, U4, f27_in, U5, f19_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F26_IN, F27_IN, F19_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))

(17) 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: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: overlap(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(member(X14, T7), member(X14, T8))\n\nT7 is ground\nT8 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: member(X14, T7)\n\nT7 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: member(T9, T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: member(X14, T7), SCOPE: 2, CLAUSE: 1 | member(X14, T7), SCOPE: 2, CLAUSE: 2 | member(X14, T7), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_2, failure(a)) | member(X14, []), SCOPE: 2, CLAUSE: 2 | member(X14, []), SCOPE: 2, CLAUSE: 3\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: member(X14, T7), SCOPE: 2, CLAUSE: 2 | member(X14, T7), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX14 is free\nX23 is free\nmember(X14, T7) !=? member(X23, [])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: failure(a), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: head(T12, X33) | member(X14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX14 is free\nX23 is free\nX33 is free\nmember(X14, T12) !=? member(X23, [])", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: head(T12, X33), SCOPE: 4, CLAUSE: 4 | head(T12, X33), SCOPE: 4, CLAUSE: 5 | ?_4 | member(X14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX14 is free\nX23 is free\nX33 is free\nmember(X14, T12) !=? member(X23, [])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: head(T12, X33), SCOPE: 4, CLAUSE: 5 | ?_4 | member(X14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX14 is free\nX23 is free\nX33 is free\nmember(X14, T12) !=? member(X23, [])", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true | ?_4 | member(X14, .(T17, T18)), SCOPE: 2, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ?_4 | member(X14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX14 is free\nX23 is free\nX33 is free\nX40 is free\nX41 is free\nmember(X14, T12) !=? member(X23, [])\nhead(T12, X33) !=? head(.(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ?_4 | member(X14, .(T17, T18)), SCOPE: 2, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: member(X14, .(T17, T18)), SCOPE: 2, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tail(.(T23, T24), X53), member(X54, X53))\n\nT23 is ground\nT24 is ground\nX54 is free\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(tail(.(T23, T24), X53), member(X54, X53)), SCOPE: 5, CLAUSE: 6 | ','(tail(.(T23, T24), X53), member(X54, X53)), SCOPE: 5, CLAUSE: 7\n\nT23 is ground\nT24 is ground\nX54 is free\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(tail(.(T23, T24), X53), member(X54, X53)), SCOPE: 5, CLAUSE: 7\n\nT23 is ground\nT24 is ground\nX54 is free\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: member(X54, T30)\n\nT30 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: member(X14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX14 is free\nX23 is free\nX33 is free\nX40 is free\nX41 is free\nmember(X14, T12) !=? member(X23, [])\nhead(T12, X33) !=? head(.(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(T33, X78), member(X79, X78))\n\nT33 is ground\nX14 is free\nX23 is free\nX33 is free\nX40 is free\nX41 is free\nX79 is free\nX78 is free\nmember(X14, T33) !=? member(X23, [])\nhead(T33, X33) !=? head(.(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail(T33, X78), member(X79, X78)), SCOPE: 6, CLAUSE: 6 | ','(tail(T33, X78), member(X79, X78)), SCOPE: 6, CLAUSE: 7\n\nT33 is ground\nX14 is free\nX23 is free\nX33 is free\nX40 is free\nX41 is free\nX79 is free\nX78 is free\nmember(X14, T33) !=? member(X23, [])\nhead(T33, X33) !=? head(.(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail(T33, X78), member(X79, X78)), SCOPE: 6, CLAUSE: 7\n\nT33 is ground\nX14 is free\nX23 is free\nX33 is free\nX40 is free\nX41 is free\nX79 is free\nX78 is free\nmember(X14, T33) !=? member(X23, [])\nhead(T33, X33) !=? head(.(X40, X41), X40)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: member(T9, T8), SCOPE: 7, CLAUSE: 1 | member(T9, T8), SCOPE: 7, CLAUSE: 2 | member(T9, T8), SCOPE: 7, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(!_7, failure(a)) | member(T9, []), SCOPE: 7, CLAUSE: 2 | member(T9, []), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: member(T9, T8), SCOPE: 7, CLAUSE: 2 | member(T9, T8), SCOPE: 7, CLAUSE: 3\n\nT8 is ground\nX87 is free\nmember(T9, T8) !=? member(X87, [])", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: failure(a), SCOPE: 8, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: head(T46, T47) | member(T9, T46), SCOPE: 7, CLAUSE: 3\n\nT46 is ground\nX87 is free\nmember(T9, T46) !=? member(X87, [])", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: head(T46, T47), SCOPE: 9, CLAUSE: 4 | head(T46, T47), SCOPE: 9, CLAUSE: 5 | ?_9 | member(T9, T46), SCOPE: 7, CLAUSE: 3\n\nT46 is ground\nX87 is free\nmember(T9, T46) !=? member(X87, [])", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: head(T46, T47), SCOPE: 9, CLAUSE: 5 | ?_9 | member(T9, T46), SCOPE: 7, CLAUSE: 3\n\nT46 is ground\nX87 is free\nmember(T9, T46) !=? member(X87, [])", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true | ?_9 | member(T9, .(T53, T54)), SCOPE: 7, CLAUSE: 3\n\nT53 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ?_9 | member(T9, T46), SCOPE: 7, CLAUSE: 3\n\nT46 is ground\nX87 is free\nX99 is free\nX100 is free\nmember(T9, T46) !=? member(X87, [])\nhead(T46, T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ?_9 | member(T9, .(T53, T54)), SCOPE: 7, CLAUSE: 3\n\nT53 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: member(T9, .(T53, T54)), SCOPE: 7, CLAUSE: 3\n\nT53 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail(.(T64, T65), X109), member(T66, X109))\n\nT64 is ground\nT65 is ground\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(tail(.(T64, T65), X109), member(T66, X109)), SCOPE: 10, CLAUSE: 6 | ','(tail(.(T64, T65), X109), member(T66, X109)), SCOPE: 10, CLAUSE: 7\n\nT64 is ground\nT65 is ground\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(tail(.(T64, T65), X109), member(T66, X109)), SCOPE: 10, CLAUSE: 7\n\nT64 is ground\nT65 is ground\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: member(T66, T74)\n\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: member(T9, T46), SCOPE: 7, CLAUSE: 3\n\nT46 is ground\nX87 is free\nX99 is free\nX100 is free\nmember(T9, T46) !=? member(X87, [])\nhead(T46, T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(tail(T83, X127), member(T84, X127))\n\nT83 is ground\nX87 is free\nX99 is free\nX100 is free\nX127 is free\nmember(T9, T83) !=? member(X87, [])\nhead(T83, T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(tail(T83, X127), member(T84, X127)), SCOPE: 11, CLAUSE: 6 | ','(tail(T83, X127), member(T84, X127)), SCOPE: 11, CLAUSE: 7\n\nT83 is ground\nX87 is free\nX99 is free\nX100 is free\nX127 is free\nmember(T9, T83) !=? member(X87, [])\nhead(T83, T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(tail(T83, X127), member(T84, X127)), SCOPE: 11, CLAUSE: 7\n\nT83 is ground\nX87 is free\nX99 is free\nX100 is free\nX127 is free\nmember(T9, T83) !=? member(X87, [])\nhead(T83, T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: member(T84, T91)\n\nT90 is ground\nT91 is ground\nX99 is free\nX100 is free\nhead(.(T90, T91), T47) !=? head(.(X99, X100), X99)", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 100 [arrowhead = none ];
100 -> 4;

101 [label="ONLY EVAL with clause\noverlap(X12, X13) :- ','(member(X14, X12), member(X14, X13)).\nand substitutionT1 -> T7,\nX12 -> T7,\nT2 -> T8,\nX13 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 101 [arrowhead = none ];
101 -> 7;

102 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 102 [arrowhead = none ];
102 -> 13;

103 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:X14 -> T9", fontsize=14, style = filled, fillcolor = violet];
7 -> 103 [arrowhead = none ];
103 -> 15;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 104 [arrowhead = none ];
104 -> 18;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 105 [arrowhead = none ];
105 -> 65;

106 [label="EVAL with clause\nmember(X23, []) :- ','(!_2, failure(a)).\nand substitutionX14 -> X24,\nX23 -> X24,\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 106 [arrowhead = none ];
106 -> 20;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 107 [arrowhead = none ];
107 -> 21;

108 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 108 [arrowhead = none ];
108 -> 22;

109 [label="ONLY EVAL with clause\nmember(X31, X32) :- head(X32, X31).\nand substitutionX14 -> X33,\nX31 -> X33,\nT7 -> T12,\nX32 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 109 [arrowhead = none ];
109 -> 28;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 110 [arrowhead = none ];
110 -> 24;

111 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 111 [arrowhead = none ];
111 -> 25;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 112 [arrowhead = none ];
112 -> 29;

113 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(X14, T12), member(X23, []))", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 113 [arrowhead = none ];
113 -> 31;

114 [label="EVAL with clause\nhead(.(X40, X41), X40).\nand substitutionX40 -> T17,\nX41 -> T18,\nT12 -> .(T17, T18),\nX33 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 114 [arrowhead = none ];
114 -> 37;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 115 [arrowhead = none ];
115 -> 38;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 116 [arrowhead = none ];
116 -> 39;

117 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 117 [arrowhead = none ];
117 -> 55;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 118 [arrowhead = none ];
118 -> 40;

119 [label="ONLY EVAL with clause\nmember(X51, X52) :- ','(tail(X52, X53), member(X51, X53)).\nand substitutionX14 -> X54,\nX51 -> X54,\nT17 -> T23,\nT18 -> T24,\nX52 -> .(T23, T24)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 119 [arrowhead = none ];
119 -> 45;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 120 [arrowhead = none ];
120 -> 49;

121 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 121 [arrowhead = none ];
121 -> 50;

122 [label="ONLY EVAL with clause\ntail(.(X63, X64), X64).\nand substitutionT23 -> T29,\nX63 -> T29,\nT24 -> T30,\nX64 -> T30,\nX53 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 122 [arrowhead = none ];
122 -> 52;

123 [label="INSTANCE with matching:\nX14 -> X54\nT7 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 123 [arrowhead = none , style=dashed];
123 -> 13[style=dashed];

124 [label="ONLY EVAL with clause\nmember(X76, X77) :- ','(tail(X77, X78), member(X76, X78)).\nand substitutionX14 -> X79,\nX76 -> X79,\nT12 -> T33,\nX77 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 124 [arrowhead = none ];
124 -> 58;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 125 [arrowhead = none ];
125 -> 59;

126 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(X14, T33), member(X23, []))", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 126 [arrowhead = none ];
126 -> 61;

127 [label="BACKTRACK\nfor clause: tail(.(X4, T), T)\nwith clash: (head(T33, X33), head(.(X40, X41), X40))", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 127 [arrowhead = none ];
127 -> 63;

128 [label="EVAL with clause\nmember(X87, []) :- ','(!_7, failure(a)).\nand substitutionT9 -> T39,\nX87 -> T39,\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 128 [arrowhead = none ];
128 -> 68;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 129 [arrowhead = none ];
129 -> 69;

130 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 130 [arrowhead = none ];
130 -> 70;

131 [label="ONLY EVAL with clause\nmember(X92, X93) :- head(X93, X92).\nand substitutionT9 -> T47,\nX92 -> T47,\nT8 -> T46,\nX93 -> T46,\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 131 [arrowhead = none ];
131 -> 74;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 132 [arrowhead = none ];
132 -> 71;

133 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 133 [arrowhead = none ];
133 -> 72;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 134 [arrowhead = none ];
134 -> 75;

135 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(T9, T46), member(X87, []))", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 135 [arrowhead = none ];
135 -> 76;

136 [label="EVAL with clause\nhead(.(X99, X100), X99).\nand substitutionX99 -> T53,\nX100 -> T54,\nT46 -> .(T53, T54),\nT47 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 136 [arrowhead = none ];
136 -> 80;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 137 [arrowhead = none ];
137 -> 81;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 138 [arrowhead = none ];
138 -> 82;

139 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 139 [arrowhead = none ];
139 -> 92;

140 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 140 [arrowhead = none ];
140 -> 83;

141 [label="ONLY EVAL with clause\nmember(X107, X108) :- ','(tail(X108, X109), member(X107, X109)).\nand substitutionT9 -> T66,\nX107 -> T66,\nT53 -> T64,\nT54 -> T65,\nX108 -> .(T64, T65),\nT63 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 141 [arrowhead = none ];
141 -> 85;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 142 [arrowhead = none ];
142 -> 87;

143 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 143 [arrowhead = none ];
143 -> 88;

144 [label="ONLY EVAL with clause\ntail(.(X116, X117), X117).\nand substitutionT64 -> T73,\nX116 -> T73,\nT65 -> T74,\nX117 -> T74,\nX109 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 144 [arrowhead = none ];
144 -> 89;

145 [label="INSTANCE with matching:\nT9 -> T66\nT8 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 145 [arrowhead = none , style=dashed];
145 -> 15[style=dashed];

146 [label="ONLY EVAL with clause\nmember(X125, X126) :- ','(tail(X126, X127), member(X125, X127)).\nand substitutionT9 -> T84,\nX125 -> T84,\nT46 -> T83,\nX126 -> T83,\nT82 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 146 [arrowhead = none ];
146 -> 95;

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
95 -> 147 [arrowhead = none ];
147 -> 96;

148 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(T9, T83), member(X87, []))", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 148 [arrowhead = none ];
148 -> 97;

149 [label="EVAL with clause\ntail(.(X133, X134), X134).\nand substitutionX133 -> T90,\nX134 -> T91,\nT83 -> .(T90, T91),\nX127 -> T91", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 149 [arrowhead = none ];
149 -> 98;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 150 [arrowhead = none ];
150 -> 99;

151 [label="INSTANCE with matching:\nT9 -> T84\nT8 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 151 [arrowhead = none , style=dashed];
151 -> 15[style=dashed];

}

(18) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

(19) MaxProof (BOTH BOUNDS(ID, ID) transformation)

Took the maximum complexity of the problems.

(20) Complex Obligation (MAX)

(21) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(23) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(U10'(f15_in(z1), z0, z1))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))

S tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F13_IN, U5', F15_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(25) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

S tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
U2'(f13_out1, z0, z1) → c3
F2_IN(z0, z1) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F13_IN, U5', F15_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(26) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F2_IN(z0, z1) → c7
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U2'(f13_out1, z0, z1) → c3
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U9'(f13_out1, z0, z1) → c7
U5'(f13_out1, z0, z1) → c4

Now S is empty

(27) BOUNDS(O(1), O(1))

(28) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

(29) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)

Multiplied the complexity of the problems.

(30) Complex Obligation (MULT)

(31) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(33) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(U10'(f15_in(z1), z0, z1))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))

S tuples:

F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F13_IN, U5', F15_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(35) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

S tuples:

F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F13_IN, U5', F15_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(36) 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.

F7_IN(z0, z1) → c7(F13_IN(z0))
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4

We considered the (Usable) Rules:

f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F13_IN(x₁)) = x₁   
POL(F15_IN(x₁)) = 0   
POL(F2_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [1] + [2]x₁ + [3]x₂   
POL(U2'(x₁, x₂, x₃)) = x₂   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [1] + x₂   
POL(U9'(x₁, x₂, x₃)) = x₂ + x₃   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f13_in(x₁)) = 0   
POL(f13_out1) = 0   

(37) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

S tuples:

F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))

K tuples:

F7_IN(z0, z1) → c7(F13_IN(z0))
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F13_IN, U5', F15_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(38) 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.

F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))

We considered the (Usable) Rules:

f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

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

POL(.(x₁, x₂)) = 0   
POL(F13_IN(x₁)) = [1]   
POL(F15_IN(x₁)) = 0   
POL(F2_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [1] + [2]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = 0   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2]x₃   
POL(U9'(x₁, x₂, x₃)) = [2]x₂ + [2]x₃   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f13_in(x₁)) = 0   
POL(f13_out1) = 0   

(39) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

S tuples:none
K tuples:

F7_IN(z0, z1) → c7(F13_IN(z0))
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F13_IN, U5', F15_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(40) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(41) BOUNDS(O(1), O(1))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
F7_IN(z0, z1) → c3(U5'(f13_in(z0), z0, z1), F13_IN(z0))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

S tuples:

F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c6(U10'(f15_in(z1), z0, z1), F15_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F13_IN, U5', F15_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
U2'(f13_out1, z0, z1) → c3(U3'(f15_in(z1), z0, z1))
F13_IN(.(z0, z1)) → c1(U4'(f13_in(z1), .(z0, z1)), F13_IN(z1))
U5'(f13_out1, z0, z1) → c4(U6'(f15_in(z1), z0, z1))
F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(U10'(f15_in(z1), z0, z1))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))

S tuples:

F15_IN(.(z0, z1)) → c1(U7'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F15_IN(.(z0, z1)) → c2(U8'(f15_in(z1), .(z0, z1)), F15_IN(z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c7(U10'(f15_in(z1), z0, z1))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F13_IN, U5', F15_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1, z0, z1) → f2_out1
f7_in(z0, z1) → U2(f13_in(z0), z0, z1)
f7_in(z0, z1) → U5(f13_in(z0), z0, z1)
f7_in(z0, z1) → U9(f13_in(z0), z0, z1)
U2(f13_out1, z0, z1) → U3(f15_in(z1), z0, z1)
U3(f15_out1, z0, z1) → f7_out1
f13_in(.(z0, z1)) → f13_out1
f13_in(.(z0, z1)) → U4(f13_in(z1), .(z0, z1))
U4(f13_out1, .(z0, z1)) → f13_out1
U5(f13_out1, z0, z1) → U6(f15_in(z1), z0, z1)
U6(f15_out1, z0, z1) → f7_out1
f15_in(.(z0, z1)) → f15_out1
f15_in(.(z0, z1)) → U7(f15_in(z1), .(z0, z1))
f15_in(.(z0, z1)) → U8(f15_in(z1), .(z0, z1))
U7(f15_out1, .(z0, z1)) → f15_out1
U8(f15_out1, .(z0, z1)) → f15_out1
U9(f13_out1, z0, z1) → U10(f15_in(z1), z0, z1)
U10(f15_out1, z0, z1) → f7_out1

Tuples:

F7_IN(z0, z1) → c2(U2'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f13_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F13_IN(z0))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
U2'(f13_out1, z0, z1) → c3
F13_IN(.(z0, z1)) → c1(F13_IN(z1))
U5'(f13_out1, z0, z1) → c4
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
F2_IN(z0, z1) → c7
U9'(f13_out1, z0, z1) → c7

S tuples:

F7_IN(z0, z1) → c5(U9'(f13_in(z0), z0, z1))
U9'(f13_out1, z0, z1) → c7(F15_IN(z1))
F15_IN(.(z0, z1)) → c1(F15_IN(z1))
F15_IN(.(z0, z1)) → c2(F15_IN(z1))
U9'(f13_out1, z0, z1) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f13_in, U4, U5, U6, f15_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F13_IN, U5', F15_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7