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

(0) Obligation:

Clauses:

p(a, b).
p(b, c).
tc(X, X).
tc(X, Y) :- ','(p(X, Z), tc(Z, Y)).

Query: tc(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];
1 [label="1: tc(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: tc(T1, T2), SCOPE: 1, CLAUSE: 2 | tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | tc(T5, T2), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX3 is free\ntc(T1, T2) !=? tc(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: tc(T5, T2), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T12, X13), tc(X13, T14))\n\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T12, X13), tc(X13, T14)), SCOPE: 2, CLAUSE: 0 | ','(p(T12, X13), tc(X13, T14)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: tc(b, T14) | ','(p(a, X13), tc(X13, T14)), SCOPE: 2, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T12, X13), tc(X13, T14)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground\nX13 is free\np(T12, X13) !=? p(a, b)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: tc(b, T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(a, X13), tc(X13, T14)), SCOPE: 2, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: tc(b, T14), SCOPE: 3, CLAUSE: 2 | tc(b, T14), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | tc(b, T14), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: tc(b, T14), SCOPE: 3, CLAUSE: 3\n\nX22 is free\ntc(b, T14) !=? tc(X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: tc(b, T14), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(b, X32), tc(X32, T21))\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(b, X32), tc(X32, T21)), SCOPE: 4, CLAUSE: 0 | ','(p(b, X32), tc(X32, T21)), SCOPE: 4, CLAUSE: 1\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(b, X32), tc(X32, T21)), SCOPE: 4, CLAUSE: 1\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: tc(c, T21)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: tc(c, T21), SCOPE: 5, CLAUSE: 2 | tc(c, T21), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true | tc(c, T21), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: tc(c, T21), SCOPE: 5, CLAUSE: 3\n\nX41 is free\ntc(c, T21) !=? tc(X41, X41)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: tc(c, T21), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(c, X51), tc(X51, T28))\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(p(c, X51), tc(X51, T28)), SCOPE: 6, CLAUSE: 0 | ','(p(c, X51), tc(X51, T28)), SCOPE: 6, CLAUSE: 1\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(c, X51), tc(X51, T28)), SCOPE: 6, CLAUSE: 1\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(c, X63), tc(X63, T35))\n\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(c, X63), tc(X63, T35)), SCOPE: 7, CLAUSE: 0 | ','(p(c, X63), tc(X63, T35)), SCOPE: 7, CLAUSE: 1\n\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(c, X63), tc(X63, T35)), SCOPE: 7, CLAUSE: 1\n\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(b, X75), tc(X75, T42))\n\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(p(b, X75), tc(X75, T42)), SCOPE: 8, CLAUSE: 0 | ','(p(b, X75), tc(X75, T42)), SCOPE: 8, CLAUSE: 1\n\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(b, X75), tc(X75, T42)), SCOPE: 8, CLAUSE: 1\n\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: tc(c, T42)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: tc(c, T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(p(T50, X93), tc(X93, T52))\n\nT50 is ground\nX3 is free\nX93 is free\ntc(T50, T2) !=? tc(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(p(T50, X93), tc(X93, T52)), SCOPE: 9, CLAUSE: 0 | ','(p(T50, X93), tc(X93, T52)), SCOPE: 9, CLAUSE: 1\n\nT50 is ground\nX3 is free\nX93 is free\ntc(T50, T2) !=? tc(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: tc(b, T52) | ','(p(a, X93), tc(X93, T52)), SCOPE: 9, CLAUSE: 1\n\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(p(T50, X93), tc(X93, T52)), SCOPE: 9, CLAUSE: 1\n\nT50 is ground\nX3 is free\nX93 is free\ntc(T50, T2) !=? tc(X3, X3)\np(T50, X93) !=? p(a, b)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: tc(b, T52)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(p(a, X93), tc(X93, T52)), SCOPE: 9, CLAUSE: 1\n\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: tc(c, T52)\n\n", 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="EVAL with clause\ntc(X3, X3).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 97 [arrowhead = none ];
97 -> 8;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 98 [arrowhead = none ];
98 -> 9;

99 [label="ONLY EVAL with clause\ntc(X91, X92) :- ','(p(X91, X93), tc(X93, X92)).\nand substitutionT1 -> T50,\nX91 -> T50,\nT2 -> T52,\nX92 -> T52,\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 99 [arrowhead = none ];
99 -> 80;

100 [label="ONLY EVAL with clause\ntc(X11, X12) :- ','(p(X11, X13), tc(X13, X12)).\nand substitutionT5 -> T12,\nX11 -> T12,\nT2 -> T14,\nX12 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 100 [arrowhead = none ];
100 -> 13;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 101 [arrowhead = none ];
101 -> 15;

102 [label="EVAL with clause\np(a, b).\nand substitutionT12 -> a,\nX13 -> b", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 102 [arrowhead = none ];
102 -> 17;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 103 [arrowhead = none ];
103 -> 18;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 105 [arrowhead = none ];
105 -> 21;

106 [label="EVAL with clause\np(b, c).\nand substitutionT12 -> b,\nX13 -> c", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 106 [arrowhead = none ];
106 -> 73;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 107 [arrowhead = none ];
107 -> 75;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 108 [arrowhead = none ];
108 -> 23;

109 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 109 [arrowhead = none ];
109 -> 71;

110 [label="EVAL with clause\ntc(X22, X22).\nand substitutionX22 -> b,\nT14 -> b", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 111 [arrowhead = none ];
111 -> 27;

112 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 112 [arrowhead = none ];
112 -> 29;

113 [label="ONLY EVAL with clause\ntc(X73, X74) :- ','(p(X73, X75), tc(X75, X74)).\nand substitutionX73 -> b,\nT14 -> T42,\nX74 -> T42,\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 113 [arrowhead = none ];
113 -> 64;

114 [label="ONLY EVAL with clause\ntc(X30, X31) :- ','(p(X30, X32), tc(X32, X31)).\nand substitutionX30 -> b,\nT14 -> T21,\nX31 -> T21,\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 114 [arrowhead = none ];
114 -> 32;

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

116 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 116 [arrowhead = none ];
116 -> 36;

117 [label="ONLY EVAL with clause\np(b, c).\nand substitutionX32 -> c", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 117 [arrowhead = none ];
117 -> 37;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 118 [arrowhead = none ];
118 -> 39;

119 [label="EVAL with clause\ntc(X41, X41).\nand substitutionX41 -> c,\nT21 -> c", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 119 [arrowhead = none ];
119 -> 41;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 120 [arrowhead = none ];
120 -> 43;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 121 [arrowhead = none ];
121 -> 45;

122 [label="ONLY EVAL with clause\ntc(X61, X62) :- ','(p(X61, X63), tc(X63, X62)).\nand substitutionX61 -> c,\nT21 -> T35,\nX62 -> T35,\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 122 [arrowhead = none ];
122 -> 56;

123 [label="ONLY EVAL with clause\ntc(X49, X50) :- ','(p(X49, X51), tc(X51, X50)).\nand substitutionX49 -> c,\nT21 -> T28,\nX50 -> T28,\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 126 [arrowhead = none ];
126 -> 54;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 127 [arrowhead = none ];
127 -> 60;

128 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 128 [arrowhead = none ];
128 -> 61;

129 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 129 [arrowhead = none ];
129 -> 62;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 130 [arrowhead = none ];
130 -> 67;

131 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 131 [arrowhead = none ];
131 -> 68;

132 [label="ONLY EVAL with clause\np(b, c).\nand substitutionX75 -> c", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 132 [arrowhead = none ];
132 -> 70;

133 [label="INSTANCE with matching:\nT21 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 133 [arrowhead = none , style=dashed];
133 -> 37[style=dashed];

134 [label="INSTANCE with matching:\nT21 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 134 [arrowhead = none , style=dashed];
134 -> 37[style=dashed];

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 135 [arrowhead = none ];
135 -> 82;

136 [label="EVAL with clause\np(a, b).\nand substitutionT50 -> a,\nX93 -> b", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 136 [arrowhead = none ];
136 -> 85;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 137 [arrowhead = none ];
137 -> 86;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 138 [arrowhead = none ];
138 -> 88;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 139 [arrowhead = none ];
139 -> 90;

140 [label="EVAL with clause\np(b, c).\nand substitutionT50 -> b,\nX93 -> c", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 140 [arrowhead = none ];
140 -> 93;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 141 [arrowhead = none ];
141 -> 94;

142 [label="INSTANCE with matching:\nT14 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 142 [arrowhead = none , style=dashed];
142 -> 19[style=dashed];

143 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 143 [arrowhead = none ];
143 -> 92;

144 [label="INSTANCE with matching:\nT21 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 144 [arrowhead = none , style=dashed];
144 -> 37[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0)
f1_in(a) → U1(f17_in, a)
f1_in(b) → U2(f37_in, b)
f1_in(a) → U3(f85_in, a)
f1_in(b) → U4(f37_in, b)
U1(f17_out1(z0), a) → f1_out1(z0)
U1(f17_out2(z0, z1), a) → f1_out1(z1)
U2(f37_out1(z0), b) → f1_out1(z0)
U3(f85_out1(z0), a) → f1_out1(z0)
U3(f85_out2(z0, z1), a) → f1_out1(z1)
U4(f37_out1(z0), b) → f1_out1(z0)
f19_in → f19_out1(b)
f19_in → U5(f37_in)
f19_in → U6(f37_in)
U5(f37_out1(z0)) → f19_out1(z0)
U6(f37_out1(z0)) → f19_out1(z0)
f37_in → f37_out1(c)
f17_in → U7(f19_in, f21_in)
U7(f19_out1(z0), z1) → f17_out1(z0)
U7(z0, f21_out1(z1, z2)) → f17_out2(z1, z2)
f85_in → U8(f19_in, f90_in)
U8(f19_out1(z0), z1) → f85_out1(z0)
U8(z0, f90_out1(z1, z2)) → f85_out2(z1, z2)

Tuples:

F1_IN(a) → c2(U1'(f17_in, a), F17_IN)
F1_IN(b) → c3(U2'(f37_in, b), F37_IN)
F1_IN(a) → c4(U3'(f85_in, a), F85_IN)
F1_IN(b) → c5(U4'(f37_in, b), F37_IN)
F19_IN → c13(U5'(f37_in), F37_IN)
F19_IN → c14(U6'(f37_in), F37_IN)
F17_IN → c18(U7'(f19_in, f21_in), F19_IN)
F85_IN → c21(U8'(f19_in, f90_in), F19_IN)

S tuples:

F1_IN(a) → c2(U1'(f17_in, a), F17_IN)
F1_IN(b) → c3(U2'(f37_in, b), F37_IN)
F1_IN(a) → c4(U3'(f85_in, a), F85_IN)
F1_IN(b) → c5(U4'(f37_in, b), F37_IN)
F19_IN → c13(U5'(f37_in), F37_IN)
F19_IN → c14(U6'(f37_in), F37_IN)
F17_IN → c18(U7'(f19_in, f21_in), F19_IN)
F85_IN → c21(U8'(f19_in, f90_in), F19_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f19_in, U5, U6, f37_in, f17_in, U7, f85_in, U8

Defined Pair Symbols:

F1_IN, F19_IN, F17_IN, F85_IN

Compound Symbols:

c2, c3, c4, c5, c13, c14, c18, c21

(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) → f1_out1(z0)
f1_in(a) → U1(f17_in, a)
f1_in(b) → U2(f37_in, b)
f1_in(a) → U3(f85_in, a)
f1_in(b) → U4(f37_in, b)
U1(f17_out1(z0), a) → f1_out1(z0)
U1(f17_out2(z0, z1), a) → f1_out1(z1)
U2(f37_out1(z0), b) → f1_out1(z0)
U3(f85_out1(z0), a) → f1_out1(z0)
U3(f85_out2(z0, z1), a) → f1_out1(z1)
U4(f37_out1(z0), b) → f1_out1(z0)
f19_in → f19_out1(b)
f19_in → U5(f37_in)
f19_in → U6(f37_in)
U5(f37_out1(z0)) → f19_out1(z0)
U6(f37_out1(z0)) → f19_out1(z0)
f37_in → f37_out1(c)
f17_in → U7(f19_in, f21_in)
U7(f19_out1(z0), z1) → f17_out1(z0)
U7(z0, f21_out1(z1, z2)) → f17_out2(z1, z2)
f85_in → U8(f19_in, f90_in)
U8(f19_out1(z0), z1) → f85_out1(z0)
U8(z0, f90_out1(z1, z2)) → f85_out2(z1, z2)

Tuples:

F1_IN(a) → c1(U1'(f17_in, a))
F1_IN(a) → c1(F17_IN)
F1_IN(b) → c1(U2'(f37_in, b))
F1_IN(b) → c1(F37_IN)
F1_IN(a) → c1(U3'(f85_in, a))
F1_IN(a) → c1(F85_IN)
F1_IN(b) → c1(U4'(f37_in, b))
F19_IN → c1(U5'(f37_in))
F19_IN → c1(F37_IN)
F19_IN → c1(U6'(f37_in))
F17_IN → c1(U7'(f19_in, f21_in))
F17_IN → c1(F19_IN)
F85_IN → c1(U8'(f19_in, f90_in))
F85_IN → c1(F19_IN)

S tuples:

F1_IN(a) → c1(U1'(f17_in, a))
F1_IN(a) → c1(F17_IN)
F1_IN(b) → c1(U2'(f37_in, b))
F1_IN(b) → c1(F37_IN)
F1_IN(a) → c1(U3'(f85_in, a))
F1_IN(a) → c1(F85_IN)
F1_IN(b) → c1(U4'(f37_in, b))
F19_IN → c1(U5'(f37_in))
F19_IN → c1(F37_IN)
F19_IN → c1(U6'(f37_in))
F17_IN → c1(U7'(f19_in, f21_in))
F17_IN → c1(F19_IN)
F85_IN → c1(U8'(f19_in, f90_in))
F85_IN → c1(F19_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f19_in, U5, U6, f37_in, f17_in, U7, f85_in, U8

Defined Pair Symbols:

F1_IN, F19_IN, F17_IN, F85_IN

Compound Symbols:

c1

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

Removed 12 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0)
f1_in(a) → U1(f17_in, a)
f1_in(b) → U2(f37_in, b)
f1_in(a) → U3(f85_in, a)
f1_in(b) → U4(f37_in, b)
U1(f17_out1(z0), a) → f1_out1(z0)
U1(f17_out2(z0, z1), a) → f1_out1(z1)
U2(f37_out1(z0), b) → f1_out1(z0)
U3(f85_out1(z0), a) → f1_out1(z0)
U3(f85_out2(z0, z1), a) → f1_out1(z1)
U4(f37_out1(z0), b) → f1_out1(z0)
f19_in → f19_out1(b)
f19_in → U5(f37_in)
f19_in → U6(f37_in)
U5(f37_out1(z0)) → f19_out1(z0)
U6(f37_out1(z0)) → f19_out1(z0)
f37_in → f37_out1(c)
f17_in → U7(f19_in, f21_in)
U7(f19_out1(z0), z1) → f17_out1(z0)
U7(z0, f21_out1(z1, z2)) → f17_out2(z1, z2)
f85_in → U8(f19_in, f90_in)
U8(f19_out1(z0), z1) → f85_out1(z0)
U8(z0, f90_out1(z1, z2)) → f85_out2(z1, z2)

Tuples:

F1_IN(a) → c1(F17_IN)
F1_IN(a) → c1(F85_IN)
F17_IN → c1(F19_IN)
F85_IN → c1(F19_IN)
F1_IN(a) → c1
F1_IN(b) → c1
F19_IN → c1
F17_IN → c1
F85_IN → c1

S tuples:

F1_IN(a) → c1(F17_IN)
F1_IN(a) → c1(F85_IN)
F17_IN → c1(F19_IN)
F85_IN → c1(F19_IN)
F1_IN(a) → c1
F1_IN(b) → c1
F19_IN → c1
F17_IN → c1
F85_IN → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f19_in, U5, U6, f37_in, f17_in, U7, f85_in, U8

Defined Pair Symbols:

F1_IN, F17_IN, F85_IN, F19_IN

Compound Symbols:

c1, c1

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(a) → c1(F17_IN)
F1_IN(a) → c1(F85_IN)
F17_IN → c1(F19_IN)
F85_IN → c1(F19_IN)
F1_IN(a) → c1
F1_IN(a) → c1
F1_IN(b) → c1
F1_IN(b) → c1
F1_IN(b) → c1
F1_IN(b) → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F17_IN → c1
F85_IN → c1
F17_IN → c1(F19_IN)
F17_IN → c1
F85_IN → c1(F19_IN)
F85_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1
F19_IN → c1

Now 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];
2 [label="2: tc(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: tc(T1, T2), SCOPE: 1, CLAUSE: 2 | tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | tc(T4, T2), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: tc(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX2 is free\ntc(T1, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: tc(T4, T2), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T7, X7), tc(X7, T9))\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 0 | ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: tc(b, T9) | ','(p(a, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T7, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free\np(T7, X7) !=? p(a, b)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: tc(b, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(a, X7), tc(X7, T9)), SCOPE: 2, CLAUSE: 1\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: tc(b, T9), SCOPE: 3, CLAUSE: 2 | tc(b, T9), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | tc(b, T9), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: tc(b, T9), SCOPE: 3, CLAUSE: 3\n\nX14 is free\ntc(b, T9) !=? tc(X14, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: tc(b, T9), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(b, X24), tc(X24, T15))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 0 | ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 1\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(b, X24), tc(X24, T15)), SCOPE: 4, CLAUSE: 1\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tc(c, T15)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: tc(c, T15), SCOPE: 5, CLAUSE: 2 | tc(c, T15), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true | tc(c, T15), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: tc(c, T15), SCOPE: 5, CLAUSE: 3\n\nX33 is free\ntc(c, T15) !=? tc(X33, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: tc(c, T15), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(c, X43), tc(X43, T22))\n\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(c, X43), tc(X43, T22)), SCOPE: 6, CLAUSE: 0 | ','(p(c, X43), tc(X43, T22)), SCOPE: 6, CLAUSE: 1\n\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(c, X43), tc(X43, T22)), SCOPE: 6, CLAUSE: 1\n\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(c, X55), tc(X55, T29))\n\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(c, X55), tc(X55, T29)), SCOPE: 7, CLAUSE: 0 | ','(p(c, X55), tc(X55, T29)), SCOPE: 7, CLAUSE: 1\n\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(c, X55), tc(X55, T29)), SCOPE: 7, CLAUSE: 1\n\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(b, X67), tc(X67, T36))\n\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(p(b, X67), tc(X67, T36)), SCOPE: 8, CLAUSE: 0 | ','(p(b, X67), tc(X67, T36)), SCOPE: 8, CLAUSE: 1\n\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(p(b, X67), tc(X67, T36)), SCOPE: 8, CLAUSE: 1\n\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: tc(c, T36)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: tc(c, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(p(T40, X80), tc(X80, T42))\n\nT40 is ground\nX2 is free\nX80 is free\ntc(T40, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(p(T40, X80), tc(X80, T42)), SCOPE: 9, CLAUSE: 0 | ','(p(T40, X80), tc(X80, T42)), SCOPE: 9, CLAUSE: 1\n\nT40 is ground\nX2 is free\nX80 is free\ntc(T40, T2) !=? tc(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: tc(b, T42) | ','(p(a, X80), tc(X80, T42)), SCOPE: 9, CLAUSE: 1\n\nX80 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(T40, X80), tc(X80, T42)), SCOPE: 9, CLAUSE: 1\n\nT40 is ground\nX2 is free\nX80 is free\ntc(T40, T2) !=? tc(X2, X2)\np(T40, X80) !=? p(a, b)", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: tc(b, T42)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(p(a, X80), tc(X80, T42)), SCOPE: 9, CLAUSE: 1\n\nX80 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: tc(c, T42)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 92 [arrowhead = none ];
92 -> 4;

93 [label="EVAL with clause\ntc(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 93 [arrowhead = none ];
93 -> 6;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 94 [arrowhead = none ];
94 -> 7;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 95 [arrowhead = none ];
95 -> 10;

96 [label="ONLY EVAL with clause\ntc(X78, X79) :- ','(p(X78, X80), tc(X80, X79)).\nand substitutionT1 -> T40,\nX78 -> T40,\nT2 -> T42,\nX79 -> T42,\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 96 [arrowhead = none ];
96 -> 77;

97 [label="ONLY EVAL with clause\ntc(X5, X6) :- ','(p(X5, X7), tc(X7, X6)).\nand substitutionT4 -> T7,\nX5 -> T7,\nT2 -> T9,\nX6 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 97 [arrowhead = none ];
97 -> 11;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 98 [arrowhead = none ];
98 -> 12;

99 [label="EVAL with clause\np(a, b).\nand substitutionT7 -> a,\nX7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 99 [arrowhead = none ];
99 -> 14;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 100 [arrowhead = none ];
100 -> 16;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 101 [arrowhead = none ];
101 -> 20;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 102 [arrowhead = none ];
102 -> 22;

103 [label="EVAL with clause\np(b, c).\nand substitutionT7 -> b,\nX7 -> c", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 103 [arrowhead = none ];
103 -> 74;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 104 [arrowhead = none ];
104 -> 76;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 105 [arrowhead = none ];
105 -> 24;

106 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 106 [arrowhead = none ];
106 -> 72;

107 [label="EVAL with clause\ntc(X14, X14).\nand substitutionX14 -> b,\nT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 107 [arrowhead = none ];
107 -> 26;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 108 [arrowhead = none ];
108 -> 28;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 109 [arrowhead = none ];
109 -> 30;

110 [label="ONLY EVAL with clause\ntc(X65, X66) :- ','(p(X65, X67), tc(X67, X66)).\nand substitutionX65 -> b,\nT9 -> T36,\nX66 -> T36,\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 110 [arrowhead = none ];
110 -> 63;

111 [label="ONLY EVAL with clause\ntc(X22, X23) :- ','(p(X22, X24), tc(X24, X23)).\nand substitutionX22 -> b,\nT9 -> T15,\nX23 -> T15,\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 111 [arrowhead = none ];
111 -> 31;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 113 [arrowhead = none ];
113 -> 35;

114 [label="ONLY EVAL with clause\np(b, c).\nand substitutionX24 -> c", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\ntc(X33, X33).\nand substitutionX33 -> c,\nT15 -> c", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 117 [arrowhead = none ];
117 -> 44;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 118 [arrowhead = none ];
118 -> 46;

119 [label="ONLY EVAL with clause\ntc(X53, X54) :- ','(p(X53, X55), tc(X55, X54)).\nand substitutionX53 -> c,\nT15 -> T29,\nX54 -> T29,\nT28 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 119 [arrowhead = none ];
119 -> 55;

120 [label="ONLY EVAL with clause\ntc(X41, X42) :- ','(p(X41, X43), tc(X43, X42)).\nand substitutionX41 -> c,\nT15 -> T22,\nX42 -> T22,\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 120 [arrowhead = none ];
120 -> 47;

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

122 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 122 [arrowhead = none ];
122 -> 51;

123 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 123 [arrowhead = none ];
123 -> 53;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 124 [arrowhead = none ];
124 -> 57;

125 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 125 [arrowhead = none ];
125 -> 58;

126 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 126 [arrowhead = none ];
126 -> 59;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 127 [arrowhead = none ];
127 -> 65;

128 [label="BACKTRACK\nfor clause: p(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 128 [arrowhead = none ];
128 -> 66;

129 [label="ONLY EVAL with clause\np(b, c).\nand substitutionX67 -> c", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 129 [arrowhead = none ];
129 -> 69;

130 [label="INSTANCE with matching:\nT15 -> T36", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 130 [arrowhead = none , style=dashed];
130 -> 38[style=dashed];

131 [label="INSTANCE with matching:\nT15 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 131 [arrowhead = none , style=dashed];
131 -> 38[style=dashed];

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 132 [arrowhead = none ];
132 -> 78;

133 [label="EVAL with clause\np(a, b).\nand substitutionT40 -> a,\nX80 -> b", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 133 [arrowhead = none ];
133 -> 79;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 134 [arrowhead = none ];
134 -> 81;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 135 [arrowhead = none ];
135 -> 83;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 136 [arrowhead = none ];
136 -> 84;

137 [label="EVAL with clause\np(b, c).\nand substitutionT40 -> b,\nX80 -> c", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 137 [arrowhead = none ];
137 -> 89;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 138 [arrowhead = none ];
138 -> 91;

139 [label="INSTANCE with matching:\nT9 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 139 [arrowhead = none , style=dashed];
139 -> 20[style=dashed];

140 [label="BACKTRACK\nfor clause: p(b, c)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 140 [arrowhead = none ];
140 -> 87;

141 [label="INSTANCE with matching:\nT15 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 141 [arrowhead = none , style=dashed];
141 -> 38[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(a) → U1(f14_in, a)
f2_in(b) → U2(f38_in, b)
f2_in(a) → U3(f79_in, a)
f2_in(b) → U4(f38_in, b)
U1(f14_out1(z0), a) → f2_out1(z0)
U1(f14_out2(z0, z1), a) → f2_out1(z1)
U2(f38_out1(z0), b) → f2_out1(z0)
U3(f79_out1(z0), a) → f2_out1(z0)
U3(f79_out2(z0, z1), a) → f2_out1(z1)
U4(f38_out1(z0), b) → f2_out1(z0)
f20_in → f20_out1(b)
f20_in → U5(f38_in)
f20_in → U6(f38_in)
U5(f38_out1(z0)) → f20_out1(z0)
U6(f38_out1(z0)) → f20_out1(z0)
f38_in → f38_out1(c)
f14_in → U7(f20_in, f22_in)
U7(f20_out1(z0), z1) → f14_out1(z0)
U7(z0, f22_out1(z1, z2)) → f14_out2(z1, z2)
f79_in → U8(f20_in, f84_in)
U8(f20_out1(z0), z1) → f79_out1(z0)
U8(z0, f84_out1(z1, z2)) → f79_out2(z1, z2)

Tuples:

F2_IN(a) → c2(U1'(f14_in, a), F14_IN)
F2_IN(b) → c3(U2'(f38_in, b), F38_IN)
F2_IN(a) → c4(U3'(f79_in, a), F79_IN)
F2_IN(b) → c5(U4'(f38_in, b), F38_IN)
F20_IN → c13(U5'(f38_in), F38_IN)
F20_IN → c14(U6'(f38_in), F38_IN)
F14_IN → c18(U7'(f20_in, f22_in), F20_IN)
F79_IN → c21(U8'(f20_in, f84_in), F20_IN)

S tuples:

F2_IN(a) → c2(U1'(f14_in, a), F14_IN)
F2_IN(b) → c3(U2'(f38_in, b), F38_IN)
F2_IN(a) → c4(U3'(f79_in, a), F79_IN)
F2_IN(b) → c5(U4'(f38_in, b), F38_IN)
F20_IN → c13(U5'(f38_in), F38_IN)
F20_IN → c14(U6'(f38_in), F38_IN)
F14_IN → c18(U7'(f20_in, f22_in), F20_IN)
F79_IN → c21(U8'(f20_in, f84_in), F20_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4, f20_in, U5, U6, f38_in, f14_in, U7, f79_in, U8

Defined Pair Symbols:

F2_IN, F20_IN, F14_IN, F79_IN

Compound Symbols:

c2, c3, c4, c5, c13, c14, c18, c21

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

Split RHS of tuples not part of any SCC

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(a) → U1(f14_in, a)
f2_in(b) → U2(f38_in, b)
f2_in(a) → U3(f79_in, a)
f2_in(b) → U4(f38_in, b)
U1(f14_out1(z0), a) → f2_out1(z0)
U1(f14_out2(z0, z1), a) → f2_out1(z1)
U2(f38_out1(z0), b) → f2_out1(z0)
U3(f79_out1(z0), a) → f2_out1(z0)
U3(f79_out2(z0, z1), a) → f2_out1(z1)
U4(f38_out1(z0), b) → f2_out1(z0)
f20_in → f20_out1(b)
f20_in → U5(f38_in)
f20_in → U6(f38_in)
U5(f38_out1(z0)) → f20_out1(z0)
U6(f38_out1(z0)) → f20_out1(z0)
f38_in → f38_out1(c)
f14_in → U7(f20_in, f22_in)
U7(f20_out1(z0), z1) → f14_out1(z0)
U7(z0, f22_out1(z1, z2)) → f14_out2(z1, z2)
f79_in → U8(f20_in, f84_in)
U8(f20_out1(z0), z1) → f79_out1(z0)
U8(z0, f84_out1(z1, z2)) → f79_out2(z1, z2)

Tuples:

F2_IN(a) → c1(U1'(f14_in, a))
F2_IN(a) → c1(F14_IN)
F2_IN(b) → c1(U2'(f38_in, b))
F2_IN(b) → c1(F38_IN)
F2_IN(a) → c1(U3'(f79_in, a))
F2_IN(a) → c1(F79_IN)
F2_IN(b) → c1(U4'(f38_in, b))
F20_IN → c1(U5'(f38_in))
F20_IN → c1(F38_IN)
F20_IN → c1(U6'(f38_in))
F14_IN → c1(U7'(f20_in, f22_in))
F14_IN → c1(F20_IN)
F79_IN → c1(U8'(f20_in, f84_in))
F79_IN → c1(F20_IN)

S tuples:

F2_IN(a) → c1(U1'(f14_in, a))
F2_IN(a) → c1(F14_IN)
F2_IN(b) → c1(U2'(f38_in, b))
F2_IN(b) → c1(F38_IN)
F2_IN(a) → c1(U3'(f79_in, a))
F2_IN(a) → c1(F79_IN)
F2_IN(b) → c1(U4'(f38_in, b))
F20_IN → c1(U5'(f38_in))
F20_IN → c1(F38_IN)
F20_IN → c1(U6'(f38_in))
F14_IN → c1(U7'(f20_in, f22_in))
F14_IN → c1(F20_IN)
F79_IN → c1(U8'(f20_in, f84_in))
F79_IN → c1(F20_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4, f20_in, U5, U6, f38_in, f14_in, U7, f79_in, U8

Defined Pair Symbols:

F2_IN, F20_IN, F14_IN, F79_IN

Compound Symbols:

c1

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

Removed 12 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(a) → U1(f14_in, a)
f2_in(b) → U2(f38_in, b)
f2_in(a) → U3(f79_in, a)
f2_in(b) → U4(f38_in, b)
U1(f14_out1(z0), a) → f2_out1(z0)
U1(f14_out2(z0, z1), a) → f2_out1(z1)
U2(f38_out1(z0), b) → f2_out1(z0)
U3(f79_out1(z0), a) → f2_out1(z0)
U3(f79_out2(z0, z1), a) → f2_out1(z1)
U4(f38_out1(z0), b) → f2_out1(z0)
f20_in → f20_out1(b)
f20_in → U5(f38_in)
f20_in → U6(f38_in)
U5(f38_out1(z0)) → f20_out1(z0)
U6(f38_out1(z0)) → f20_out1(z0)
f38_in → f38_out1(c)
f14_in → U7(f20_in, f22_in)
U7(f20_out1(z0), z1) → f14_out1(z0)
U7(z0, f22_out1(z1, z2)) → f14_out2(z1, z2)
f79_in → U8(f20_in, f84_in)
U8(f20_out1(z0), z1) → f79_out1(z0)
U8(z0, f84_out1(z1, z2)) → f79_out2(z1, z2)

Tuples:

F2_IN(a) → c1(F14_IN)
F2_IN(a) → c1(F79_IN)
F14_IN → c1(F20_IN)
F79_IN → c1(F20_IN)
F2_IN(a) → c1
F2_IN(b) → c1
F20_IN → c1
F14_IN → c1
F79_IN → c1

S tuples:

F2_IN(a) → c1(F14_IN)
F2_IN(a) → c1(F79_IN)
F14_IN → c1(F20_IN)
F79_IN → c1(F20_IN)
F2_IN(a) → c1
F2_IN(b) → c1
F20_IN → c1
F14_IN → c1
F79_IN → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4, f20_in, U5, U6, f38_in, f14_in, U7, f79_in, U8

Defined Pair Symbols:

F2_IN, F14_IN, F79_IN, F20_IN

Compound Symbols:

c1, c1