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

(0) Obligation:

Clauses:

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

Query: fold(g,g,a)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, eq(T8, T10)) | fold(T8, [], T3), SCOPE: 1, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX8 is free\nX9 is free\nfold(T1, T2, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(T8, T10)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T8, T10), SCOPE: 2, CLAUSE: 7\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25))))\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 3 | ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 4\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(head(T23, X28), ','(tail(T23, X29), ','(myop(T22, X28, X30), fold(X30, X29, T25)))), SCOPE: 3, CLAUSE: 4\n\nT22 is ground\nT23 is ground\nX8 is free\nX9 is free\nX28 is free\nX29 is free\nX30 is free\nfold(T22, T23, T3) !=? fold(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25)))\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 6\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T31, T32), X29), ','(myop(T22, T31, X30), fold(X30, X29, T25))), SCOPE: 4, CLAUSE: 6\n\nT22 is ground\nT31 is ground\nT32 is ground\nX29 is free\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(myop(T22, T39, X30), fold(X30, T40, T25))\n\nT22 is ground\nT39 is ground\nT40 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(myop(T22, T39, X30), fold(X30, T40, T25)), SCOPE: 5, CLAUSE: 2\n\nT22 is ground\nT39 is ground\nT40 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: fold(a, T40, T25)\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: fold(a, T40, T25), SCOPE: 6, CLAUSE: 0 | fold(a, T40, T25), SCOPE: 6, CLAUSE: 1\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(!_6, eq(a, T46)) | fold(a, [], T25), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: fold(a, T40, T25), SCOPE: 6, CLAUSE: 1\n\nT40 is ground\nX63 is free\nX64 is free\nfold(a, T40, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(a, T46)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: eq(a, T46), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(a, X83, X85), fold(X85, X84, T55))))\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(a, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(a, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 3 | ','(head(T53, X83), ','(tail(T53, X84), ','(myop(a, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 4\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(a, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(head(T53, X83), ','(tail(T53, X84), ','(myop(a, X83, X85), fold(X85, X84, T55)))), SCOPE: 8, CLAUSE: 4\n\nT53 is ground\nX63 is free\nX64 is free\nX83 is free\nX84 is free\nX85 is free\nfold(a, T53, T25) !=? fold(X63, [], X64)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(T61, T62), X84), ','(myop(a, T61, X85), fold(X85, X84, T55)))\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(tail(.(T61, T62), X84), ','(myop(a, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T61, T62), X84), ','(myop(a, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 6\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(T61, T62), X84), ','(myop(a, T61, X85), fold(X85, X84, T55))), SCOPE: 9, CLAUSE: 6\n\nT61 is ground\nT62 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(myop(a, T69, X85), fold(X85, T70, T55))\n\nT69 is ground\nT70 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(myop(a, T69, X85), fold(X85, T70, T55)), SCOPE: 10, CLAUSE: 2\n\nT69 is ground\nT70 is ground\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: fold(a, T70, T55)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="EVAL with clause\nfold(X8, [], X9) :- ','(!_1, eq(X8, X9)).\nand substitutionT1 -> T8,\nX8 -> T8,\nT2 -> [],\nT3 -> T10,\nX9 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 81 [arrowhead = none ];
81 -> 8;

82 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 82 [arrowhead = none ];
82 -> 9;

83 [label="ONLY EVAL with clause\nfold(X25, X26, X27) :- ','(head(X26, X28), ','(tail(X26, X29), ','(myop(X25, X28, X30), fold(X30, X29, X27)))).\nand substitutionT1 -> T22,\nX25 -> T22,\nT2 -> T23,\nX26 -> T23,\nT3 -> T25,\nX27 -> T25,\nT24 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 83 [arrowhead = none ];
83 -> 21;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 84 [arrowhead = none ];
84 -> 12;

85 [label="EVAL with clause\neq(X12, X12).\nand substitutionT8 -> T13,\nX12 -> T13,\nT10 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 85 [arrowhead = none ];
85 -> 14;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 87 [arrowhead = none ];
87 -> 18;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 88 [arrowhead = none ];
88 -> 23;

89 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T22, T23, T3), fold(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 89 [arrowhead = none ];
89 -> 24;

90 [label="EVAL with clause\nhead(.(X40, X41), X40).\nand substitutionX40 -> T31,\nX41 -> T32,\nT23 -> .(T31, T32),\nX28 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 90 [arrowhead = none ];
90 -> 29;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 91 [arrowhead = none ];
91 -> 30;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="ONLY EVAL with clause\ntail(.(X52, X53), X53).\nand substitutionT31 -> T39,\nX52 -> T39,\nT32 -> T40,\nX53 -> T40,\nX29 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 94 [arrowhead = none ];
94 -> 37;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 95 [arrowhead = none ];
95 -> 38;

96 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT22 -> a,\nT39 -> b,\nX30 -> a", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 96 [arrowhead = none ];
96 -> 39;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 97 [arrowhead = none ];
97 -> 40;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 98 [arrowhead = none ];
98 -> 42;

99 [label="EVAL with clause\nfold(X63, [], X64) :- ','(!_6, eq(X63, X64)).\nand substitutionX63 -> a,\nT40 -> [],\nT25 -> T46,\nX64 -> T46,\nT45 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 99 [arrowhead = none ];
99 -> 44;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 100 [arrowhead = none ];
100 -> 47;

101 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 101 [arrowhead = none ];
101 -> 48;

102 [label="ONLY EVAL with clause\nfold(X80, X81, X82) :- ','(head(X81, X83), ','(tail(X81, X84), ','(myop(X80, X83, X85), fold(X85, X84, X82)))).\nand substitutionX80 -> a,\nT40 -> T53,\nX81 -> T53,\nT25 -> T55,\nX82 -> T55,\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 102 [arrowhead = none ];
102 -> 58;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 103 [arrowhead = none ];
103 -> 50;

104 [label="EVAL with clause\neq(X67, X67).\nand substitutionX67 -> a,\nT46 -> a", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 104 [arrowhead = none ];
104 -> 53;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 105 [arrowhead = none ];
105 -> 55;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 106 [arrowhead = none ];
106 -> 56;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 107 [arrowhead = none ];
107 -> 60;

108 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(a, T53, T25), fold(X63, [], X64))", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 108 [arrowhead = none ];
108 -> 62;

109 [label="EVAL with clause\nhead(.(X95, X96), X95).\nand substitutionX95 -> T61,\nX96 -> T62,\nT53 -> .(T61, T62),\nX83 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 109 [arrowhead = none ];
109 -> 65;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 110 [arrowhead = none ];
110 -> 66;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 111 [arrowhead = none ];
111 -> 69;

112 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 112 [arrowhead = none ];
112 -> 70;

113 [label="ONLY EVAL with clause\ntail(.(X107, X108), X108).\nand substitutionT61 -> T69,\nX107 -> T69,\nT62 -> T70,\nX108 -> T70,\nX84 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 113 [arrowhead = none ];
113 -> 73;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 114 [arrowhead = none ];
114 -> 76;

115 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT69 -> b,\nX85 -> a", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 115 [arrowhead = none ];
115 -> 77;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 116 [arrowhead = none ];
116 -> 78;

117 [label="INSTANCE with matching:\nT40 -> T70\nT25 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 117 [arrowhead = none , style=dashed];
117 -> 39[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, z0)) → U1(f39_in(z0), a, .(b, z0))
U1(f39_out1(z0), a, .(b, z1)) → f2_out1(z0)
f39_in([]) → f39_out1(a)
f39_in(.(b, z0)) → U2(f39_in(z0), .(b, z0))
U2(f39_out1(z0), .(b, z1)) → f39_out1(z0)

Tuples:

F2_IN(a, .(b, z0)) → c1(U1'(f39_in(z0), a, .(b, z0)), F39_IN(z0))
F39_IN(.(b, z0)) → c4(U2'(f39_in(z0), .(b, z0)), F39_IN(z0))

S tuples:

F2_IN(a, .(b, z0)) → c1(U1'(f39_in(z0), a, .(b, z0)), F39_IN(z0))
F39_IN(.(b, z0)) → c4(U2'(f39_in(z0), .(b, z0)), F39_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f39_in, U2

Defined Pair Symbols:

F2_IN, F39_IN

Compound Symbols:

c1, c4

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, z0)) → U1(f39_in(z0), a, .(b, z0))
U1(f39_out1(z0), a, .(b, z1)) → f2_out1(z0)
f39_in([]) → f39_out1(a)
f39_in(.(b, z0)) → U2(f39_in(z0), .(b, z0))
U2(f39_out1(z0), .(b, z1)) → f39_out1(z0)

Tuples:

F39_IN(.(b, z0)) → c4(U2'(f39_in(z0), .(b, z0)), F39_IN(z0))
F2_IN(a, .(b, z0)) → c(U1'(f39_in(z0), a, .(b, z0)))
F2_IN(a, .(b, z0)) → c(F39_IN(z0))

S tuples:

F39_IN(.(b, z0)) → c4(U2'(f39_in(z0), .(b, z0)), F39_IN(z0))
F2_IN(a, .(b, z0)) → c(U1'(f39_in(z0), a, .(b, z0)))
F2_IN(a, .(b, z0)) → c(F39_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f39_in, U2

Defined Pair Symbols:

F39_IN, F2_IN

Compound Symbols:

c4, c

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, z0)) → U1(f39_in(z0), a, .(b, z0))
U1(f39_out1(z0), a, .(b, z1)) → f2_out1(z0)
f39_in([]) → f39_out1(a)
f39_in(.(b, z0)) → U2(f39_in(z0), .(b, z0))
U2(f39_out1(z0), .(b, z1)) → f39_out1(z0)

Tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F39_IN(.(b, z0)) → c4(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

S tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F39_IN(.(b, z0)) → c4(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f39_in, U2

Defined Pair Symbols:

F2_IN, F39_IN

Compound Symbols:

c, c4, c

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, z0)) → U1(f39_in(z0), a, .(b, z0))
U1(f39_out1(z0), a, .(b, z1)) → f2_out1(z0)
f39_in([]) → f39_out1(a)
f39_in(.(b, z0)) → U2(f39_in(z0), .(b, z0))
U2(f39_out1(z0), .(b, z1)) → f39_out1(z0)

Tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F39_IN(.(b, z0)) → c4(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

S tuples:

F39_IN(.(b, z0)) → c4(F39_IN(z0))

K tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

Defined Rule Symbols:

f2_in, U1, f39_in, U2

Defined Pair Symbols:

F2_IN, F39_IN

Compound Symbols:

c, c4, c

(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.

F39_IN(.(b, z0)) → c4(F39_IN(z0))

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

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F39_IN(.(b, z0)) → c4(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F2_IN(x₁, x₂)) = x₁ + [2]x₂   
POL(F39_IN(x₁)) = [3] + x₁   
POL(a) = [2]   
POL(b) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c4(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1(z0)
f2_in(a, .(b, z0)) → U1(f39_in(z0), a, .(b, z0))
U1(f39_out1(z0), a, .(b, z1)) → f2_out1(z0)
f39_in([]) → f39_out1(a)
f39_in(.(b, z0)) → U2(f39_in(z0), .(b, z0))
U2(f39_out1(z0), .(b, z1)) → f39_out1(z0)

Tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F39_IN(.(b, z0)) → c4(F39_IN(z0))
F2_IN(a, .(b, z0)) → c

S tuples:none
K tuples:

F2_IN(a, .(b, z0)) → c(F39_IN(z0))
F2_IN(a, .(b, z0)) → c
F39_IN(.(b, z0)) → c4(F39_IN(z0))

Defined Rule Symbols:

f2_in, U1, f39_in, U2

Defined Pair Symbols:

F2_IN, F39_IN

Compound Symbols:

c, c4, c

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))

(13) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, eq(T6, T8)) | fold(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX6 is free\nX7 is free\nfold(T1, T2, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(T6, T8)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(T6, T8), SCOPE: 2, CLAUSE: 7\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18))))\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 3 | ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18)))\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(myop(T15, T31, X19), fold(X19, T32, T18))\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(myop(T15, T31, X19), fold(X19, T32, T18)), SCOPE: 5, CLAUSE: 2\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: fold(a, T32, T18)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: fold(a, T32, T18), SCOPE: 6, CLAUSE: 0 | fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(!_6, eq(a, T38)) | fold(a, [], T18), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground\nX49 is free\nX50 is free\nfold(a, T32, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: eq(a, T38)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: eq(a, T38), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47))))\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 3 | ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47)))\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(myop(a, T61, X71), fold(X71, T62, T47))\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(myop(a, T61, X71), fold(X71, T62, T47)), SCOPE: 10, CLAUSE: 2\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: fold(a, T62, T47)\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 76 [arrowhead = none ];
76 -> 3;

77 [label="EVAL with clause\nfold(X6, [], X7) :- ','(!_1, eq(X6, X7)).\nand substitutionT1 -> T6,\nX6 -> T6,\nT2 -> [],\nT3 -> T8,\nX7 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 79 [arrowhead = none ];
79 -> 10;

80 [label="ONLY EVAL with clause\nfold(X14, X15, X16) :- ','(head(X15, X17), ','(tail(X15, X18), ','(myop(X14, X17, X19), fold(X19, X18, X16)))).\nand substitutionT1 -> T15,\nX14 -> T15,\nT2 -> T16,\nX15 -> T16,\nT3 -> T18,\nX16 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 80 [arrowhead = none ];
80 -> 19;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 81 [arrowhead = none ];
81 -> 11;

82 [label="EVAL with clause\neq(X10, X10).\nand substitutionT6 -> T11,\nX10 -> T11,\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 82 [arrowhead = none ];
82 -> 13;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 83 [arrowhead = none ];
83 -> 15;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 85 [arrowhead = none ];
85 -> 20;

86 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T15, T16, T3), fold(X6, [], X7))", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="EVAL with clause\nhead(.(X26, X27), X26).\nand substitutionX26 -> T23,\nX27 -> T24,\nT16 -> .(T23, T24),\nX17 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 89 [arrowhead = none ];
89 -> 27;

90 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 90 [arrowhead = none ];
90 -> 28;

91 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitutionT23 -> T31,\nX38 -> T31,\nT24 -> T32,\nX39 -> T32,\nX18 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 91 [arrowhead = none ];
91 -> 33;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 92 [arrowhead = none ];
92 -> 34;

93 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT15 -> a,\nT31 -> b,\nX19 -> a", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 93 [arrowhead = none ];
93 -> 35;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 94 [arrowhead = none ];
94 -> 36;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 95 [arrowhead = none ];
95 -> 41;

96 [label="EVAL with clause\nfold(X49, [], X50) :- ','(!_6, eq(X49, X50)).\nand substitutionX49 -> a,\nT32 -> [],\nT18 -> T38,\nX50 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 96 [arrowhead = none ];
96 -> 43;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 97 [arrowhead = none ];
97 -> 45;

98 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 98 [arrowhead = none ];
98 -> 46;

99 [label="ONLY EVAL with clause\nfold(X66, X67, X68) :- ','(head(X67, X69), ','(tail(X67, X70), ','(myop(X66, X69, X71), fold(X71, X70, X68)))).\nand substitutionX66 -> a,\nT32 -> T45,\nX67 -> T45,\nT18 -> T47,\nX68 -> T47,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 99 [arrowhead = none ];
99 -> 57;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 100 [arrowhead = none ];
100 -> 49;

101 [label="EVAL with clause\neq(X53, X53).\nand substitutionX53 -> a,\nT38 -> a", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 101 [arrowhead = none ];
101 -> 51;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 102 [arrowhead = none ];
102 -> 52;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 103 [arrowhead = none ];
103 -> 54;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 104 [arrowhead = none ];
104 -> 59;

105 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(a, T45, T18), fold(X49, [], X50))", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 105 [arrowhead = none ];
105 -> 61;

106 [label="EVAL with clause\nhead(.(X81, X82), X81).\nand substitutionX81 -> T53,\nX82 -> T54,\nT45 -> .(T53, T54),\nX69 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 106 [arrowhead = none ];
106 -> 63;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 107 [arrowhead = none ];
107 -> 64;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 108 [arrowhead = none ];
108 -> 67;

109 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 109 [arrowhead = none ];
109 -> 68;

110 [label="ONLY EVAL with clause\ntail(.(X93, X94), X94).\nand substitutionT53 -> T61,\nX93 -> T61,\nT54 -> T62,\nX94 -> T62,\nX70 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 110 [arrowhead = none ];
110 -> 71;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 111 [arrowhead = none ];
111 -> 72;

112 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT61 -> b,\nX71 -> a", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 112 [arrowhead = none ];
112 -> 74;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 113 [arrowhead = none ];
113 -> 75;

114 [label="INSTANCE with matching:\nT32 -> T62\nT18 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 114 [arrowhead = none , style=dashed];
114 -> 35[style=dashed];

}

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, z0)) → U1(f35_in(z0), a, .(b, z0))
U1(f35_out1(z0), a, .(b, z1)) → f1_out1(z0)
f35_in([]) → f35_out1(a)
f35_in(.(b, z0)) → U2(f35_in(z0), .(b, z0))
U2(f35_out1(z0), .(b, z1)) → f35_out1(z0)

Tuples:

F1_IN(a, .(b, z0)) → c1(U1'(f35_in(z0), a, .(b, z0)), F35_IN(z0))
F35_IN(.(b, z0)) → c4(U2'(f35_in(z0), .(b, z0)), F35_IN(z0))

S tuples:

F1_IN(a, .(b, z0)) → c1(U1'(f35_in(z0), a, .(b, z0)), F35_IN(z0))
F35_IN(.(b, z0)) → c4(U2'(f35_in(z0), .(b, z0)), F35_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f35_in, U2

Defined Pair Symbols:

F1_IN, F35_IN

Compound Symbols:

c1, c4

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, z0)) → U1(f35_in(z0), a, .(b, z0))
U1(f35_out1(z0), a, .(b, z1)) → f1_out1(z0)
f35_in([]) → f35_out1(a)
f35_in(.(b, z0)) → U2(f35_in(z0), .(b, z0))
U2(f35_out1(z0), .(b, z1)) → f35_out1(z0)

Tuples:

F35_IN(.(b, z0)) → c4(U2'(f35_in(z0), .(b, z0)), F35_IN(z0))
F1_IN(a, .(b, z0)) → c(U1'(f35_in(z0), a, .(b, z0)))
F1_IN(a, .(b, z0)) → c(F35_IN(z0))

S tuples:

F35_IN(.(b, z0)) → c4(U2'(f35_in(z0), .(b, z0)), F35_IN(z0))
F1_IN(a, .(b, z0)) → c(U1'(f35_in(z0), a, .(b, z0)))
F1_IN(a, .(b, z0)) → c(F35_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f35_in, U2

Defined Pair Symbols:

F35_IN, F1_IN

Compound Symbols:

c4, c

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

Removed 2 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, z0)) → U1(f35_in(z0), a, .(b, z0))
U1(f35_out1(z0), a, .(b, z1)) → f1_out1(z0)
f35_in([]) → f35_out1(a)
f35_in(.(b, z0)) → U2(f35_in(z0), .(b, z0))
U2(f35_out1(z0), .(b, z1)) → f35_out1(z0)

Tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F35_IN(.(b, z0)) → c4(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

S tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F35_IN(.(b, z0)) → c4(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f35_in, U2

Defined Pair Symbols:

F1_IN, F35_IN

Compound Symbols:

c, c4, c

(19) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, z0)) → U1(f35_in(z0), a, .(b, z0))
U1(f35_out1(z0), a, .(b, z1)) → f1_out1(z0)
f35_in([]) → f35_out1(a)
f35_in(.(b, z0)) → U2(f35_in(z0), .(b, z0))
U2(f35_out1(z0), .(b, z1)) → f35_out1(z0)

Tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F35_IN(.(b, z0)) → c4(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

S tuples:

F35_IN(.(b, z0)) → c4(F35_IN(z0))

K tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

Defined Rule Symbols:

f1_in, U1, f35_in, U2

Defined Pair Symbols:

F1_IN, F35_IN

Compound Symbols:

c, c4, c

(21) 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.

F35_IN(.(b, z0)) → c4(F35_IN(z0))

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

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F35_IN(.(b, z0)) → c4(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F1_IN(x₁, x₂)) = x₁ + [2]x₂   
POL(F35_IN(x₁)) = [3] + x₁   
POL(a) = [2]   
POL(b) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c4(x₁)) = x₁   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1(z0)
f1_in(a, .(b, z0)) → U1(f35_in(z0), a, .(b, z0))
U1(f35_out1(z0), a, .(b, z1)) → f1_out1(z0)
f35_in([]) → f35_out1(a)
f35_in(.(b, z0)) → U2(f35_in(z0), .(b, z0))
U2(f35_out1(z0), .(b, z1)) → f35_out1(z0)

Tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F35_IN(.(b, z0)) → c4(F35_IN(z0))
F1_IN(a, .(b, z0)) → c

S tuples:none
K tuples:

F1_IN(a, .(b, z0)) → c(F35_IN(z0))
F1_IN(a, .(b, z0)) → c
F35_IN(.(b, z0)) → c4(F35_IN(z0))

Defined Rule Symbols:

f1_in, U1, f35_in, U2

Defined Pair Symbols:

F1_IN, F35_IN

Compound Symbols:

c, c4, c