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

(0) Obligation:

Clauses:

max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Query: max(a,g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

less(0, s(X1)).
max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(s(X), s(Y)) :- less(X, Y).

Query: max(a,g,a)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: max(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T13, T14)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T13, T14), SCOPE: 2, CLAUSE: 0 | less(T13, T14), SCOPE: 2, CLAUSE: 3\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | less(0, T14), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: less(T13, T14), SCOPE: 2, CLAUSE: 3\n\nT13 is ground\nX16 is free\nless(T13, T14) !=? less(0, s(X16))", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: less(0, T14), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T24, T26)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: less(T35, s(T34))\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: less(T35, s(T34)), SCOPE: 3, CLAUSE: 0 | less(T35, s(T34)), SCOPE: 3, CLAUSE: 3\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true | less(T35, s(T40)), SCOPE: 3, CLAUSE: 3\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T35, s(T34)), SCOPE: 3, CLAUSE: 3\n\nT34 is ground\nX37 is free\nless(T35, s(T34)) !=? less(0, s(X37))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T35, s(T40)), SCOPE: 3, CLAUSE: 3\n\nT40 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: less(T47, T46)\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(T47, T46), SCOPE: 4, CLAUSE: 0 | less(T47, T46), SCOPE: 4, CLAUSE: 3\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | less(T47, s(T52)), SCOPE: 4, CLAUSE: 3\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(T47, T46), SCOPE: 4, CLAUSE: 3\n\nT46 is ground\nX48 is free\nless(T47, T46) !=? less(0, s(X48))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T47, s(T52)), SCOPE: 4, CLAUSE: 3\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T59, T58)\n\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: less(T68, T67)\n\nT67 is ground\nX48 is free\nless(T47, s(T67)) !=? less(0, s(X48))", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(T77, T76)\n\nT76 is ground\nX37 is free\nless(T35, s(T76)) !=? less(0, s(X37))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 67 [arrowhead = none ];
67 -> 3;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
3 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
3 -> 69 [arrowhead = none ];
69 -> 6;

70 [label="EVAL with clause\nmax(X10, X11, X10) :- less(X11, X10).\nand substitutionT1 -> T14,\nX10 -> T14,\nT2 -> T13,\nX11 -> T13,\nT3 -> T14,\nT12 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 70 [arrowhead = none ];
70 -> 11;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 71 [arrowhead = none ];
71 -> 12;

72 [label="EVAL with clause\nmax(X31, X32, X32) :- less(X31, s(X32)).\nand substitutionT1 -> T35,\nX31 -> T35,\nT2 -> T34,\nX32 -> T34,\nT3 -> T34,\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 72 [arrowhead = none ];
72 -> 37;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 73 [arrowhead = none ];
73 -> 38;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 74 [arrowhead = none ];
74 -> 17;

75 [label="EVAL with clause\nless(0, s(X16)).\nand substitutionT13 -> 0,\nX16 -> T19,\nT14 -> s(T19)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 75 [arrowhead = none ];
75 -> 19;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 76 [arrowhead = none ];
76 -> 21;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 77 [arrowhead = none ];
77 -> 22;

78 [label="EVAL with clause\nless(s(X23), s(X24)) :- less(X23, X24).\nand substitutionX23 -> T24,\nT13 -> s(T24),\nX24 -> T26,\nT14 -> s(T26),\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 78 [arrowhead = none ];
78 -> 28;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 79 [arrowhead = none ];
79 -> 29;

80 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 80 [arrowhead = none ];
80 -> 23;

81 [label="INSTANCE with matching:\nT13 -> T24\nT14 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 81 [arrowhead = none , style=dashed];
81 -> 11[style=dashed];

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 82 [arrowhead = none ];
82 -> 39;

83 [label="EVAL with clause\nless(0, s(X37)).\nand substitutionT35 -> 0,\nT34 -> T40,\nX37 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 83 [arrowhead = none ];
83 -> 42;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 84 [arrowhead = none ];
84 -> 43;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 85 [arrowhead = none ];
85 -> 44;

86 [label="EVAL with clause\nless(s(X69), s(X70)) :- less(X69, X70).\nand substitutionX69 -> T77,\nT35 -> s(T77),\nT34 -> T76,\nX70 -> T76,\nT75 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 86 [arrowhead = none ];
86 -> 65;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 87 [arrowhead = none ];
87 -> 66;

88 [label="EVAL with clause\nless(s(X42), s(X43)) :- less(X42, X43).\nand substitutionX42 -> T47,\nT35 -> s(T47),\nT40 -> T46,\nX43 -> T46,\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 88 [arrowhead = none ];
88 -> 49;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 89 [arrowhead = none ];
89 -> 51;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 90 [arrowhead = none ];
90 -> 54;

91 [label="EVAL with clause\nless(0, s(X48)).\nand substitutionT47 -> 0,\nX48 -> T52,\nT46 -> s(T52)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 91 [arrowhead = none ];
91 -> 55;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 92 [arrowhead = none ];
92 -> 57;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 93 [arrowhead = none ];
93 -> 59;

94 [label="EVAL with clause\nless(s(X61), s(X62)) :- less(X61, X62).\nand substitutionX61 -> T68,\nT47 -> s(T68),\nX62 -> T67,\nT46 -> s(T67),\nT66 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 94 [arrowhead = none ];
94 -> 62;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 95 [arrowhead = none ];
95 -> 63;

96 [label="EVAL with clause\nless(s(X53), s(X54)) :- less(X53, X54).\nand substitutionX53 -> T59,\nT47 -> s(T59),\nT52 -> T58,\nX54 -> T58,\nT57 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 96 [arrowhead = none ];
96 -> 60;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 97 [arrowhead = none ];
97 -> 61;

98 [label="INSTANCE with matching:\nT47 -> T59\nT46 -> T58", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 98 [arrowhead = none , style=dashed];
98 -> 49[style=dashed];

99 [label="INSTANCE with matching:\nT47 -> T68\nT46 -> T67", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 99 [arrowhead = none , style=dashed];
99 -> 49[style=dashed];

100 [label="INSTANCE with matching:\nT47 -> T77\nT46 -> T76", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 100 [arrowhead = none , style=dashed];
100 -> 49[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F2_IN(z0) → c(U1'(f3_in(z0), z0), F3_IN(z0))
F11_IN(s(z0)) → c4(U2'(f11_in(z0), s(z0)), F11_IN(z0))
F49_IN(s(z0)) → c7(U3'(f49_in(z0), s(z0)), F49_IN(z0))
F49_IN(s(z0)) → c8(U4'(f49_in(z0), s(z0)), F49_IN(z0))
F5_IN(z0) → c11(U5'(f11_in(z0), z0), F11_IN(z0))
F6_IN(z0) → c14(U6'(f49_in(z0), z0), F49_IN(z0))
F6_IN(z0) → c15(U7'(f49_in(z0), z0), F49_IN(z0))
F3_IN(z0) → c18(U8'(f5_in(z0), f6_in(z0), z0), F5_IN(z0), F6_IN(z0))

S tuples:

F2_IN(z0) → c(U1'(f3_in(z0), z0), F3_IN(z0))
F11_IN(s(z0)) → c4(U2'(f11_in(z0), s(z0)), F11_IN(z0))
F49_IN(s(z0)) → c7(U3'(f49_in(z0), s(z0)), F49_IN(z0))
F49_IN(s(z0)) → c8(U4'(f49_in(z0), s(z0)), F49_IN(z0))
F5_IN(z0) → c11(U5'(f11_in(z0), z0), F11_IN(z0))
F6_IN(z0) → c14(U6'(f49_in(z0), z0), F49_IN(z0))
F6_IN(z0) → c15(U7'(f49_in(z0), z0), F49_IN(z0))
F3_IN(z0) → c18(U8'(f5_in(z0), f6_in(z0), z0), F5_IN(z0), F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F2_IN, F11_IN, F49_IN, F5_IN, F6_IN, F3_IN

Compound Symbols:

c, c4, c7, c8, c11, c14, c15, c18

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F11_IN(s(z0)) → c4(U2'(f11_in(z0), s(z0)), F11_IN(z0))
F49_IN(s(z0)) → c7(U3'(f49_in(z0), s(z0)), F49_IN(z0))
F49_IN(s(z0)) → c8(U4'(f49_in(z0), s(z0)), F49_IN(z0))
F2_IN(z0) → c1(U1'(f3_in(z0), z0))
F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(U5'(f11_in(z0), z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(U6'(f49_in(z0), z0))
F6_IN(z0) → c1(F49_IN(z0))
F6_IN(z0) → c1(U7'(f49_in(z0), z0))
F3_IN(z0) → c1(U8'(f5_in(z0), f6_in(z0), z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))

S tuples:

F11_IN(s(z0)) → c4(U2'(f11_in(z0), s(z0)), F11_IN(z0))
F49_IN(s(z0)) → c7(U3'(f49_in(z0), s(z0)), F49_IN(z0))
F49_IN(s(z0)) → c8(U4'(f49_in(z0), s(z0)), F49_IN(z0))
F2_IN(z0) → c1(U1'(f3_in(z0), z0))
F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(U5'(f11_in(z0), z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(U6'(f49_in(z0), z0))
F6_IN(z0) → c1(F49_IN(z0))
F6_IN(z0) → c1(U7'(f49_in(z0), z0))
F3_IN(z0) → c1(U8'(f5_in(z0), f6_in(z0), z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F11_IN, F49_IN, F2_IN, F5_IN, F6_IN, F3_IN

Compound Symbols:

c4, c7, c8, c1

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

Removed 8 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

S tuples:

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F3_IN, F11_IN, F49_IN

Compound Symbols:

c1, c4, c7, c8, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F3_IN(z0) → c1
F5_IN(z0) → c1(F11_IN(z0))
F5_IN(z0) → c1
F6_IN(z0) → c1(F49_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F6_IN(z0) → c1
F6_IN(z0) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

S tuples:

F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))

K tuples:

F2_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))

Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F3_IN, F11_IN, F49_IN

Compound Symbols:

c1, c4, c7, c8, c1

(11) 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.

F11_IN(s(z0)) → c4(F11_IN(z0))

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

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

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

POL(F11_IN(x₁)) = [1] + x₁   
POL(F2_IN(x₁)) = [3] + x₁   
POL(F3_IN(x₁)) = [2] + x₁   
POL(F49_IN(x₁)) = 0   
POL(F5_IN(x₁)) = [2] + x₁   
POL(F6_IN(x₁)) = [1]   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

S tuples:

F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))

K tuples:

F2_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))

Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F3_IN, F11_IN, F49_IN

Compound Symbols:

c1, c4, c7, c8, c1

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

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

F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))

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

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

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

POL(F11_IN(x₁)) = [1]   
POL(F2_IN(x₁)) = [3] + [2]x₁   
POL(F3_IN(x₁)) = [2] + [2]x₁   
POL(F49_IN(x₁)) = x₁   
POL(F5_IN(x₁)) = [2] + x₁   
POL(F6_IN(x₁)) = [1] + x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f2_out1
U1(f3_out2, z0) → f2_out1
f11_in(0) → f11_out1
f11_in(s(z0)) → U2(f11_in(z0), s(z0))
U2(f11_out1, s(z0)) → f11_out1
f49_in(s(z0)) → f49_out1(0)
f49_in(s(z0)) → U3(f49_in(z0), s(z0))
f49_in(s(z0)) → U4(f49_in(z0), s(z0))
U3(f49_out1(z0), s(z1)) → f49_out1(s(z0))
U4(f49_out1(z0), s(z1)) → f49_out1(s(z0))
f5_in(z0) → U5(f11_in(z0), z0)
U5(f11_out1, z0) → f5_out1
f6_in(z0) → f6_out1
f6_in(z0) → U6(f49_in(z0), z0)
f6_in(z0) → U7(f49_in(z0), z0)
U6(f49_out1(z0), z1) → f6_out1
U7(f49_out1(z0), z1) → f6_out1
f3_in(z0) → U8(f5_in(z0), f6_in(z0), z0)
U8(f5_out1, z0, z1) → f3_out1
U8(z0, f6_out1, z1) → f3_out2

Tuples:

F2_IN(z0) → c1(F3_IN(z0))
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1

S tuples:none
K tuples:

F2_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F5_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F3_IN(z0) → c1
F5_IN(z0) → c1(F11_IN(z0))
F6_IN(z0) → c1(F49_IN(z0))
F11_IN(s(z0)) → c4(F11_IN(z0))
F49_IN(s(z0)) → c7(F49_IN(z0))
F49_IN(s(z0)) → c8(F49_IN(z0))

Defined Rule Symbols:

f2_in, U1, f11_in, U2, f49_in, U3, U4, f5_in, U5, f6_in, U6, U7, f3_in, U8

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F3_IN, F11_IN, F49_IN

Compound Symbols:

c1, c4, c7, c8, 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];
1 [label="1: max(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: less(T7, T8) | max(T1, T7, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT2 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: less(T7, T8), SCOPE: 2, CLAUSE: 0 | less(T7, T8), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T7, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | less(0, T8), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, 0, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T7, T8), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T7, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nX8 is free\nless(T7, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(0, T8), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, 0, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ?_2 | max(T1, 0, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: max(T1, 0, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T15, s(0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T15, s(0)), SCOPE: 3, CLAUSE: 0 | less(T15, s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | less(T15, s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T15, s(0)), SCOPE: 3, CLAUSE: 3\n\nX21 is free\nless(T15, s(0)) !=? less(0, s(X21))", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T15, s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T19, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(T19, 0), SCOPE: 4, CLAUSE: 0 | less(T19, 0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: less(T19, 0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T24, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T7, T8), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX8 is free\nless(T7, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_2 | max(T1, T7, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nX8 is free\nless(T7, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T33, T35)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T33, T35), SCOPE: 5, CLAUSE: 0 | less(T33, T35), SCOPE: 5, CLAUSE: 3\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true | less(0, T35), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T33, T35), SCOPE: 5, CLAUSE: 3\n\nT33 is ground\nX55 is free\nless(T33, T35) !=? less(0, s(X55))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(0, T35), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T45, T47)\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: max(T1, T7, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nX8 is free\nless(T7, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(T56, s(T55))\n\nT55 is ground\nX8 is free\nless(T55, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: less(T56, s(T55)), SCOPE: 6, CLAUSE: 0 | less(T56, s(T55)), SCOPE: 6, CLAUSE: 3\n\nT55 is ground\nX8 is free\nless(T55, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: true | less(T56, s(T61)), SCOPE: 6, CLAUSE: 3\n\nT61 is ground\nX8 is free\nless(T61, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T56, s(T55)), SCOPE: 6, CLAUSE: 3\n\nT55 is ground\nX8 is free\nX76 is free\nless(T55, T8) !=? less(0, s(X8))\nless(T56, s(T55)) !=? less(0, s(X76))", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: less(T56, s(T61)), SCOPE: 6, CLAUSE: 3\n\nT61 is ground\nX8 is free\nless(T61, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T68, T67)\n\nT67 is ground\nX8 is free\nless(T67, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T68, T67), SCOPE: 7, CLAUSE: 0 | less(T68, T67), SCOPE: 7, CLAUSE: 3\n\nT67 is ground\nX8 is free\nless(T67, T8) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true | less(T68, s(T73)), SCOPE: 7, CLAUSE: 3\n\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(T68, T67), SCOPE: 7, CLAUSE: 3\n\nT67 is ground\nX8 is free\nX87 is free\nless(T67, T8) !=? less(0, s(X8))\nless(T68, T67) !=? less(0, s(X87))", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: less(T68, s(T73)), SCOPE: 7, CLAUSE: 3\n\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: less(T80, T79)\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: less(T80, T79), SCOPE: 8, CLAUSE: 0 | less(T80, T79), SCOPE: 8, CLAUSE: 3\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true | less(T80, s(T85)), SCOPE: 8, CLAUSE: 3\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: less(T80, T79), SCOPE: 8, CLAUSE: 3\n\nT79 is ground\nX98 is free\nless(T80, T79) !=? less(0, s(X98))", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: less(T80, s(T85)), SCOPE: 8, CLAUSE: 3\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: less(T92, T91)\n\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: less(T101, T100)\n\nT100 is ground\nX98 is free\nless(T80, s(T100)) !=? less(0, s(X98))", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: less(T110, T109)\n\nT109 is ground\nX87 is free\nless(T68, s(T109)) !=? less(0, s(X87))", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: less(T119, T118)\n\nT118 is ground\nX8 is free\nX76 is free\nless(T118, T8) !=? less(0, s(X8))\nless(T56, s(T118)) !=? less(0, s(X76))", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: less(T126, s(T125))\n\nT125 is ground\nX4 is free\nX5 is free\nmax(T1, T125, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: less(T126, s(T125)), SCOPE: 9, CLAUSE: 0 | less(T126, s(T125)), SCOPE: 9, CLAUSE: 3\n\nT125 is ground\nX4 is free\nX5 is free\nmax(T1, T125, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: true | less(T126, s(T129)), SCOPE: 9, CLAUSE: 3\n\nT129 is ground\nX4 is free\nX5 is free\nmax(T1, T129, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: less(T126, s(T125)), SCOPE: 9, CLAUSE: 3\n\nT125 is ground\nX4 is free\nX5 is free\nX137 is free\nmax(T1, T125, T3) !=? max(X4, X5, X4)\nless(T126, s(T125)) !=? less(0, s(X137))", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: less(T126, s(T129)), SCOPE: 9, CLAUSE: 3\n\nT129 is ground\nX4 is free\nX5 is free\nmax(T1, T129, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: less(T136, T135)\n\nT135 is ground\nX4 is free\nX5 is free\nmax(T1, T135, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: less(T145, T144)\n\nT144 is ground\nX4 is free\nX5 is free\nX137 is free\nmax(T1, T144, T3) !=? max(X4, X5, X4)\nless(T126, s(T144)) !=? less(0, s(X137))", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 111 [arrowhead = none ];
111 -> 4;

112 [label="EVAL with clause\nmax(X4, X5, X4) :- less(X5, X4).\nand substitutionT1 -> T8,\nX4 -> T8,\nT2 -> T7,\nX5 -> T7,\nT3 -> T8,\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 112 [arrowhead = none ];
112 -> 7;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 113 [arrowhead = none ];
113 -> 8;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 114 [arrowhead = none ];
114 -> 9;

115 [label="EVAL with clause\nmax(X133, X134, X134) :- less(X133, s(X134)).\nand substitutionT1 -> T126,\nX133 -> T126,\nT2 -> T125,\nX134 -> T125,\nT3 -> T125,\nT124 -> T126", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 115 [arrowhead = none ];
115 -> 101;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 116 [arrowhead = none ];
116 -> 102;

117 [label="EVAL with clause\nless(0, s(X8)).\nand substitutionT7 -> 0,\nX8 -> T11,\nT8 -> s(T11)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 117 [arrowhead = none ];
117 -> 10;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 118 [arrowhead = none ];
118 -> 13;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 119 [arrowhead = none ];
119 -> 14;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 120 [arrowhead = none ];
120 -> 40;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 122 [arrowhead = none ];
122 -> 15;

123 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 123 [arrowhead = none ];
123 -> 16;

124 [label="EVAL with clause\nmax(X15, X16, X16) :- less(X15, s(X16)).\nand substitutionT1 -> T15,\nX15 -> T15,\nX16 -> 0,\nT3 -> 0,\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 124 [arrowhead = none ];
124 -> 18;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 125 [arrowhead = none ];
125 -> 20;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 126 [arrowhead = none ];
126 -> 24;

127 [label="EVAL with clause\nless(0, s(X21)).\nand substitutionT15 -> 0,\nX21 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 127 [arrowhead = none ];
127 -> 25;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 128 [arrowhead = none ];
128 -> 26;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 129 [arrowhead = none ];
129 -> 27;

130 [label="EVAL with clause\nless(s(X37), s(X38)) :- less(X37, X38).\nand substitutionX37 -> T24,\nT15 -> s(T24),\nX38 -> 0,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 130 [arrowhead = none ];
130 -> 35;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 131 [arrowhead = none ];
131 -> 36;

132 [label="EVAL with clause\nless(s(X26), s(X27)) :- less(X26, X27).\nand substitutionX26 -> T19,\nT15 -> s(T19),\nX27 -> 0,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 132 [arrowhead = none ];
132 -> 30;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 133 [arrowhead = none ];
133 -> 31;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 134 [arrowhead = none ];
134 -> 32;

135 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 135 [arrowhead = none ];
135 -> 33;

136 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 136 [arrowhead = none ];
136 -> 34;

137 [label="INSTANCE with matching:\nT19 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 137 [arrowhead = none , style=dashed];
137 -> 30[style=dashed];

138 [label="EVAL with clause\nless(s(X49), s(X50)) :- less(X49, X50).\nand substitutionX49 -> T33,\nT7 -> s(T33),\nX50 -> T35,\nT8 -> s(T35),\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 138 [arrowhead = none ];
138 -> 45;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 139 [arrowhead = none ];
139 -> 46;

140 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 140 [arrowhead = none ];
140 -> 64;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 141 [arrowhead = none ];
141 -> 47;

142 [label="EVAL with clause\nless(0, s(X55)).\nand substitutionT33 -> 0,\nX55 -> T40,\nT35 -> s(T40)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 142 [arrowhead = none ];
142 -> 48;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 143 [arrowhead = none ];
143 -> 50;

144 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 144 [arrowhead = none ];
144 -> 52;

145 [label="EVAL with clause\nless(s(X62), s(X63)) :- less(X62, X63).\nand substitutionX62 -> T45,\nT33 -> s(T45),\nX63 -> T47,\nT35 -> s(T47),\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 145 [arrowhead = none ];
145 -> 56;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 146 [arrowhead = none ];
146 -> 58;

147 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 147 [arrowhead = none ];
147 -> 53;

148 [label="INSTANCE with matching:\nT33 -> T45\nT35 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 148 [arrowhead = none , style=dashed];
148 -> 45[style=dashed];

149 [label="EVAL with clause\nmax(X70, X71, X71) :- less(X70, s(X71)).\nand substitutionT1 -> T56,\nX70 -> T56,\nT7 -> T55,\nX71 -> T55,\nT3 -> T55,\nT54 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 149 [arrowhead = none ];
149 -> 67;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 150 [arrowhead = none ];
150 -> 68;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="EVAL with clause\nless(0, s(X76)).\nand substitutionT56 -> 0,\nT55 -> T61,\nX76 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 152 [arrowhead = none ];
152 -> 70;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 153 [arrowhead = none ];
153 -> 71;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 154 [arrowhead = none ];
154 -> 72;

155 [label="EVAL with clause\nless(s(X127), s(X128)) :- less(X127, X128).\nand substitutionX127 -> T119,\nT56 -> s(T119),\nT55 -> T118,\nX128 -> T118,\nT117 -> T119", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 155 [arrowhead = none ];
155 -> 99;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 156 [arrowhead = none ];
156 -> 100;

157 [label="EVAL with clause\nless(s(X81), s(X82)) :- less(X81, X82).\nand substitutionX81 -> T68,\nT56 -> s(T68),\nT61 -> T67,\nX82 -> T67,\nT66 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 157 [arrowhead = none ];
157 -> 73;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 158 [arrowhead = none ];
158 -> 74;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 159 [arrowhead = none ];
159 -> 75;

160 [label="EVAL with clause\nless(0, s(X87)).\nand substitutionT68 -> 0,\nX87 -> T73,\nT67 -> s(T73)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 160 [arrowhead = none ];
160 -> 76;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 161 [arrowhead = none ];
161 -> 77;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 162 [arrowhead = none ];
162 -> 78;

163 [label="EVAL with clause\nless(s(X119), s(X120)) :- less(X119, X120).\nand substitutionX119 -> T110,\nT68 -> s(T110),\nX120 -> T109,\nT67 -> s(T109),\nT108 -> T110", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 163 [arrowhead = none ];
163 -> 89;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 164 [arrowhead = none ];
164 -> 90;

165 [label="EVAL with clause\nless(s(X92), s(X93)) :- less(X92, X93).\nand substitutionX92 -> T80,\nT68 -> s(T80),\nT73 -> T79,\nX93 -> T79,\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 165 [arrowhead = none ];
165 -> 79;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 166 [arrowhead = none ];
166 -> 80;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 167 [arrowhead = none ];
167 -> 81;

168 [label="EVAL with clause\nless(0, s(X98)).\nand substitutionT80 -> 0,\nX98 -> T85,\nT79 -> s(T85)", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 168 [arrowhead = none ];
168 -> 82;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 169 [arrowhead = none ];
169 -> 83;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 170 [arrowhead = none ];
170 -> 84;

171 [label="EVAL with clause\nless(s(X111), s(X112)) :- less(X111, X112).\nand substitutionX111 -> T101,\nT80 -> s(T101),\nX112 -> T100,\nT79 -> s(T100),\nT99 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 171 [arrowhead = none ];
171 -> 87;

172 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 172 [arrowhead = none ];
172 -> 88;

173 [label="EVAL with clause\nless(s(X103), s(X104)) :- less(X103, X104).\nand substitutionX103 -> T92,\nT80 -> s(T92),\nT85 -> T91,\nX104 -> T91,\nT90 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 173 [arrowhead = none ];
173 -> 85;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 174 [arrowhead = none ];
174 -> 86;

175 [label="INSTANCE with matching:\nT80 -> T92\nT79 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 175 [arrowhead = none , style=dashed];
175 -> 79[style=dashed];

176 [label="INSTANCE with matching:\nT80 -> T101\nT79 -> T100", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 176 [arrowhead = none , style=dashed];
176 -> 79[style=dashed];

177 [label="INSTANCE with matching:\nT80 -> T110\nT79 -> T109", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 177 [arrowhead = none , style=dashed];
177 -> 79[style=dashed];

178 [label="INSTANCE with matching:\nT68 -> T119\nT67 -> T118", fontsize=14, style = filled, fillcolor = lightgrey];
99 -> 178 [arrowhead = none , style=dashed];
178 -> 73[style=dashed];

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
101 -> 179 [arrowhead = none ];
179 -> 103;

180 [label="EVAL with clause\nless(0, s(X137)).\nand substitutionT126 -> 0,\nT125 -> T129,\nX137 -> T129", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 180 [arrowhead = none ];
180 -> 104;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 181 [arrowhead = none ];
181 -> 105;

182 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 182 [arrowhead = none ];
182 -> 106;

183 [label="EVAL with clause\nless(s(X150), s(X151)) :- less(X150, X151).\nand substitutionX150 -> T145,\nT126 -> s(T145),\nT125 -> T144,\nX151 -> T144,\nT143 -> T145", fontsize=14, style = filled, fillcolor = lightblue];
105 -> 183 [arrowhead = none ];
183 -> 109;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 184 [arrowhead = none ];
184 -> 110;

185 [label="EVAL with clause\nless(s(X142), s(X143)) :- less(X142, X143).\nand substitutionX142 -> T136,\nT126 -> s(T136),\nT129 -> T135,\nX143 -> T135,\nT134 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 185 [arrowhead = none ];
185 -> 107;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 186 [arrowhead = none ];
186 -> 108;

187 [label="INSTANCE with matching:\nT80 -> T136\nT79 -> T135", fontsize=14, style = filled, fillcolor = lightgrey];
107 -> 187 [arrowhead = none , style=dashed];
187 -> 79[style=dashed];

188 [label="INSTANCE with matching:\nT80 -> T145\nT79 -> T144", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 188 [arrowhead = none , style=dashed];
188 -> 79[style=dashed];

}

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F1_IN(0) → c1(U1'(f30_in, 0))
F1_IN(0) → c2(U2'(f30_in, 0))
F1_IN(z0) → c3(U3'(f13_in(z0), z0), F13_IN(z0))
F1_IN(z0) → c5(U4'(f79_in(z0), z0), F79_IN(z0))
F1_IN(z0) → c6(U5'(f79_in(z0), z0), F79_IN(z0))
F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F73_IN(s(z0)) → c22(U9'(f79_in(z0), s(z0)), F79_IN(z0))
F73_IN(s(z0)) → c23(U10'(f79_in(z0), s(z0)), F79_IN(z0))
F40_IN(s(z0)) → c26(U11'(f45_in(z0), s(z0)), F45_IN(z0))
F41_IN(z0) → c29(U12'(f73_in(z0), z0), F73_IN(z0))
F41_IN(z0) → c30(U13'(f73_in(z0), z0), F73_IN(z0))
F13_IN(z0) → c33(U14'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))

S tuples:

F1_IN(0) → c1(U1'(f30_in, 0))
F1_IN(0) → c2(U2'(f30_in, 0))
F1_IN(z0) → c3(U3'(f13_in(z0), z0), F13_IN(z0))
F1_IN(z0) → c5(U4'(f79_in(z0), z0), F79_IN(z0))
F1_IN(z0) → c6(U5'(f79_in(z0), z0), F79_IN(z0))
F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F73_IN(s(z0)) → c22(U9'(f79_in(z0), s(z0)), F79_IN(z0))
F73_IN(s(z0)) → c23(U10'(f79_in(z0), s(z0)), F79_IN(z0))
F40_IN(s(z0)) → c26(U11'(f45_in(z0), s(z0)), F45_IN(z0))
F41_IN(z0) → c29(U12'(f73_in(z0), z0), F73_IN(z0))
F41_IN(z0) → c30(U13'(f73_in(z0), z0), F73_IN(z0))
F13_IN(z0) → c33(U14'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F1_IN, F79_IN, F45_IN, F73_IN, F40_IN, F41_IN, F13_IN

Compound Symbols:

c1, c2, c3, c5, c6, c14, c15, c19, c22, c23, c26, c29, c30, c33

(19) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 14 dangling nodes:

F1_IN(0) → c1(U1'(f30_in, 0))
F1_IN(0) → c2(U2'(f30_in, 0))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F1_IN(z0) → c3(U3'(f13_in(z0), z0), F13_IN(z0))
F1_IN(z0) → c5(U4'(f79_in(z0), z0), F79_IN(z0))
F1_IN(z0) → c6(U5'(f79_in(z0), z0), F79_IN(z0))
F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F73_IN(s(z0)) → c22(U9'(f79_in(z0), s(z0)), F79_IN(z0))
F73_IN(s(z0)) → c23(U10'(f79_in(z0), s(z0)), F79_IN(z0))
F40_IN(s(z0)) → c26(U11'(f45_in(z0), s(z0)), F45_IN(z0))
F41_IN(z0) → c29(U12'(f73_in(z0), z0), F73_IN(z0))
F41_IN(z0) → c30(U13'(f73_in(z0), z0), F73_IN(z0))
F13_IN(z0) → c33(U14'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))

S tuples:

F1_IN(z0) → c3(U3'(f13_in(z0), z0), F13_IN(z0))
F1_IN(z0) → c5(U4'(f79_in(z0), z0), F79_IN(z0))
F1_IN(z0) → c6(U5'(f79_in(z0), z0), F79_IN(z0))
F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F73_IN(s(z0)) → c22(U9'(f79_in(z0), s(z0)), F79_IN(z0))
F73_IN(s(z0)) → c23(U10'(f79_in(z0), s(z0)), F79_IN(z0))
F40_IN(s(z0)) → c26(U11'(f45_in(z0), s(z0)), F45_IN(z0))
F41_IN(z0) → c29(U12'(f73_in(z0), z0), F73_IN(z0))
F41_IN(z0) → c30(U13'(f73_in(z0), z0), F73_IN(z0))
F13_IN(z0) → c33(U14'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F1_IN, F79_IN, F45_IN, F73_IN, F40_IN, F41_IN, F13_IN

Compound Symbols:

c3, c5, c6, c14, c15, c19, c22, c23, c26, c29, c30, c33

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

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F1_IN(z0) → c(U3'(f13_in(z0), z0))
F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(U4'(f79_in(z0), z0))
F1_IN(z0) → c(F79_IN(z0))
F1_IN(z0) → c(U5'(f79_in(z0), z0))
F73_IN(s(z0)) → c(U9'(f79_in(z0), s(z0)))
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c(U10'(f79_in(z0), s(z0)))
F40_IN(s(z0)) → c(U11'(f45_in(z0), s(z0)))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(U12'(f73_in(z0), z0))
F41_IN(z0) → c(F73_IN(z0))
F41_IN(z0) → c(U13'(f73_in(z0), z0))
F13_IN(z0) → c(U14'(f40_in(z0), f41_in(z0), z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))

S tuples:

F79_IN(s(z0)) → c14(U6'(f79_in(z0), s(z0)), F79_IN(z0))
F79_IN(s(z0)) → c15(U7'(f79_in(z0), s(z0)), F79_IN(z0))
F45_IN(s(z0)) → c19(U8'(f45_in(z0), s(z0)), F45_IN(z0))
F1_IN(z0) → c(U3'(f13_in(z0), z0))
F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(U4'(f79_in(z0), z0))
F1_IN(z0) → c(F79_IN(z0))
F1_IN(z0) → c(U5'(f79_in(z0), z0))
F73_IN(s(z0)) → c(U9'(f79_in(z0), s(z0)))
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c(U10'(f79_in(z0), s(z0)))
F40_IN(s(z0)) → c(U11'(f45_in(z0), s(z0)))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(U12'(f73_in(z0), z0))
F41_IN(z0) → c(F73_IN(z0))
F41_IN(z0) → c(U13'(f73_in(z0), z0))
F13_IN(z0) → c(U14'(f40_in(z0), f41_in(z0), z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F79_IN, F45_IN, F1_IN, F73_IN, F40_IN, F41_IN, F13_IN

Compound Symbols:

c14, c15, c19, c

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

Removed 12 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))
F1_IN(z0) → c
F73_IN(s(z0)) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c

S tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))
F1_IN(z0) → c
F73_IN(s(z0)) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F1_IN, F73_IN, F40_IN, F41_IN, F13_IN, F79_IN, F45_IN

Compound Symbols:

c, c14, c15, c19, c

(25) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F1_IN(z0) → c
F1_IN(z0) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F41_IN(z0) → c
F13_IN(z0) → c
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c
F40_IN(s(z0)) → c(F45_IN(z0))
F40_IN(s(z0)) → c
F41_IN(z0) → c(F73_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F41_IN(z0) → c
F41_IN(z0) → c
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c
F73_IN(s(z0)) → c

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))
F1_IN(z0) → c
F73_IN(s(z0)) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c

S tuples:

F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))

K tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c

Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F1_IN, F73_IN, F40_IN, F41_IN, F13_IN, F79_IN, F45_IN

Compound Symbols:

c, c14, c15, c19, c

(27) 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.

F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))

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

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))
F1_IN(z0) → c
F73_IN(s(z0)) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c

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

POL(F13_IN(x₁)) = [2] + x₁   
POL(F1_IN(x₁)) = [2] + x₁   
POL(F40_IN(x₁)) = [1]   
POL(F41_IN(x₁)) = [2] + x₁   
POL(F45_IN(x₁)) = 0   
POL(F73_IN(x₁)) = [1] + x₁   
POL(F79_IN(x₁)) = [2] + x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(0) → U1(f30_in, 0)
f1_in(0) → U2(f30_in, 0)
f1_in(z0) → U3(f13_in(z0), z0)
f1_in(z0) → f1_out1
f1_in(z0) → U4(f79_in(z0), z0)
f1_in(z0) → U5(f79_in(z0), z0)
U1(f30_out1(z0), 0) → f1_out1
U2(f30_out1(z0), 0) → f1_out1
U3(f13_out1, z0) → f1_out1
U3(f13_out3, z0) → f1_out1
U4(f79_out1(z0), z1) → f1_out1
U5(f79_out1(z0), z1) → f1_out1
f79_in(s(z0)) → f79_out1(0)
f79_in(s(z0)) → U6(f79_in(z0), s(z0))
f79_in(s(z0)) → U7(f79_in(z0), s(z0))
U6(f79_out1(z0), s(z1)) → f79_out1(s(z0))
U7(f79_out1(z0), s(z1)) → f79_out1(s(z0))
f45_in(0) → f45_out1
f45_in(s(z0)) → U8(f45_in(z0), s(z0))
U8(f45_out1, s(z0)) → f45_out1
f73_in(s(z0)) → f73_out1(0)
f73_in(s(z0)) → U9(f79_in(z0), s(z0))
f73_in(s(z0)) → U10(f79_in(z0), s(z0))
U9(f79_out1(z0), s(z1)) → f73_out1(s(z0))
U10(f79_out1(z0), s(z1)) → f73_out1(s(z0))
f40_in(s(z0)) → U11(f45_in(z0), s(z0))
U11(f45_out1, s(z0)) → f40_out1
f41_in(z0) → f41_out2
f41_in(z0) → U12(f73_in(z0), z0)
f41_in(z0) → U13(f73_in(z0), z0)
U12(f73_out1(z0), z1) → f41_out2
U13(f73_out1(z0), z1) → f41_out2
f13_in(z0) → U14(f40_in(z0), f41_in(z0), z0)
U14(f40_out1, z0, z1) → f13_out1
U14(z0, f41_out2, z1) → f13_out3

Tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))
F45_IN(s(z0)) → c19(F45_IN(z0))
F1_IN(z0) → c
F73_IN(s(z0)) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c

S tuples:

F45_IN(s(z0)) → c19(F45_IN(z0))

K tuples:

F1_IN(z0) → c(F13_IN(z0))
F1_IN(z0) → c(F79_IN(z0))
F13_IN(z0) → c(F40_IN(z0))
F13_IN(z0) → c(F41_IN(z0))
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F13_IN(z0) → c
F40_IN(s(z0)) → c(F45_IN(z0))
F41_IN(z0) → c(F73_IN(z0))
F73_IN(s(z0)) → c(F79_IN(z0))
F73_IN(s(z0)) → c
F79_IN(s(z0)) → c14(F79_IN(z0))
F79_IN(s(z0)) → c15(F79_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, U3, U4, U5, f79_in, U6, U7, f45_in, U8, f73_in, U9, U10, f40_in, U11, f41_in, U12, U13, f13_in, U14

Defined Pair Symbols:

F1_IN, F73_IN, F40_IN, F41_IN, F13_IN, F79_IN, F45_IN

Compound Symbols:

c, c14, c15, c19, c