Runtime Complexity of the query pattern
max(g,a,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(g,a,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(g,a,a)

(3) 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\nT1 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\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: less(T14, T12)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(T14, T12), SCOPE: 2, CLAUSE: 0 | less(T14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | less(T14, s(T19)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX16 is free\nless(T14, T12) !=? less(0, s(X16))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T14, s(T19)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: less(T26, T25)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T35, T34)\n\nT34 is ground\nX16 is free\nless(T14, s(T34)) !=? less(0, s(X16))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: less(T42, s(T44))\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T42, s(T44)), SCOPE: 3, CLAUSE: 0 | less(T42, s(T44)), SCOPE: 3, CLAUSE: 3\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true | less(0, s(T44)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: less(T42, s(T44)), SCOPE: 3, CLAUSE: 3\n\nT42 is ground\nX43 is free\nless(T42, s(T44)) !=? less(0, s(X43))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: less(0, s(T44)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T54, T56)\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T54, T56), SCOPE: 4, CLAUSE: 0 | less(T54, T56), SCOPE: 4, CLAUSE: 3\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true | less(0, T56), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T54, T56), SCOPE: 4, CLAUSE: 3\n\nT54 is ground\nX56 is free\nless(T54, T56) !=? less(0, s(X56))", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(0, T56), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T66, T68)\n\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
3 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
3 -> 54 [arrowhead = none ];
54 -> 6;

55 [label="EVAL with clause\nmax(X10, X11, X10) :- less(X11, X10).\nand substitutionT1 -> T12,\nX10 -> T12,\nT2 -> T14,\nX11 -> T14,\nT3 -> T12,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 7;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 9;

57 [label="EVAL with clause\nmax(X37, X38, X38) :- less(X37, s(X38)).\nand substitutionT1 -> T42,\nX37 -> T42,\nT2 -> T44,\nX38 -> T44,\nT3 -> T44,\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 57 [arrowhead = none ];
57 -> 33;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 58 [arrowhead = none ];
58 -> 34;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 59 [arrowhead = none ];
59 -> 15;

60 [label="EVAL with clause\nless(0, s(X16)).\nand substitutionT14 -> 0,\nX16 -> T19,\nT12 -> s(T19)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 62 [arrowhead = none ];
62 -> 18;

63 [label="EVAL with clause\nless(s(X29), s(X30)) :- less(X29, X30).\nand substitutionX29 -> T35,\nT14 -> s(T35),\nX30 -> T34,\nT12 -> s(T34),\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 63 [arrowhead = none ];
63 -> 29;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 64 [arrowhead = none ];
64 -> 30;

65 [label="EVAL with clause\nless(s(X21), s(X22)) :- less(X21, X22).\nand substitutionX21 -> T26,\nT14 -> s(T26),\nT19 -> T25,\nX22 -> T25,\nT24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="INSTANCE with matching:\nT14 -> T26\nT12 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 67 [arrowhead = none , style=dashed];
67 -> 7[style=dashed];

68 [label="INSTANCE with matching:\nT14 -> T35\nT12 -> T34", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 68 [arrowhead = none , style=dashed];
68 -> 7[style=dashed];

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 69 [arrowhead = none ];
69 -> 35;

70 [label="EVAL with clause\nless(0, s(X43)).\nand substitutionT42 -> 0,\nT44 -> T49,\nX43 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 70 [arrowhead = none ];
70 -> 36;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 71 [arrowhead = none ];
71 -> 37;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 72 [arrowhead = none ];
72 -> 38;

73 [label="EVAL with clause\nless(s(X50), s(X51)) :- less(X50, X51).\nand substitutionX50 -> T54,\nT42 -> s(T54),\nT44 -> T56,\nX51 -> T56,\nT55 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 73 [arrowhead = none ];
73 -> 42;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 74 [arrowhead = none ];
74 -> 44;

75 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 75 [arrowhead = none ];
75 -> 39;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 76 [arrowhead = none ];
76 -> 45;

77 [label="EVAL with clause\nless(0, s(X56)).\nand substitutionT54 -> 0,\nX56 -> T61,\nT56 -> s(T61)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 77 [arrowhead = none ];
77 -> 46;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 78 [arrowhead = none ];
78 -> 47;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 79 [arrowhead = none ];
79 -> 48;

80 [label="EVAL with clause\nless(s(X63), s(X64)) :- less(X63, X64).\nand substitutionX63 -> T66,\nT54 -> s(T66),\nX64 -> T68,\nT56 -> s(T68),\nT67 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 80 [arrowhead = none ];
80 -> 50;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 81 [arrowhead = none ];
81 -> 51;

82 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 82 [arrowhead = none ];
82 -> 49;

83 [label="INSTANCE with matching:\nT54 -> T66\nT56 -> T68", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 83 [arrowhead = none , style=dashed];
83 -> 42[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f1_out1
U1(f3_out2, z0) → f1_out1
f7_in(s(z0)) → f7_out1(0)
f7_in(s(z0)) → U2(f7_in(z0), s(z0))
f7_in(s(z0)) → U3(f7_in(z0), s(z0))
U2(f7_out1(z0), s(z1)) → f7_out1(s(z0))
U3(f7_out1(z0), s(z1)) → f7_out1(s(z0))
f42_in(0) → f42_out1
f42_in(s(z0)) → U4(f42_in(z0), s(z0))
U4(f42_out1, s(z0)) → f42_out1
f4_in(z0) → U5(f7_in(z0), z0)
U5(f7_out1(z0), z1) → f4_out1
f6_in(0) → f6_out1
f6_in(s(z0)) → U6(f42_in(z0), s(z0))
U6(f42_out1, s(z0)) → f6_out1
f3_in(z0) → U7(f4_in(z0), f6_in(z0), z0)
U7(f4_out1, z0, z1) → f3_out1
U7(z0, f6_out1, z1) → f3_out2

Tuples:

F1_IN(z0) → c(U1'(f3_in(z0), z0), F3_IN(z0))
F7_IN(s(z0)) → c4(U2'(f7_in(z0), s(z0)), F7_IN(z0))
F7_IN(s(z0)) → c5(U3'(f7_in(z0), s(z0)), F7_IN(z0))
F42_IN(s(z0)) → c9(U4'(f42_in(z0), s(z0)), F42_IN(z0))
F4_IN(z0) → c11(U5'(f7_in(z0), z0), F7_IN(z0))
F6_IN(s(z0)) → c14(U6'(f42_in(z0), s(z0)), F42_IN(z0))
F3_IN(z0) → c16(U7'(f4_in(z0), f6_in(z0), z0), F4_IN(z0), F6_IN(z0))

S tuples:

F1_IN(z0) → c(U1'(f3_in(z0), z0), F3_IN(z0))
F7_IN(s(z0)) → c4(U2'(f7_in(z0), s(z0)), F7_IN(z0))
F7_IN(s(z0)) → c5(U3'(f7_in(z0), s(z0)), F7_IN(z0))
F42_IN(s(z0)) → c9(U4'(f42_in(z0), s(z0)), F42_IN(z0))
F4_IN(z0) → c11(U5'(f7_in(z0), z0), F7_IN(z0))
F6_IN(s(z0)) → c14(U6'(f42_in(z0), s(z0)), F42_IN(z0))
F3_IN(z0) → c16(U7'(f4_in(z0), f6_in(z0), z0), F4_IN(z0), F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f42_in, U4, f4_in, U5, f6_in, U6, f3_in, U7

Defined Pair Symbols:

F1_IN, F7_IN, F42_IN, F4_IN, F6_IN, F3_IN

Compound Symbols:

c, c4, c5, c9, c11, c14, c16

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f1_out1
U1(f3_out2, z0) → f1_out1
f7_in(s(z0)) → f7_out1(0)
f7_in(s(z0)) → U2(f7_in(z0), s(z0))
f7_in(s(z0)) → U3(f7_in(z0), s(z0))
U2(f7_out1(z0), s(z1)) → f7_out1(s(z0))
U3(f7_out1(z0), s(z1)) → f7_out1(s(z0))
f42_in(0) → f42_out1
f42_in(s(z0)) → U4(f42_in(z0), s(z0))
U4(f42_out1, s(z0)) → f42_out1
f4_in(z0) → U5(f7_in(z0), z0)
U5(f7_out1(z0), z1) → f4_out1
f6_in(0) → f6_out1
f6_in(s(z0)) → U6(f42_in(z0), s(z0))
U6(f42_out1, s(z0)) → f6_out1
f3_in(z0) → U7(f4_in(z0), f6_in(z0), z0)
U7(f4_out1, z0, z1) → f3_out1
U7(z0, f6_out1, z1) → f3_out2

Tuples:

F7_IN(s(z0)) → c4(U2'(f7_in(z0), s(z0)), F7_IN(z0))
F7_IN(s(z0)) → c5(U3'(f7_in(z0), s(z0)), F7_IN(z0))
F42_IN(s(z0)) → c9(U4'(f42_in(z0), s(z0)), F42_IN(z0))
F1_IN(z0) → c1(U1'(f3_in(z0), z0))
F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(U5'(f7_in(z0), z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(U6'(f42_in(z0), s(z0)))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(U7'(f4_in(z0), f6_in(z0), z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))

S tuples:

F7_IN(s(z0)) → c4(U2'(f7_in(z0), s(z0)), F7_IN(z0))
F7_IN(s(z0)) → c5(U3'(f7_in(z0), s(z0)), F7_IN(z0))
F42_IN(s(z0)) → c9(U4'(f42_in(z0), s(z0)), F42_IN(z0))
F1_IN(z0) → c1(U1'(f3_in(z0), z0))
F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(U5'(f7_in(z0), z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(U6'(f42_in(z0), s(z0)))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(U7'(f4_in(z0), f6_in(z0), z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f42_in, U4, f4_in, U5, f6_in, U6, f3_in, U7

Defined Pair Symbols:

F7_IN, F42_IN, F1_IN, F4_IN, F6_IN, F3_IN

Compound Symbols:

c4, c5, c9, c1

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

Removed 7 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f1_out1
U1(f3_out2, z0) → f1_out1
f7_in(s(z0)) → f7_out1(0)
f7_in(s(z0)) → U2(f7_in(z0), s(z0))
f7_in(s(z0)) → U3(f7_in(z0), s(z0))
U2(f7_out1(z0), s(z1)) → f7_out1(s(z0))
U3(f7_out1(z0), s(z1)) → f7_out1(s(z0))
f42_in(0) → f42_out1
f42_in(s(z0)) → U4(f42_in(z0), s(z0))
U4(f42_out1, s(z0)) → f42_out1
f4_in(z0) → U5(f7_in(z0), z0)
U5(f7_out1(z0), z1) → f4_out1
f6_in(0) → f6_out1
f6_in(s(z0)) → U6(f42_in(z0), s(z0))
U6(f42_out1, s(z0)) → f6_out1
f3_in(z0) → U7(f4_in(z0), f6_in(z0), z0)
U7(f4_out1, z0, z1) → f3_out1
U7(z0, f6_out1, z1) → f3_out2

Tuples:

F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1

S tuples:

F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f42_in, U4, f4_in, U5, f6_in, U6, f3_in, U7

Defined Pair Symbols:

F1_IN, F4_IN, F6_IN, F3_IN, F7_IN, F42_IN

Compound Symbols:

c1, c4, c5, c9, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F3_IN(z0) → c1
F4_IN(z0) → c1(F7_IN(z0))
F4_IN(z0) → c1
F6_IN(s(z0)) → c1(F42_IN(z0))
F6_IN(s(z0)) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f1_out1
U1(f3_out2, z0) → f1_out1
f7_in(s(z0)) → f7_out1(0)
f7_in(s(z0)) → U2(f7_in(z0), s(z0))
f7_in(s(z0)) → U3(f7_in(z0), s(z0))
U2(f7_out1(z0), s(z1)) → f7_out1(s(z0))
U3(f7_out1(z0), s(z1)) → f7_out1(s(z0))
f42_in(0) → f42_out1
f42_in(s(z0)) → U4(f42_in(z0), s(z0))
U4(f42_out1, s(z0)) → f42_out1
f4_in(z0) → U5(f7_in(z0), z0)
U5(f7_out1(z0), z1) → f4_out1
f6_in(0) → f6_out1
f6_in(s(z0)) → U6(f42_in(z0), s(z0))
U6(f42_out1, s(z0)) → f6_out1
f3_in(z0) → U7(f4_in(z0), f6_in(z0), z0)
U7(f4_out1, z0, z1) → f3_out1
U7(z0, f6_out1, z1) → f3_out2

Tuples:

F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1

S tuples:

F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))

K tuples:

F1_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f42_in, U4, f4_in, U5, f6_in, U6, f3_in, U7

Defined Pair Symbols:

F1_IN, F4_IN, F6_IN, F3_IN, F7_IN, F42_IN

Compound Symbols:

c1, c4, c5, c9, 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.

F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))

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

F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1

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

POL(F1_IN(x₁)) = [3] + x₁   
POL(F3_IN(x₁)) = [2] + x₁   
POL(F42_IN(x₁)) = [3] + x₁   
POL(F4_IN(x₁)) = [1] + x₁   
POL(F6_IN(x₁)) = [2] + x₁   
POL(F7_IN(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f3_in(z0), z0)
U1(f3_out1, z0) → f1_out1
U1(f3_out2, z0) → f1_out1
f7_in(s(z0)) → f7_out1(0)
f7_in(s(z0)) → U2(f7_in(z0), s(z0))
f7_in(s(z0)) → U3(f7_in(z0), s(z0))
U2(f7_out1(z0), s(z1)) → f7_out1(s(z0))
U3(f7_out1(z0), s(z1)) → f7_out1(s(z0))
f42_in(0) → f42_out1
f42_in(s(z0)) → U4(f42_in(z0), s(z0))
U4(f42_out1, s(z0)) → f42_out1
f4_in(z0) → U5(f7_in(z0), z0)
U5(f7_out1(z0), z1) → f4_out1
f6_in(0) → f6_out1
f6_in(s(z0)) → U6(f42_in(z0), s(z0))
U6(f42_out1, s(z0)) → f6_out1
f3_in(z0) → U7(f4_in(z0), f6_in(z0), z0)
U7(f4_out1, z0, z1) → f3_out1
U7(z0, f6_out1, z1) → f3_out2

Tuples:

F1_IN(z0) → c1(F3_IN(z0))
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1

S tuples:none
K tuples:

F1_IN(z0) → c1(F3_IN(z0))
F3_IN(z0) → c1(F4_IN(z0))
F3_IN(z0) → c1(F6_IN(z0))
F1_IN(z0) → c1
F4_IN(z0) → c1
F6_IN(s(z0)) → c1
F3_IN(z0) → c1
F4_IN(z0) → c1(F7_IN(z0))
F6_IN(s(z0)) → c1(F42_IN(z0))
F7_IN(s(z0)) → c4(F7_IN(z0))
F7_IN(s(z0)) → c5(F7_IN(z0))
F42_IN(s(z0)) → c9(F42_IN(z0))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, U3, f42_in, U4, f4_in, U5, f6_in, U6, f3_in, U7

Defined Pair Symbols:

F1_IN, F4_IN, F6_IN, F3_IN, F7_IN, F42_IN

Compound Symbols:

c1, c4, c5, c9, c1

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))

(15) 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\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: less(T8, T6) | max(T6, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T8, T6), SCOPE: 2, CLAUSE: 0 | less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true | less(T8, s(T11)), SCOPE: 2, CLAUSE: 3 | ?_2 | max(s(T11), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: less(T8, s(T11)), SCOPE: 2, CLAUSE: 3 | ?_2 | max(s(T11), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: less(T8, s(T11)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ?_2 | max(s(T11), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T22, T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(T22, T21), SCOPE: 3, CLAUSE: 0 | less(T22, T21), SCOPE: 3, CLAUSE: 3\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | less(T22, s(T27)), SCOPE: 3, CLAUSE: 3\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T22, T21), SCOPE: 3, CLAUSE: 3\n\nT21 is ground\nX23 is free\nless(T22, T21) !=? less(0, s(X23))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T22, s(T27)), SCOPE: 3, CLAUSE: 3\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: less(T34, T33)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T43, T42)\n\nT42 is ground\nX23 is free\nless(T22, s(T42)) !=? less(0, s(X23))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: max(s(T11), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(s(T50), s(T52))\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(s(T50), s(T52)), SCOPE: 4, CLAUSE: 0 | less(s(T50), s(T52)), SCOPE: 4, CLAUSE: 3\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(s(T50), s(T52)), SCOPE: 4, CLAUSE: 3\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T60, T62)\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(T60, T62), SCOPE: 5, CLAUSE: 0 | less(T60, T62), SCOPE: 5, CLAUSE: 3\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true | less(0, T62), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T60, T62), SCOPE: 5, CLAUSE: 3\n\nT60 is ground\nX61 is free\nless(T60, T62) !=? less(0, s(X61))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(0, T62), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: less(T72, T74)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T8, T6), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: less(T87, T86)\n\nT86 is ground\nX8 is free\nless(T8, s(T86)) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: max(T6, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: less(T94, s(T96))\n\nT94 is ground\nX8 is free\nless(T8, T94) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T101, s(T103))\n\nT101 is ground\nX4 is free\nX5 is free\nmax(T101, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T101, s(T103)), SCOPE: 6, CLAUSE: 0 | less(T101, s(T103)), SCOPE: 6, CLAUSE: 3\n\nT101 is ground\nX4 is free\nX5 is free\nmax(T101, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true | less(0, s(T103)), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T101, s(T103)), SCOPE: 6, CLAUSE: 3\n\nT101 is ground\nX4 is free\nX5 is free\nX98 is free\nmax(T101, T2, T3) !=? max(X4, X5, X4)\nless(T101, s(T103)) !=? less(0, s(X98))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(0, s(T103)), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: less(T111, T113)\n\nT111 is ground\nX4 is free\nX5 is free\nmax(s(T111), T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="EVAL with clause\nmax(X4, X5, X4) :- less(X5, X4).\nand substitutionT1 -> T6,\nX4 -> T6,\nT2 -> T8,\nX5 -> T8,\nT3 -> T6,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 81 [arrowhead = none ];
81 -> 8;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 82 [arrowhead = none ];
82 -> 10;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 83 [arrowhead = none ];
83 -> 11;

84 [label="EVAL with clause\nmax(X94, X95, X95) :- less(X94, s(X95)).\nand substitutionT1 -> T101,\nX94 -> T101,\nT2 -> T103,\nX95 -> T103,\nT3 -> T103,\nT102 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 84 [arrowhead = none ];
84 -> 71;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 85 [arrowhead = none ];
85 -> 72;

86 [label="EVAL with clause\nless(0, s(X8)).\nand substitutionT8 -> 0,\nX8 -> T11,\nT6 -> s(T11)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 86 [arrowhead = none ];
86 -> 12;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 87 [arrowhead = none ];
87 -> 13;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 88 [arrowhead = none ];
88 -> 14;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 89 [arrowhead = none ];
89 -> 64;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 90 [arrowhead = none ];
90 -> 65;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 91 [arrowhead = none ];
91 -> 19;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 92 [arrowhead = none ];
92 -> 20;

93 [label="EVAL with clause\nless(s(X17), s(X18)) :- less(X17, X18).\nand substitutionX17 -> T22,\nT8 -> s(T22),\nT11 -> T21,\nX18 -> T21,\nT20 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 93 [arrowhead = none ];
93 -> 23;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 94 [arrowhead = none ];
94 -> 24;

95 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 95 [arrowhead = none ];
95 -> 43;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 96 [arrowhead = none ];
96 -> 25;

97 [label="EVAL with clause\nless(0, s(X23)).\nand substitutionT22 -> 0,\nX23 -> T27,\nT21 -> s(T27)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 97 [arrowhead = none ];
97 -> 26;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 98 [arrowhead = none ];
98 -> 27;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="EVAL with clause\nless(s(X36), s(X37)) :- less(X36, X37).\nand substitutionX36 -> T43,\nT22 -> s(T43),\nX37 -> T42,\nT21 -> s(T42),\nT41 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 100 [arrowhead = none ];
100 -> 40;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 101 [arrowhead = none ];
101 -> 41;

102 [label="EVAL with clause\nless(s(X28), s(X29)) :- less(X28, X29).\nand substitutionX28 -> T34,\nT22 -> s(T34),\nT27 -> T33,\nX29 -> T33,\nT32 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 103 [arrowhead = none ];
103 -> 32;

104 [label="INSTANCE with matching:\nT22 -> T34\nT21 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 104 [arrowhead = none , style=dashed];
104 -> 23[style=dashed];

105 [label="INSTANCE with matching:\nT22 -> T43\nT21 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 105 [arrowhead = none , style=dashed];
105 -> 23[style=dashed];

106 [label="EVAL with clause\nmax(X44, X45, X45) :- less(X44, s(X45)).\nand substitutionT11 -> T50,\nX44 -> s(T50),\nT2 -> T52,\nX45 -> T52,\nT3 -> T52,\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 106 [arrowhead = none ];
106 -> 52;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 107 [arrowhead = none ];
107 -> 53;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 108 [arrowhead = none ];
108 -> 54;

109 [label="BACKTRACK\nfor clause: less(0, s(X1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 109 [arrowhead = none ];
109 -> 55;

110 [label="ONLY EVAL with clause\nless(s(X55), s(X56)) :- less(X55, X56).\nand substitutionT50 -> T60,\nX55 -> T60,\nT52 -> T62,\nX56 -> T62,\nT61 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 110 [arrowhead = none ];
110 -> 56;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 111 [arrowhead = none ];
111 -> 57;

112 [label="EVAL with clause\nless(0, s(X61)).\nand substitutionT60 -> 0,\nX61 -> T67,\nT62 -> s(T67)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 112 [arrowhead = none ];
112 -> 58;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 113 [arrowhead = none ];
113 -> 59;

114 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 114 [arrowhead = none ];
114 -> 60;

115 [label="EVAL with clause\nless(s(X68), s(X69)) :- less(X68, X69).\nand substitutionX68 -> T72,\nT60 -> s(T72),\nX69 -> T74,\nT62 -> s(T74),\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 115 [arrowhead = none ];
115 -> 62;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 116 [arrowhead = none ];
116 -> 63;

117 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 117 [arrowhead = none ];
117 -> 61;

118 [label="INSTANCE with matching:\nT60 -> T72\nT62 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 118 [arrowhead = none , style=dashed];
118 -> 56[style=dashed];

119 [label="EVAL with clause\nless(s(X80), s(X81)) :- less(X80, X81).\nand substitutionX80 -> T87,\nT8 -> s(T87),\nX81 -> T86,\nT6 -> s(T86),\nT85 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 119 [arrowhead = none ];
119 -> 66;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 120 [arrowhead = none ];
120 -> 67;

121 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 121 [arrowhead = none ];
121 -> 68;

122 [label="INSTANCE with matching:\nT22 -> T87\nT21 -> T86", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 122 [arrowhead = none , style=dashed];
122 -> 23[style=dashed];

123 [label="EVAL with clause\nmax(X88, X89, X89) :- less(X88, s(X89)).\nand substitutionT6 -> T94,\nX88 -> T94,\nT2 -> T96,\nX89 -> T96,\nT3 -> T96,\nT95 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 123 [arrowhead = none ];
123 -> 69;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 124 [arrowhead = none ];
124 -> 70;

125 [label="INSTANCE with matching:\nT60 -> T94\nT62 -> s(T96)", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 125 [arrowhead = none , style=dashed];
125 -> 56[style=dashed];

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 126 [arrowhead = none ];
126 -> 73;

127 [label="EVAL with clause\nless(0, s(X98)).\nand substitutionT101 -> 0,\nT103 -> T106,\nX98 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 127 [arrowhead = none ];
127 -> 74;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 128 [arrowhead = none ];
128 -> 75;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 129 [arrowhead = none ];
129 -> 76;

130 [label="EVAL with clause\nless(s(X105), s(X106)) :- less(X105, X106).\nand substitutionX105 -> T111,\nT101 -> s(T111),\nT103 -> T113,\nX106 -> T113,\nT112 -> T113", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 130 [arrowhead = none ];
130 -> 78;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 131 [arrowhead = none ];
131 -> 79;

132 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 132 [arrowhead = none ];
132 -> 77;

133 [label="INSTANCE with matching:\nT60 -> T111\nT62 -> T113", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 133 [arrowhead = none , style=dashed];
133 -> 56[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(s(z0)) → f2_out1
f2_in(s(z0)) → U1(f14_in(z0), s(z0))
f2_in(z0) → U2(f13_in(z0), z0)
f2_in(0) → f2_out1
f2_in(s(z0)) → U3(f56_in(z0), s(z0))
U1(f14_out1(z0), s(z1)) → f2_out1
U1(f14_out3, s(z0)) → f2_out1
U2(f13_out1(z0), z1) → f2_out1
U2(f13_out3, z0) → f2_out1
U3(f56_out1, s(z0)) → f2_out1
f23_in(s(z0)) → f23_out1(0)
f23_in(s(z0)) → U4(f23_in(z0), s(z0))
f23_in(s(z0)) → U5(f23_in(z0), s(z0))
U4(f23_out1(z0), s(z1)) → f23_out1(s(z0))
U5(f23_out1(z0), s(z1)) → f23_out1(s(z0))
f56_in(0) → f56_out1
f56_in(s(z0)) → U6(f56_in(z0), s(z0))
U6(f56_out1, s(z0)) → f56_out1
f64_in(s(z0)) → U7(f23_in(z0), s(z0))
U7(f23_out1(z0), s(z1)) → f64_out1(s(z0))
f65_in(z0) → U8(f56_in(z0), z0)
U8(f56_out1, z0) → f65_out2
f19_in(z0) → U9(f23_in(z0), z0)
U9(f23_out1(z0), z1) → f19_out1(s(z0))
f20_in(z0) → U10(f56_in(z0), z0)
U10(f56_out1, z0) → f20_out2
f13_in(z0) → U11(f64_in(z0), f65_in(z0), z0)
U11(f64_out1(z0), z1, z2) → f13_out1(z0)
U11(z0, f65_out2, z1) → f13_out3
f14_in(z0) → U12(f19_in(z0), f20_in(z0), z0)
U12(f19_out1(z0), z1, z2) → f14_out1(z0)
U12(z0, f20_out2, z1) → f14_out3

Tuples:

F2_IN(s(z0)) → c1(U1'(f14_in(z0), s(z0)), F14_IN(z0))
F2_IN(z0) → c2(U2'(f13_in(z0), z0), F13_IN(z0))
F2_IN(s(z0)) → c4(U3'(f56_in(z0), s(z0)), F56_IN(z0))
F23_IN(s(z0)) → c11(U4'(f23_in(z0), s(z0)), F23_IN(z0))
F23_IN(s(z0)) → c12(U5'(f23_in(z0), s(z0)), F23_IN(z0))
F56_IN(s(z0)) → c16(U6'(f56_in(z0), s(z0)), F56_IN(z0))
F64_IN(s(z0)) → c18(U7'(f23_in(z0), s(z0)), F23_IN(z0))
F65_IN(z0) → c20(U8'(f56_in(z0), z0), F56_IN(z0))
F19_IN(z0) → c22(U9'(f23_in(z0), z0), F23_IN(z0))
F20_IN(z0) → c24(U10'(f56_in(z0), z0), F56_IN(z0))
F13_IN(z0) → c26(U11'(f64_in(z0), f65_in(z0), z0), F64_IN(z0), F65_IN(z0))
F14_IN(z0) → c29(U12'(f19_in(z0), f20_in(z0), z0), F19_IN(z0), F20_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(U1'(f14_in(z0), s(z0)), F14_IN(z0))
F2_IN(z0) → c2(U2'(f13_in(z0), z0), F13_IN(z0))
F2_IN(s(z0)) → c4(U3'(f56_in(z0), s(z0)), F56_IN(z0))
F23_IN(s(z0)) → c11(U4'(f23_in(z0), s(z0)), F23_IN(z0))
F23_IN(s(z0)) → c12(U5'(f23_in(z0), s(z0)), F23_IN(z0))
F56_IN(s(z0)) → c16(U6'(f56_in(z0), s(z0)), F56_IN(z0))
F64_IN(s(z0)) → c18(U7'(f23_in(z0), s(z0)), F23_IN(z0))
F65_IN(z0) → c20(U8'(f56_in(z0), z0), F56_IN(z0))
F19_IN(z0) → c22(U9'(f23_in(z0), z0), F23_IN(z0))
F20_IN(z0) → c24(U10'(f56_in(z0), z0), F56_IN(z0))
F13_IN(z0) → c26(U11'(f64_in(z0), f65_in(z0), z0), F64_IN(z0), F65_IN(z0))
F14_IN(z0) → c29(U12'(f19_in(z0), f20_in(z0), z0), F19_IN(z0), F20_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, f23_in, U4, U5, f56_in, U6, f64_in, U7, f65_in, U8, f19_in, U9, f20_in, U10, f13_in, U11, f14_in, U12

Defined Pair Symbols:

F2_IN, F23_IN, F56_IN, F64_IN, F65_IN, F19_IN, F20_IN, F13_IN, F14_IN

Compound Symbols:

c1, c2, c4, c11, c12, c16, c18, c20, c22, c24, c26, c29

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

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(s(z0)) → f2_out1
f2_in(s(z0)) → U1(f14_in(z0), s(z0))
f2_in(z0) → U2(f13_in(z0), z0)
f2_in(0) → f2_out1
f2_in(s(z0)) → U3(f56_in(z0), s(z0))
U1(f14_out1(z0), s(z1)) → f2_out1
U1(f14_out3, s(z0)) → f2_out1
U2(f13_out1(z0), z1) → f2_out1
U2(f13_out3, z0) → f2_out1
U3(f56_out1, s(z0)) → f2_out1
f23_in(s(z0)) → f23_out1(0)
f23_in(s(z0)) → U4(f23_in(z0), s(z0))
f23_in(s(z0)) → U5(f23_in(z0), s(z0))
U4(f23_out1(z0), s(z1)) → f23_out1(s(z0))
U5(f23_out1(z0), s(z1)) → f23_out1(s(z0))
f56_in(0) → f56_out1
f56_in(s(z0)) → U6(f56_in(z0), s(z0))
U6(f56_out1, s(z0)) → f56_out1
f64_in(s(z0)) → U7(f23_in(z0), s(z0))
U7(f23_out1(z0), s(z1)) → f64_out1(s(z0))
f65_in(z0) → U8(f56_in(z0), z0)
U8(f56_out1, z0) → f65_out2
f19_in(z0) → U9(f23_in(z0), z0)
U9(f23_out1(z0), z1) → f19_out1(s(z0))
f20_in(z0) → U10(f56_in(z0), z0)
U10(f56_out1, z0) → f20_out2
f13_in(z0) → U11(f64_in(z0), f65_in(z0), z0)
U11(f64_out1(z0), z1, z2) → f13_out1(z0)
U11(z0, f65_out2, z1) → f13_out3
f14_in(z0) → U12(f19_in(z0), f20_in(z0), z0)
U12(f19_out1(z0), z1, z2) → f14_out1(z0)
U12(z0, f20_out2, z1) → f14_out3

Tuples:

F23_IN(s(z0)) → c11(U4'(f23_in(z0), s(z0)), F23_IN(z0))
F23_IN(s(z0)) → c12(U5'(f23_in(z0), s(z0)), F23_IN(z0))
F56_IN(s(z0)) → c16(U6'(f56_in(z0), s(z0)), F56_IN(z0))
F2_IN(s(z0)) → c(U1'(f14_in(z0), s(z0)))
F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(U2'(f13_in(z0), z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(U3'(f56_in(z0), s(z0)))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(U7'(f23_in(z0), s(z0)))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(U8'(f56_in(z0), z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(U9'(f23_in(z0), z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(U10'(f56_in(z0), z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(U11'(f64_in(z0), f65_in(z0), z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(U12'(f19_in(z0), f20_in(z0), z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))

S tuples:

F23_IN(s(z0)) → c11(U4'(f23_in(z0), s(z0)), F23_IN(z0))
F23_IN(s(z0)) → c12(U5'(f23_in(z0), s(z0)), F23_IN(z0))
F56_IN(s(z0)) → c16(U6'(f56_in(z0), s(z0)), F56_IN(z0))
F2_IN(s(z0)) → c(U1'(f14_in(z0), s(z0)))
F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(U2'(f13_in(z0), z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(U3'(f56_in(z0), s(z0)))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(U7'(f23_in(z0), s(z0)))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(U8'(f56_in(z0), z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(U9'(f23_in(z0), z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(U10'(f56_in(z0), z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(U11'(f64_in(z0), f65_in(z0), z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(U12'(f19_in(z0), f20_in(z0), z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, f23_in, U4, U5, f56_in, U6, f64_in, U7, f65_in, U8, f19_in, U9, f20_in, U10, f13_in, U11, f14_in, U12

Defined Pair Symbols:

F23_IN, F56_IN, F2_IN, F64_IN, F65_IN, F19_IN, F20_IN, F13_IN, F14_IN

Compound Symbols:

c11, c12, c16, c

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

Removed 12 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(s(z0)) → f2_out1
f2_in(s(z0)) → U1(f14_in(z0), s(z0))
f2_in(z0) → U2(f13_in(z0), z0)
f2_in(0) → f2_out1
f2_in(s(z0)) → U3(f56_in(z0), s(z0))
U1(f14_out1(z0), s(z1)) → f2_out1
U1(f14_out3, s(z0)) → f2_out1
U2(f13_out1(z0), z1) → f2_out1
U2(f13_out3, z0) → f2_out1
U3(f56_out1, s(z0)) → f2_out1
f23_in(s(z0)) → f23_out1(0)
f23_in(s(z0)) → U4(f23_in(z0), s(z0))
f23_in(s(z0)) → U5(f23_in(z0), s(z0))
U4(f23_out1(z0), s(z1)) → f23_out1(s(z0))
U5(f23_out1(z0), s(z1)) → f23_out1(s(z0))
f56_in(0) → f56_out1
f56_in(s(z0)) → U6(f56_in(z0), s(z0))
U6(f56_out1, s(z0)) → f56_out1
f64_in(s(z0)) → U7(f23_in(z0), s(z0))
U7(f23_out1(z0), s(z1)) → f64_out1(s(z0))
f65_in(z0) → U8(f56_in(z0), z0)
U8(f56_out1, z0) → f65_out2
f19_in(z0) → U9(f23_in(z0), z0)
U9(f23_out1(z0), z1) → f19_out1(s(z0))
f20_in(z0) → U10(f56_in(z0), z0)
U10(f56_out1, z0) → f20_out2
f13_in(z0) → U11(f64_in(z0), f65_in(z0), z0)
U11(f64_out1(z0), z1, z2) → f13_out1(z0)
U11(z0, f65_out2, z1) → f13_out3
f14_in(z0) → U12(f19_in(z0), f20_in(z0), z0)
U12(f19_out1(z0), z1, z2) → f14_out1(z0)
U12(z0, f20_out2, z1) → f14_out3

Tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c

S tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, f23_in, U4, U5, f56_in, U6, f64_in, U7, f65_in, U8, f19_in, U9, f20_in, U10, f13_in, U11, f14_in, U12

Defined Pair Symbols:

F2_IN, F64_IN, F65_IN, F19_IN, F20_IN, F13_IN, F14_IN, F23_IN, F56_IN

Compound Symbols:

c, c11, c12, c16, c

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F2_IN(s(z0)) → c
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F14_IN(z0) → c
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F13_IN(z0) → c
F64_IN(s(z0)) → c(F23_IN(z0))
F64_IN(s(z0)) → c
F65_IN(z0) → c(F56_IN(z0))
F65_IN(z0) → c
F19_IN(z0) → c(F23_IN(z0))
F19_IN(z0) → c
F20_IN(z0) → c(F56_IN(z0))
F20_IN(z0) → c

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(s(z0)) → f2_out1
f2_in(s(z0)) → U1(f14_in(z0), s(z0))
f2_in(z0) → U2(f13_in(z0), z0)
f2_in(0) → f2_out1
f2_in(s(z0)) → U3(f56_in(z0), s(z0))
U1(f14_out1(z0), s(z1)) → f2_out1
U1(f14_out3, s(z0)) → f2_out1
U2(f13_out1(z0), z1) → f2_out1
U2(f13_out3, z0) → f2_out1
U3(f56_out1, s(z0)) → f2_out1
f23_in(s(z0)) → f23_out1(0)
f23_in(s(z0)) → U4(f23_in(z0), s(z0))
f23_in(s(z0)) → U5(f23_in(z0), s(z0))
U4(f23_out1(z0), s(z1)) → f23_out1(s(z0))
U5(f23_out1(z0), s(z1)) → f23_out1(s(z0))
f56_in(0) → f56_out1
f56_in(s(z0)) → U6(f56_in(z0), s(z0))
U6(f56_out1, s(z0)) → f56_out1
f64_in(s(z0)) → U7(f23_in(z0), s(z0))
U7(f23_out1(z0), s(z1)) → f64_out1(s(z0))
f65_in(z0) → U8(f56_in(z0), z0)
U8(f56_out1, z0) → f65_out2
f19_in(z0) → U9(f23_in(z0), z0)
U9(f23_out1(z0), z1) → f19_out1(s(z0))
f20_in(z0) → U10(f56_in(z0), z0)
U10(f56_out1, z0) → f20_out2
f13_in(z0) → U11(f64_in(z0), f65_in(z0), z0)
U11(f64_out1(z0), z1, z2) → f13_out1(z0)
U11(z0, f65_out2, z1) → f13_out3
f14_in(z0) → U12(f19_in(z0), f20_in(z0), z0)
U12(f19_out1(z0), z1, z2) → f14_out1(z0)
U12(z0, f20_out2, z1) → f14_out3

Tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c

S tuples:

F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))

K tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, U3, f23_in, U4, U5, f56_in, U6, f64_in, U7, f65_in, U8, f19_in, U9, f20_in, U10, f13_in, U11, f14_in, U12

Defined Pair Symbols:

F2_IN, F64_IN, F65_IN, F19_IN, F20_IN, F13_IN, F14_IN, F23_IN, F56_IN

Compound Symbols:

c, c11, c12, c16, c

(23) 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.

F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))

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

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c

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

POL(F13_IN(x₁)) = [2] + [2]x₁   
POL(F14_IN(x₁)) = [2] + [2]x₁   
POL(F19_IN(x₁)) = [1] + [2]x₁   
POL(F20_IN(x₁)) = [2] + x₁   
POL(F23_IN(x₁)) = [2]x₁   
POL(F2_IN(x₁)) = [2] + [2]x₁   
POL(F56_IN(x₁)) = [1] + x₁   
POL(F64_IN(x₁)) = [1] + [2]x₁   
POL(F65_IN(x₁)) = [1] + x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(s(z0)) → f2_out1
f2_in(s(z0)) → U1(f14_in(z0), s(z0))
f2_in(z0) → U2(f13_in(z0), z0)
f2_in(0) → f2_out1
f2_in(s(z0)) → U3(f56_in(z0), s(z0))
U1(f14_out1(z0), s(z1)) → f2_out1
U1(f14_out3, s(z0)) → f2_out1
U2(f13_out1(z0), z1) → f2_out1
U2(f13_out3, z0) → f2_out1
U3(f56_out1, s(z0)) → f2_out1
f23_in(s(z0)) → f23_out1(0)
f23_in(s(z0)) → U4(f23_in(z0), s(z0))
f23_in(s(z0)) → U5(f23_in(z0), s(z0))
U4(f23_out1(z0), s(z1)) → f23_out1(s(z0))
U5(f23_out1(z0), s(z1)) → f23_out1(s(z0))
f56_in(0) → f56_out1
f56_in(s(z0)) → U6(f56_in(z0), s(z0))
U6(f56_out1, s(z0)) → f56_out1
f64_in(s(z0)) → U7(f23_in(z0), s(z0))
U7(f23_out1(z0), s(z1)) → f64_out1(s(z0))
f65_in(z0) → U8(f56_in(z0), z0)
U8(f56_out1, z0) → f65_out2
f19_in(z0) → U9(f23_in(z0), z0)
U9(f23_out1(z0), z1) → f19_out1(s(z0))
f20_in(z0) → U10(f56_in(z0), z0)
U10(f56_out1, z0) → f20_out2
f13_in(z0) → U11(f64_in(z0), f65_in(z0), z0)
U11(f64_out1(z0), z1, z2) → f13_out1(z0)
U11(z0, f65_out2, z1) → f13_out3
f14_in(z0) → U12(f19_in(z0), f20_in(z0), z0)
U12(f19_out1(z0), z1, z2) → f14_out1(z0)
U12(z0, f20_out2, z1) → f14_out3

Tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c

S tuples:none
K tuples:

F2_IN(s(z0)) → c(F14_IN(z0))
F2_IN(z0) → c(F13_IN(z0))
F2_IN(s(z0)) → c(F56_IN(z0))
F13_IN(z0) → c(F64_IN(z0))
F13_IN(z0) → c(F65_IN(z0))
F14_IN(z0) → c(F19_IN(z0))
F14_IN(z0) → c(F20_IN(z0))
F2_IN(s(z0)) → c
F2_IN(z0) → c
F64_IN(s(z0)) → c
F65_IN(z0) → c
F19_IN(z0) → c
F20_IN(z0) → c
F13_IN(z0) → c
F14_IN(z0) → c
F64_IN(s(z0)) → c(F23_IN(z0))
F65_IN(z0) → c(F56_IN(z0))
F19_IN(z0) → c(F23_IN(z0))
F20_IN(z0) → c(F56_IN(z0))
F23_IN(s(z0)) → c11(F23_IN(z0))
F23_IN(s(z0)) → c12(F23_IN(z0))
F56_IN(s(z0)) → c16(F56_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, U3, f23_in, U4, U5, f56_in, U6, f64_in, U7, f65_in, U8, f19_in, U9, f20_in, U10, f13_in, U11, f14_in, U12

Defined Pair Symbols:

F2_IN, F64_IN, F65_IN, F19_IN, F20_IN, F13_IN, F14_IN, F23_IN, F56_IN

Compound Symbols:

c, c11, c12, c16, c