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

(0) Obligation:

Clauses:

div(X, 0, Z, R) :- ','(!, fail).
div(0, Y, Z, R) :- ','(!, ','(=(Z, 0), =(R, 0))).
div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))).
div(X, Y, 0, X).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).

Query: div(g,g,a,a)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, fail) | div(T11, 0, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T11, 0, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T11, 0, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX7 is free\nX8 is free\nX9 is free\ndiv(T1, T2, T3, T4) !=? div(X7, 0, X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(!_1, ','(=(T23, 0), =(T24, 0))) | div(0, T20, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T20, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT20 is ground\nX7 is free\nX8 is free\nX9 is free\ndiv(0, T20, T3, T4) !=? div(X7, 0, X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(=(T23, 0), =(T24, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: =(T25, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(minus(T42, T43, X41), ','(!_1, div(X41, T43, T46, T47)))\n\nT42 is ground\nT43 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\nX41 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: minus(T42, T43, X41)\n\nT42 is ground\nT43 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\nX41 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(!_1, div(T52, T43, T46, T47))\n\nT42 is ground\nT43 is ground\nT52 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: minus(T42, T43, X41), SCOPE: 2, CLAUSE: 4 | minus(T42, T43, X41), SCOPE: 2, CLAUSE: 5\n\nT42 is ground\nT43 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\nX41 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(T42, T43, X41), SCOPE: 2, CLAUSE: 5\n\nT42 is ground\nT43 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\nX41 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minus(T62, T63, X66)\n\nT62 is ground\nT63 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: minus(T62, T63, X66), SCOPE: 3, CLAUSE: 4 | minus(T62, T63, X66), SCOPE: 3, CLAUSE: 5\n\nT62 is ground\nT63 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true | minus(T68, 0, X66), SCOPE: 3, CLAUSE: 5\n\nT68 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: minus(T62, T63, X66), SCOPE: 3, CLAUSE: 5\n\nT62 is ground\nT63 is ground\nX66 is free\nX71 is free\nminus(T62, T63, X66) !=? minus(X71, 0, X71)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: minus(T68, 0, X66), SCOPE: 3, CLAUSE: 5\n\nT68 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: minus(T74, T75, X86)\n\nT74 is ground\nT75 is ground\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: div(T52, T43, T46, T47)\n\nT42 is ground\nT43 is ground\nT52 is ground\nX7 is free\nX8 is free\nX9 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T42, T43, T3, T4) !=? div(X7, 0, X8, X9)\ndiv(T42, T43, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 75 [arrowhead = none ];
75 -> 3;

76 [label="EVAL with clause\ndiv(X7, 0, X8, X9) :- ','(!_1, fail).\nand substitutionT1 -> T11,\nX7 -> T11,\nT2 -> 0,\nT3 -> T12,\nX8 -> T12,\nT4 -> T13,\nX9 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 76 [arrowhead = none ];
76 -> 5;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 77 [arrowhead = none ];
77 -> 7;

78 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 78 [arrowhead = none ];
78 -> 8;

79 [label="EVAL with clause\ndiv(0, X16, X17, X18) :- ','(!_1, ','(=(X17, 0), =(X18, 0))).\nand substitutionT1 -> 0,\nT2 -> T20,\nX16 -> T20,\nT3 -> T23,\nX17 -> T23,\nT4 -> T24,\nX18 -> T24,\nT21 -> T23,\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 79 [arrowhead = none ];
79 -> 14;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 80 [arrowhead = none ];
80 -> 17;

81 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 81 [arrowhead = none ];
81 -> 10;

82 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 82 [arrowhead = none ];
82 -> 19;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="UNIFY CASE with substitutionT23 -> 0,\nT24 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 85 [arrowhead = none ];
85 -> 21;

86 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="UNIFY CASE with substitutionT25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 87 [arrowhead = none ];
87 -> 23;

88 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 88 [arrowhead = none ];
88 -> 25;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 89 [arrowhead = none ];
89 -> 27;

90 [label="EVAL with clause\ndiv(X37, X38, s(X39), X40) :- ','(minus(X37, X38, X41), ','(!_1, div(X41, X38, X39, X40))).\nand substitutionT1 -> T42,\nX37 -> T42,\nT2 -> T43,\nX38 -> T43,\nX39 -> T46,\nT3 -> s(T46),\nT4 -> T47,\nX40 -> T47,\nT44 -> T46,\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 90 [arrowhead = none ];
90 -> 35;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 91 [arrowhead = none ];
91 -> 36;

92 [label="EVAL with clause\ndiv(X99, X100, 0, X99).\nand substitutionT1 -> T88,\nX99 -> T88,\nT2 -> T89,\nX100 -> T89,\nT3 -> 0,\nT4 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 92 [arrowhead = none ];
92 -> 72;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 93 [arrowhead = none ];
93 -> 73;

94 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="SPLIT 2\nnew knowledge:\nT42 is ground\nT43 is ground\nT52 is ground\nreplacements:X41 -> T52", fontsize=14, style = filled, fillcolor = violet];
35 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 96 [arrowhead = none ];
96 -> 49;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 97 [arrowhead = none ];
97 -> 69;

98 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T42, T43, T3, T4), div(X7, 0, X8, X9))", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 98 [arrowhead = none ];
98 -> 51;

99 [label="EVAL with clause\nminus(s(X63), s(X64), X65) :- minus(X63, X64, X65).\nand substitutionX63 -> T62,\nT42 -> s(T62),\nX64 -> T63,\nT43 -> s(T63),\nX41 -> X66,\nX65 -> X66", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 99 [arrowhead = none ];
99 -> 54;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 100 [arrowhead = none ];
100 -> 55;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 101 [arrowhead = none ];
101 -> 56;

102 [label="EVAL with clause\nminus(X71, 0, X71).\nand substitutionT62 -> T68,\nX71 -> T68,\nT63 -> 0,\nX66 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 102 [arrowhead = none ];
102 -> 58;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 103 [arrowhead = none ];
103 -> 59;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 104 [arrowhead = none ];
104 -> 60;

105 [label="EVAL with clause\nminus(s(X83), s(X84), X85) :- minus(X83, X84, X85).\nand substitutionX83 -> T74,\nT62 -> s(T74),\nX84 -> T75,\nT63 -> s(T75),\nX66 -> X86,\nX85 -> X86", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 105 [arrowhead = none ];
105 -> 64;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 106 [arrowhead = none ];
106 -> 67;

107 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 107 [arrowhead = none ];
107 -> 62;

108 [label="INSTANCE with matching:\nT62 -> T74\nT63 -> T75\nX66 -> X86", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 108 [arrowhead = none , style=dashed];
108 -> 54[style=dashed];

109 [label="INSTANCE with matching:\nT1 -> T52\nT2 -> T43\nT3 -> T46\nT4 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 109 [arrowhead = none , style=dashed];
109 -> 1[style=dashed];

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 110 [arrowhead = none ];
110 -> 74;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f17_in(z0, z1), z0, z1)
U1(f17_out1, z0, z1) → f1_out1
U1(f17_out2, z0, z1) → f1_out1
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)
f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
f44_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f44_out1
f29_in(z0, z1) → U5(f35_in(z0, z1), z0, z1)
U5(f35_out1(z0), z1, z2) → f29_out1
f30_in(z0, z1) → f30_out1
f35_in(z0, z1) → U6(f43_in(z0, z1), z0, z1)
U6(f43_out1(z0), z1, z2) → U7(f44_in(z0, z2), z1, z2, z0)
U7(f44_out1, z0, z1, z2) → f35_out1(z2)
f17_in(z0, z1) → U8(f29_in(z0, z1), f30_in(z0, z1), z0, z1)
U8(f29_out1, z0, z1, z2) → f17_out1
U8(z0, f30_out1, z1, z2) → f17_out2

Tuples:

F1_IN(z0, z1) → c1(U1'(f17_in(z0, z1), z0, z1), F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(U2'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c7(U3'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F44_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F29_IN(z0, z1) → c11(U5'(f35_in(z0, z1), z0, z1), F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(U7'(f44_in(z0, z2), z1, z2, z0), F44_IN(z0, z2))
F17_IN(z0, z1) → c17(U8'(f29_in(z0, z1), f30_in(z0, z1), z0, z1), F29_IN(z0, z1), F30_IN(z0, z1))

S tuples:

F1_IN(z0, z1) → c1(U1'(f17_in(z0, z1), z0, z1), F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(U2'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c7(U3'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F44_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F29_IN(z0, z1) → c11(U5'(f35_in(z0, z1), z0, z1), F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(U7'(f44_in(z0, z2), z1, z2, z0), F44_IN(z0, z2))
F17_IN(z0, z1) → c17(U8'(f29_in(z0, z1), f30_in(z0, z1), z0, z1), F29_IN(z0, z1), F30_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f54_in, U2, f43_in, U3, f44_in, U4, f29_in, U5, f30_in, f35_in, U6, U7, f17_in, U8

Defined Pair Symbols:

F1_IN, F54_IN, F43_IN, F44_IN, F29_IN, F35_IN, U6', F17_IN

Compound Symbols:

c1, c5, c7, c9, c11, c14, c15, c17

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f17_in(z0, z1), z0, z1)
U1(f17_out1, z0, z1) → f1_out1
U1(f17_out2, z0, z1) → f1_out1
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)
f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
f44_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f44_out1
f29_in(z0, z1) → U5(f35_in(z0, z1), z0, z1)
U5(f35_out1(z0), z1, z2) → f29_out1
f30_in(z0, z1) → f30_out1
f35_in(z0, z1) → U6(f43_in(z0, z1), z0, z1)
U6(f43_out1(z0), z1, z2) → U7(f44_in(z0, z2), z1, z2, z0)
U7(f44_out1, z0, z1, z2) → f35_out1(z2)
f17_in(z0, z1) → U8(f29_in(z0, z1), f30_in(z0, z1), z0, z1)
U8(f29_out1, z0, z1, z2) → f17_out1
U8(z0, f30_out1, z1, z2) → f17_out2

Tuples:

F1_IN(z0, z1) → c1(U1'(f17_in(z0, z1), z0, z1), F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(U2'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F44_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F29_IN(z0, z1) → c11(U5'(f35_in(z0, z1), z0, z1), F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(U7'(f44_in(z0, z2), z1, z2, z0), F44_IN(z0, z2))
F17_IN(z0, z1) → c17(U8'(f29_in(z0, z1), f30_in(z0, z1), z0, z1), F29_IN(z0, z1), F30_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(U3'(f54_in(z0, z1), s(z0), s(z1)))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))

S tuples:

F1_IN(z0, z1) → c1(U1'(f17_in(z0, z1), z0, z1), F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(U2'(f54_in(z0, z1), s(z0), s(z1)), F54_IN(z0, z1))
F44_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F29_IN(z0, z1) → c11(U5'(f35_in(z0, z1), z0, z1), F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(U7'(f44_in(z0, z2), z1, z2, z0), F44_IN(z0, z2))
F17_IN(z0, z1) → c17(U8'(f29_in(z0, z1), f30_in(z0, z1), z0, z1), F29_IN(z0, z1), F30_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(U3'(f54_in(z0, z1), s(z0), s(z1)))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f54_in, U2, f43_in, U3, f44_in, U4, f29_in, U5, f30_in, f35_in, U6, U7, f17_in, U8

Defined Pair Symbols:

F1_IN, F54_IN, F44_IN, F29_IN, F35_IN, U6', F17_IN, F43_IN

Compound Symbols:

c1, c5, c9, c11, c14, c15, c17, c

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

Removed 8 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f17_in(z0, z1), z0, z1)
U1(f17_out1, z0, z1) → f1_out1
U1(f17_out2, z0, z1) → f1_out1
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)
f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
f44_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f44_out1
f29_in(z0, z1) → U5(f35_in(z0, z1), z0, z1)
U5(f35_out1(z0), z1, z2) → f29_out1
f30_in(z0, z1) → f30_out1
f35_in(z0, z1) → U6(f43_in(z0, z1), z0, z1)
U6(f43_out1(z0), z1, z2) → U7(f44_in(z0, z2), z1, z2, z0)
U7(f44_out1, z0, z1, z2) → f35_out1(z2)
f17_in(z0, z1) → U8(f29_in(z0, z1), f30_in(z0, z1), z0, z1)
U8(f29_out1, z0, z1, z2) → f17_out1
U8(z0, f30_out1, z1, z2) → f17_out2

Tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F44_IN(z0, z1) → c9(F1_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

S tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F44_IN(z0, z1) → c9(F1_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f54_in, U2, f43_in, U3, f44_in, U4, f29_in, U5, f30_in, f35_in, U6, U7, f17_in, U8

Defined Pair Symbols:

F35_IN, F43_IN, F1_IN, F54_IN, F44_IN, F29_IN, U6', F17_IN

Compound Symbols:

c14, c, c1, c5, c9, c11, c15, c17, c

(7) 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.

F44_IN(z0, z1) → c9(F1_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))

We considered the (Usable) Rules:

f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)

And the Tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F44_IN(z0, z1) → c9(F1_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

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

POL(0) = 0   
POL(F17_IN(x₁, x₂)) = [2]x₁   
POL(F1_IN(x₁, x₂)) = [2]x₁   
POL(F29_IN(x₁, x₂)) = [2]x₁   
POL(F35_IN(x₁, x₂)) = [2]x₁   
POL(F43_IN(x₁, x₂)) = 0   
POL(F44_IN(x₁, x₂)) = [1] + [2]x₁   
POL(F54_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = [2] + x₁   
POL(U3(x₁, x₂, x₃)) = [1] + x₁   
POL(U6'(x₁, x₂, x₃)) = [2]x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(c15(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f43_in(x₁, x₂)) = x₁   
POL(f43_out1(x₁)) = [1] + x₁   
POL(f54_in(x₁, x₂)) = x₁   
POL(f54_out1(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f17_in(z0, z1), z0, z1)
U1(f17_out1, z0, z1) → f1_out1
U1(f17_out2, z0, z1) → f1_out1
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)
f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
f44_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f44_out1
f29_in(z0, z1) → U5(f35_in(z0, z1), z0, z1)
U5(f35_out1(z0), z1, z2) → f29_out1
f30_in(z0, z1) → f30_out1
f35_in(z0, z1) → U6(f43_in(z0, z1), z0, z1)
U6(f43_out1(z0), z1, z2) → U7(f44_in(z0, z2), z1, z2, z0)
U7(f44_out1, z0, z1, z2) → f35_out1(z2)
f17_in(z0, z1) → U8(f29_in(z0, z1), f30_in(z0, z1), z0, z1)
U8(f29_out1, z0, z1, z2) → f17_out1
U8(z0, f30_out1, z1, z2) → f17_out2

Tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F44_IN(z0, z1) → c9(F1_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

S tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

K tuples:

F44_IN(z0, z1) → c9(F1_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f54_in, U2, f43_in, U3, f44_in, U4, f29_in, U5, f30_in, f35_in, U6, U7, f17_in, U8

Defined Pair Symbols:

F35_IN, F43_IN, F1_IN, F54_IN, F44_IN, F29_IN, U6', F17_IN

Compound Symbols:

c14, c, c1, c5, c9, c11, c15, c17, c

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F43_IN(s(z0), s(z1)) → c

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f17_in(z0, z1), z0, z1)
U1(f17_out1, z0, z1) → f1_out1
U1(f17_out2, z0, z1) → f1_out1
f54_in(z0, 0) → f54_out1(z0)
f54_in(s(z0), s(z1)) → U2(f54_in(z0, z1), s(z0), s(z1))
U2(f54_out1(z0), s(z1), s(z2)) → f54_out1(z0)
f43_in(s(z0), s(z1)) → U3(f54_in(z0, z1), s(z0), s(z1))
U3(f54_out1(z0), s(z1), s(z2)) → f43_out1(z0)
f44_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f44_out1
f29_in(z0, z1) → U5(f35_in(z0, z1), z0, z1)
U5(f35_out1(z0), z1, z2) → f29_out1
f30_in(z0, z1) → f30_out1
f35_in(z0, z1) → U6(f43_in(z0, z1), z0, z1)
U6(f43_out1(z0), z1, z2) → U7(f44_in(z0, z2), z1, z2, z0)
U7(f44_out1, z0, z1, z2) → f35_out1(z2)
f17_in(z0, z1) → U8(f29_in(z0, z1), f30_in(z0, z1), z0, z1)
U8(f29_out1, z0, z1, z2) → f17_out1
U8(z0, f30_out1, z1, z2) → f17_out2

Tuples:

F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))
F44_IN(z0, z1) → c9(F1_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

S tuples:

F54_IN(s(z0), s(z1)) → c5(F54_IN(z0, z1))

K tuples:

F44_IN(z0, z1) → c9(F1_IN(z0, z1))
U6'(f43_out1(z0), z1, z2) → c15(F44_IN(z0, z2))
F1_IN(z0, z1) → c1(F17_IN(z0, z1))
F17_IN(z0, z1) → c17(F29_IN(z0, z1))
F29_IN(z0, z1) → c11(F35_IN(z0, z1))
F35_IN(z0, z1) → c14(U6'(f43_in(z0, z1), z0, z1), F43_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c(F54_IN(z0, z1))
F43_IN(s(z0), s(z1)) → c

Defined Rule Symbols:

f1_in, U1, f54_in, U2, f43_in, U3, f44_in, U4, f29_in, U5, f30_in, f35_in, U6, U7, f17_in, U8

Defined Pair Symbols:

F35_IN, F43_IN, F1_IN, F54_IN, F44_IN, F29_IN, U6', F17_IN

Compound Symbols:

c14, c, c1, c5, c9, c11, c15, c17, c

(11) 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: div(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, fail) | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, ','(=(T17, 0), =(T18, 0))) | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT14 is ground\nX4 is free\nX5 is free\nX6 is free\ndiv(0, T14, T3, T4) !=? div(X4, 0, X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T1, T2, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(=(T17, 0), =(T18, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: =(T19, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))) | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX17 is free\nX18 is free\nX19 is free\nX20 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T1, T2, T3, T4) !=? div(0, X10, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(X17, X18, s(X19), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 4 | ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5 | ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5 | ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(minus(T39, T40, X42), ','(!_1, div(X42, s(T40), T28, T29)))\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: minus(T39, T40, X42)\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(!_1, div(T43, s(T40), T28, T29))\n\nT40 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: minus(T39, T40, X42), SCOPE: 3, CLAUSE: 4 | minus(T39, T40, X42), SCOPE: 3, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true | minus(T48, 0, X42), SCOPE: 3, CLAUSE: 5\n\nT48 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: minus(T39, T40, X42), SCOPE: 3, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX42 is free\nX49 is free\nminus(T39, T40, X42) !=? minus(X49, 0, X49)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: minus(T48, 0, X42), SCOPE: 3, CLAUSE: 5\n\nT48 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: minus(T54, T55, X64)\n\nT54 is ground\nT55 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: div(T43, s(T40), T28, T29)\n\nT40 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 72 [arrowhead = none ];
72 -> 4;

73 [label="EVAL with clause\ndiv(X4, 0, X5, X6) :- ','(!_1, fail).\nand substitutionT1 -> T8,\nX4 -> T8,\nT2 -> 0,\nT3 -> T9,\nX5 -> T9,\nT4 -> T10,\nX6 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 75 [arrowhead = none ];
75 -> 11;

76 [label="EVAL with clause\ndiv(0, X10, X11, X12) :- ','(!_1, ','(=(X11, 0), =(X12, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T17,\nX11 -> T17,\nT4 -> T18,\nX12 -> T18,\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 77 [arrowhead = none ];
77 -> 15;

78 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 79 [arrowhead = none ];
79 -> 16;

80 [label="EVAL with clause\ndiv(X17, X18, s(X19), X20) :- ','(minus(X17, X18, X21), ','(!_1, div(X21, X18, X19, X20))).\nand substitutionT1 -> T24,\nX17 -> T24,\nT2 -> T25,\nX18 -> T25,\nX19 -> T28,\nT3 -> s(T28),\nT4 -> T29,\nX20 -> T29,\nT26 -> T28,\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 80 [arrowhead = none ];
80 -> 31;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 81 [arrowhead = none ];
81 -> 32;

82 [label="UNIFY CASE with substitutionT17 -> 0,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="UNIFY CASE with substitutionT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 84 [arrowhead = none ];
84 -> 24;

85 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 85 [arrowhead = none ];
85 -> 26;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 86 [arrowhead = none ];
86 -> 28;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 87 [arrowhead = none ];
87 -> 33;

88 [label="EVAL with clause\ndiv(X78, X79, 0, X78).\nand substitutionT1 -> T67,\nX78 -> T67,\nT2 -> T68,\nX79 -> T68,\nT3 -> 0,\nT4 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 88 [arrowhead = none ];
88 -> 68;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 89 [arrowhead = none ];
89 -> 70;

90 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T24, T25, T3, T4), div(X4, 0, X5, X6))", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 90 [arrowhead = none ];
90 -> 34;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 91 [arrowhead = none ];
91 -> 37;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 92 [arrowhead = none ];
92 -> 38;

93 [label="EVAL with clause\nminus(s(X39), s(X40), X41) :- minus(X39, X40, X41).\nand substitutionX39 -> T39,\nT24 -> s(T39),\nX40 -> T40,\nT25 -> s(T40),\nX21 -> X42,\nX41 -> X42", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 94 [arrowhead = none ];
94 -> 40;

95 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 95 [arrowhead = none ];
95 -> 61;

96 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
39 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT40 is ground\nT43 is ground\nreplacements:X42 -> T43", fontsize=14, style = filled, fillcolor = violet];
39 -> 97 [arrowhead = none ];
97 -> 42;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 98 [arrowhead = none ];
98 -> 45;

99 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 99 [arrowhead = none ];
99 -> 57;

100 [label="EVAL with clause\nminus(X49, 0, X49).\nand substitutionT39 -> T48,\nX49 -> T48,\nT40 -> 0,\nX42 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 101 [arrowhead = none ];
101 -> 47;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="EVAL with clause\nminus(s(X61), s(X62), X63) :- minus(X61, X62, X63).\nand substitutionX61 -> T54,\nT39 -> s(T54),\nX62 -> T55,\nT40 -> s(T55),\nX42 -> X64,\nX63 -> X64", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 103 [arrowhead = none ];
103 -> 52;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 104 [arrowhead = none ];
104 -> 53;

105 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="INSTANCE with matching:\nT39 -> T54\nT40 -> T55\nX42 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 106 [arrowhead = none , style=dashed];
106 -> 41[style=dashed];

107 [label="INSTANCE with matching:\nT1 -> T43\nT2 -> s(T40)\nT3 -> T28\nT4 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 107 [arrowhead = none , style=dashed];
107 -> 2[style=dashed];

108 [label="EVAL with clause\ndiv(X74, X75, 0, X74).\nand substitutionT24 -> T63,\nX74 -> T63,\nT25 -> T64,\nX75 -> T64,\nT3 -> 0,\nT4 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 108 [arrowhead = none ];
108 -> 63;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 109 [arrowhead = none ];
109 -> 65;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 110 [arrowhead = none ];
110 -> 66;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 111 [arrowhead = none ];
111 -> 71;

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f34_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f34_out1(z0), z1, z2) → f2_out1
U1(f34_out3, z0, z1) → f2_out1
f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)
f42_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f42_out1
f37_in(s(z0), s(z1)) → U4(f39_in(z0, z1), s(z0), s(z1))
U4(f39_out1(z0), s(z1), s(z2)) → f37_out1(z0)
f38_in(z0, z1) → f38_out2
f39_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → U6(f42_in(z0, z2), z1, z2, z0)
U6(f42_out1, z0, z1, z2) → f39_out1(z2)
f34_in(z0, z1) → U7(f37_in(z0, z1), f38_in(z0, z1), z0, z1)
U7(f37_out1(z0), z1, z2, z3) → f34_out1(z0)
U7(z0, f38_out2, z1, z2) → f34_out3

Tuples:

F2_IN(z0, z1) → c1(U1'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(U2'(f41_in(z0, z1), s(z0), s(z1)), F41_IN(z0, z1))
F42_IN(z0, z1) → c8(U3'(f2_in(z0, s(z1)), z0, z1), F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(U4'(f39_in(z0, z1), s(z0), s(z1)), F39_IN(z0, z1))
F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(U6'(f42_in(z0, z2), z1, z2, z0), F42_IN(z0, z2))
F34_IN(z0, z1) → c16(U7'(f37_in(z0, z1), f38_in(z0, z1), z0, z1), F37_IN(z0, z1), F38_IN(z0, z1))

S tuples:

F2_IN(z0, z1) → c1(U1'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(U2'(f41_in(z0, z1), s(z0), s(z1)), F41_IN(z0, z1))
F42_IN(z0, z1) → c8(U3'(f2_in(z0, s(z1)), z0, z1), F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(U4'(f39_in(z0, z1), s(z0), s(z1)), F39_IN(z0, z1))
F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(U6'(f42_in(z0, z2), z1, z2, z0), F42_IN(z0, z2))
F34_IN(z0, z1) → c16(U7'(f37_in(z0, z1), f38_in(z0, z1), z0, z1), F37_IN(z0, z1), F38_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f41_in, U2, f42_in, U3, f37_in, U4, f38_in, f39_in, U5, U6, f34_in, U7

Defined Pair Symbols:

F2_IN, F41_IN, F42_IN, F37_IN, F39_IN, U5', F34_IN

Compound Symbols:

c1, c6, c8, c10, c13, c14, c16

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

Removed 7 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f34_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f34_out1(z0), z1, z2) → f2_out1
U1(f34_out3, z0, z1) → f2_out1
f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)
f42_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f42_out1
f37_in(s(z0), s(z1)) → U4(f39_in(z0, z1), s(z0), s(z1))
U4(f39_out1(z0), s(z1), s(z2)) → f37_out1(z0)
f38_in(z0, z1) → f38_out2
f39_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → U6(f42_in(z0, z2), z1, z2, z0)
U6(f42_out1, z0, z1, z2) → f39_out1(z2)
f34_in(z0, z1) → U7(f37_in(z0, z1), f38_in(z0, z1), z0, z1)
U7(f37_out1(z0), z1, z2, z3) → f34_out1(z0)
U7(z0, f38_out2, z1, z2) → f34_out3

Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

S tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f41_in, U2, f42_in, U3, f37_in, U4, f38_in, f39_in, U5, U6, f34_in, U7

Defined Pair Symbols:

F39_IN, F2_IN, F41_IN, F42_IN, F37_IN, U5', F34_IN

Compound Symbols:

c13, c1, c6, c8, c10, c14, c16

(15) 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.

F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))

We considered the (Usable) Rules:

f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)

And the Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

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

POL(0) = 0   
POL(F2_IN(x₁, x₂)) = x₁   
POL(F34_IN(x₁, x₂)) = x₁   
POL(F37_IN(x₁, x₂)) = x₁   
POL(F39_IN(x₁, x₂)) = x₁   
POL(F41_IN(x₁, x₂)) = 0   
POL(F42_IN(x₁, x₂)) = x₁   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U5'(x₁, x₂, x₃)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(f41_in(x₁, x₂)) = x₁   
POL(f41_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f34_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f34_out1(z0), z1, z2) → f2_out1
U1(f34_out3, z0, z1) → f2_out1
f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)
f42_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f42_out1
f37_in(s(z0), s(z1)) → U4(f39_in(z0, z1), s(z0), s(z1))
U4(f39_out1(z0), s(z1), s(z2)) → f37_out1(z0)
f38_in(z0, z1) → f38_out2
f39_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → U6(f42_in(z0, z2), z1, z2, z0)
U6(f42_out1, z0, z1, z2) → f39_out1(z2)
f34_in(z0, z1) → U7(f37_in(z0, z1), f38_in(z0, z1), z0, z1)
U7(f37_out1(z0), z1, z2, z3) → f34_out1(z0)
U7(z0, f38_out2, z1, z2) → f34_out3

Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

S tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

K tuples:

F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f41_in, U2, f42_in, U3, f37_in, U4, f38_in, f39_in, U5, U6, f34_in, U7

Defined Pair Symbols:

F39_IN, F2_IN, F41_IN, F42_IN, F37_IN, U5', F34_IN

Compound Symbols:

c13, c1, c6, c8, c10, c14, c16

(17) CdtKnowledgeProof (EQUIVALENT transformation)

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

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f34_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f34_out1(z0), z1, z2) → f2_out1
U1(f34_out3, z0, z1) → f2_out1
f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)
f42_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f42_out1
f37_in(s(z0), s(z1)) → U4(f39_in(z0, z1), s(z0), s(z1))
U4(f39_out1(z0), s(z1), s(z2)) → f37_out1(z0)
f38_in(z0, z1) → f38_out2
f39_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → U6(f42_in(z0, z2), z1, z2, z0)
U6(f42_out1, z0, z1, z2) → f39_out1(z2)
f34_in(z0, z1) → U7(f37_in(z0, z1), f38_in(z0, z1), z0, z1)
U7(f37_out1(z0), z1, z2, z3) → f34_out1(z0)
U7(z0, f38_out2, z1, z2) → f34_out3

Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

S tuples:

F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))

K tuples:

F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f41_in, U2, f42_in, U3, f37_in, U4, f38_in, f39_in, U5, U6, f34_in, U7

Defined Pair Symbols:

F39_IN, F2_IN, F41_IN, F42_IN, F37_IN, U5', F34_IN

Compound Symbols:

c13, c1, c6, c8, c10, c14, c16

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

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

F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))

We considered the (Usable) Rules:

f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)

And the Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

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

POL(0) = 0   
POL(F2_IN(x₁, x₂)) = x₁²   
POL(F34_IN(x₁, x₂)) = x₁²   
POL(F37_IN(x₁, x₂)) = x₁²   
POL(F39_IN(x₁, x₂)) = [1] + x₁ + x₁²   
POL(F41_IN(x₁, x₂)) = x₁   
POL(F42_IN(x₁, x₂)) = x₁²   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U5'(x₁, x₂, x₃)) = x₁²   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c13(x₁, x₂)) = x₁ + x₂   
POL(c14(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(f41_in(x₁, x₂)) = x₁   
POL(f41_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f34_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f34_out1(z0), z1, z2) → f2_out1
U1(f34_out3, z0, z1) → f2_out1
f41_in(z0, 0) → f41_out1(z0)
f41_in(s(z0), s(z1)) → U2(f41_in(z0, z1), s(z0), s(z1))
U2(f41_out1(z0), s(z1), s(z2)) → f41_out1(z0)
f42_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f42_out1
f37_in(s(z0), s(z1)) → U4(f39_in(z0, z1), s(z0), s(z1))
U4(f39_out1(z0), s(z1), s(z2)) → f37_out1(z0)
f38_in(z0, z1) → f38_out2
f39_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → U6(f42_in(z0, z2), z1, z2, z0)
U6(f42_out1, z0, z1, z2) → f39_out1(z2)
f34_in(z0, z1) → U7(f37_in(z0, z1), f38_in(z0, z1), z0, z1)
U7(f37_out1(z0), z1, z2, z3) → f34_out1(z0)
U7(z0, f38_out2, z1, z2) → f34_out3

Tuples:

F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))

S tuples:none
K tuples:

F37_IN(s(z0), s(z1)) → c10(F39_IN(z0, z1))
F39_IN(z0, z1) → c13(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
U5'(f41_out1(z0), z1, z2) → c14(F42_IN(z0, z2))
F42_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F34_IN(z0, z1))
F34_IN(z0, z1) → c16(F37_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c6(F41_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f41_in, U2, f42_in, U3, f37_in, U4, f38_in, f39_in, U5, U6, f34_in, U7

Defined Pair Symbols:

F39_IN, F2_IN, F41_IN, F42_IN, F37_IN, U5', F34_IN

Compound Symbols:

c13, c1, c6, c8, c10, c14, c16

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))