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(X1, 0, X2, X3) :- ','(!, failure(a)).
div(0, X4, Z, R) :- ','(!, ','(eq(Z, 0), eq(R, 0))).
div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))).
div(X, X5, X6, X).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
failure(b).
eq(X, X).

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, failure(a)) | 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\nX13 is free\nX14 is free\nX15 is free\ndiv(T1, T2, T3, T4) !=? div(X13, 0, X14, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: failure(a), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(!_1, ','(eq(T23, 0), eq(T24, 0))) | div(0, T20, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T20, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT20 is ground\nX13 is free\nX14 is free\nX15 is free\ndiv(0, T20, T3, T4) !=? div(X13, 0, X14, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: 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\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\ndiv(T1, T2, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T1, T2, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(eq(T23, 0), eq(T24, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(eq(T23, 0), eq(T24, 0)), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq(T28, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: eq(T28, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\ndiv(T1, T2, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T1, T2, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\ndiv(T1, T2, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T1, T2, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(minus(T48, T49, X53), ','(!_1, div(X53, T49, T52, T53)))\n\nT48 is ground\nT49 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\nX53 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: minus(T48, T49, X53)\n\nT48 is ground\nT49 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\nX53 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(!_1, div(T58, T49, T52, T53))\n\nT48 is ground\nT49 is ground\nT58 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: minus(T48, T49, X53), SCOPE: 5, CLAUSE: 4 | minus(T48, T49, X53), SCOPE: 5, CLAUSE: 5\n\nT48 is ground\nT49 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\nX53 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: minus(T48, T49, X53), SCOPE: 5, CLAUSE: 5\n\nT48 is ground\nT49 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\nX53 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: minus(T68, T69, X78)\n\nT68 is ground\nT69 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: minus(T68, T69, X78), SCOPE: 6, CLAUSE: 4 | minus(T68, T69, X78), SCOPE: 6, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true | minus(T74, 0, X78), SCOPE: 6, CLAUSE: 5\n\nT74 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: minus(T68, T69, X78), SCOPE: 6, CLAUSE: 5\n\nT68 is ground\nT69 is ground\nX78 is free\nX83 is free\nminus(T68, T69, X78) !=? minus(X83, 0, X83)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: minus(T74, 0, X78), SCOPE: 6, CLAUSE: 5\n\nT74 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: minus(T80, T81, X98)\n\nT80 is ground\nT81 is ground\nX98 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: div(T58, T49, T52, T53)\n\nT48 is ground\nT49 is ground\nT58 is ground\nX13 is free\nX14 is free\nX15 is free\nX22 is free\nX23 is free\nX24 is free\ndiv(T48, T49, T3, T4) !=? div(X13, 0, X14, X15)\ndiv(T48, T49, T3, T4) !=? div(0, X22, X23, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 81 [arrowhead = none ];
81 -> 3;

82 [label="EVAL with clause\ndiv(X13, 0, X14, X15) :- ','(!_1, failure(a)).\nand substitutionT1 -> T11,\nX13 -> T11,\nT2 -> 0,\nT3 -> T12,\nX14 -> T12,\nT4 -> T13,\nX15 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 83 [arrowhead = none ];
83 -> 7;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL with clause\ndiv(0, X22, X23, X24) :- ','(!_1, ','(eq(X23, 0), eq(X24, 0))).\nand substitutionT1 -> 0,\nT2 -> T20,\nX22 -> T20,\nT3 -> T23,\nX23 -> T23,\nT4 -> T24,\nX24 -> T24,\nT21 -> T23,\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 85 [arrowhead = none ];
85 -> 16;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 19;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 88 [arrowhead = none ];
88 -> 14;

89 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 89 [arrowhead = none ];
89 -> 21;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 90 [arrowhead = none ];
90 -> 35;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 91 [arrowhead = none ];
91 -> 37;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 92 [arrowhead = none ];
92 -> 22;

93 [label="EVAL with clause\neq(X27, X27).\nand substitutionT23 -> 0,\nX27 -> 0,\nT27 -> 0,\nT24 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 93 [arrowhead = none ];
93 -> 25;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 94 [arrowhead = none ];
94 -> 27;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 95 [arrowhead = none ];
95 -> 28;

96 [label="EVAL with clause\neq(X30, X30).\nand substitutionT28 -> 0,\nX30 -> 0,\nT31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 97 [arrowhead = none ];
97 -> 33;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 98 [arrowhead = none ];
98 -> 34;

99 [label="EVAL with clause\ndiv(X49, X50, s(X51), X52) :- ','(minus(X49, X50, X53), ','(!_1, div(X53, X50, X51, X52))).\nand substitutionT1 -> T48,\nX49 -> T48,\nT2 -> T49,\nX50 -> T49,\nX51 -> T52,\nT3 -> s(T52),\nT4 -> T53,\nX52 -> T53,\nT50 -> T52,\nT51 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 99 [arrowhead = none ];
99 -> 41;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 100 [arrowhead = none ];
100 -> 44;

101 [label="EVAL with clause\ndiv(X113, X114, X115, X113).\nand substitutionT1 -> T96,\nX113 -> T96,\nT2 -> T97,\nX114 -> T97,\nT3 -> T98,\nX115 -> T98,\nT4 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 101 [arrowhead = none ];
101 -> 78;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 102 [arrowhead = none ];
102 -> 79;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 103 [arrowhead = none ];
103 -> 49;

104 [label="SPLIT 2\nnew knowledge:\nT48 is ground\nT49 is ground\nT58 is ground\nreplacements:X53 -> T58", fontsize=14, style = filled, fillcolor = violet];
41 -> 104 [arrowhead = none ];
104 -> 50;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 105 [arrowhead = none ];
105 -> 55;

106 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 106 [arrowhead = none ];
106 -> 74;

107 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T48, T49, T3, T4), div(X13, 0, X14, X15))", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 107 [arrowhead = none ];
107 -> 57;

108 [label="EVAL with clause\nminus(s(X75), s(X76), X77) :- minus(X75, X76, X77).\nand substitutionX75 -> T68,\nT48 -> s(T68),\nX76 -> T69,\nT49 -> s(T69),\nX53 -> X78,\nX77 -> X78", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 108 [arrowhead = none ];
108 -> 60;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 109 [arrowhead = none ];
109 -> 61;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 110 [arrowhead = none ];
110 -> 63;

111 [label="EVAL with clause\nminus(X83, 0, X83).\nand substitutionT68 -> T74,\nX83 -> T74,\nT69 -> 0,\nX78 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 111 [arrowhead = none ];
111 -> 64;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 112 [arrowhead = none ];
112 -> 65;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 113 [arrowhead = none ];
113 -> 66;

114 [label="EVAL with clause\nminus(s(X95), s(X96), X97) :- minus(X95, X96, X97).\nand substitutionX95 -> T80,\nT68 -> s(T80),\nX96 -> T81,\nT69 -> s(T81),\nX78 -> X98,\nX97 -> X98", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 114 [arrowhead = none ];
114 -> 69;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 115 [arrowhead = none ];
115 -> 70;

116 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 116 [arrowhead = none ];
116 -> 68;

117 [label="INSTANCE with matching:\nT68 -> T80\nT69 -> T81\nX78 -> X98", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 117 [arrowhead = none , style=dashed];
117 -> 60[style=dashed];

118 [label="INSTANCE with matching:\nT1 -> T58\nT2 -> T49\nT3 -> T52\nT4 -> T53", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 118 [arrowhead = none , style=dashed];
118 -> 1[style=dashed];

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 119 [arrowhead = none ];
119 -> 80;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f19_in(z0, z1), z0, z1)
U1(f19_out1, z0, z1) → f1_out1
U1(f19_out2, z0, z1) → f1_out1
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)
f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
f50_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f50_out1
f35_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → f35_out1
f37_in(z0, z1) → f37_out1
f41_in(z0, z1) → U6(f49_in(z0, z1), z0, z1)
U6(f49_out1(z0), z1, z2) → U7(f50_in(z0, z2), z1, z2, z0)
U7(f50_out1, z0, z1, z2) → f41_out1(z2)
f19_in(z0, z1) → U8(f35_in(z0, z1), f37_in(z0, z1), z0, z1)
U8(f35_out1, z0, z1, z2) → f19_out1
U8(z0, f37_out1, z1, z2) → f19_out2

Tuples:

F1_IN(z0, z1) → c1(U1'(f19_in(z0, z1), z0, z1), F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(U2'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c7(U3'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F50_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F35_IN(z0, z1) → c11(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(U7'(f50_in(z0, z2), z1, z2, z0), F50_IN(z0, z2))
F19_IN(z0, z1) → c17(U8'(f35_in(z0, z1), f37_in(z0, z1), z0, z1), F35_IN(z0, z1), F37_IN(z0, z1))

S tuples:

F1_IN(z0, z1) → c1(U1'(f19_in(z0, z1), z0, z1), F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(U2'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c7(U3'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F50_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F35_IN(z0, z1) → c11(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(U7'(f50_in(z0, z2), z1, z2, z0), F50_IN(z0, z2))
F19_IN(z0, z1) → c17(U8'(f35_in(z0, z1), f37_in(z0, z1), z0, z1), F35_IN(z0, z1), F37_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f60_in, U2, f49_in, U3, f50_in, U4, f35_in, U5, f37_in, f41_in, U6, U7, f19_in, U8

Defined Pair Symbols:

F1_IN, F60_IN, F49_IN, F50_IN, F35_IN, F41_IN, U6', F19_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(f19_in(z0, z1), z0, z1)
U1(f19_out1, z0, z1) → f1_out1
U1(f19_out2, z0, z1) → f1_out1
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)
f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
f50_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f50_out1
f35_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → f35_out1
f37_in(z0, z1) → f37_out1
f41_in(z0, z1) → U6(f49_in(z0, z1), z0, z1)
U6(f49_out1(z0), z1, z2) → U7(f50_in(z0, z2), z1, z2, z0)
U7(f50_out1, z0, z1, z2) → f41_out1(z2)
f19_in(z0, z1) → U8(f35_in(z0, z1), f37_in(z0, z1), z0, z1)
U8(f35_out1, z0, z1, z2) → f19_out1
U8(z0, f37_out1, z1, z2) → f19_out2

Tuples:

F1_IN(z0, z1) → c1(U1'(f19_in(z0, z1), z0, z1), F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(U2'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F50_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F35_IN(z0, z1) → c11(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(U7'(f50_in(z0, z2), z1, z2, z0), F50_IN(z0, z2))
F19_IN(z0, z1) → c17(U8'(f35_in(z0, z1), f37_in(z0, z1), z0, z1), F35_IN(z0, z1), F37_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(U3'(f60_in(z0, z1), s(z0), s(z1)))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))

S tuples:

F1_IN(z0, z1) → c1(U1'(f19_in(z0, z1), z0, z1), F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(U2'(f60_in(z0, z1), s(z0), s(z1)), F60_IN(z0, z1))
F50_IN(z0, z1) → c9(U4'(f1_in(z0, z1), z0, z1), F1_IN(z0, z1))
F35_IN(z0, z1) → c11(U5'(f41_in(z0, z1), z0, z1), F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(U7'(f50_in(z0, z2), z1, z2, z0), F50_IN(z0, z2))
F19_IN(z0, z1) → c17(U8'(f35_in(z0, z1), f37_in(z0, z1), z0, z1), F35_IN(z0, z1), F37_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(U3'(f60_in(z0, z1), s(z0), s(z1)))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f60_in, U2, f49_in, U3, f50_in, U4, f35_in, U5, f37_in, f41_in, U6, U7, f19_in, U8

Defined Pair Symbols:

F1_IN, F60_IN, F50_IN, F35_IN, F41_IN, U6', F19_IN, F49_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(f19_in(z0, z1), z0, z1)
U1(f19_out1, z0, z1) → f1_out1
U1(f19_out2, z0, z1) → f1_out1
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)
f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
f50_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f50_out1
f35_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → f35_out1
f37_in(z0, z1) → f37_out1
f41_in(z0, z1) → U6(f49_in(z0, z1), z0, z1)
U6(f49_out1(z0), z1, z2) → U7(f50_in(z0, z2), z1, z2, z0)
U7(f50_out1, z0, z1, z2) → f41_out1(z2)
f19_in(z0, z1) → U8(f35_in(z0, z1), f37_in(z0, z1), z0, z1)
U8(f35_out1, z0, z1, z2) → f19_out1
U8(z0, f37_out1, z1, z2) → f19_out2

Tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

S tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f60_in, U2, f49_in, U3, f50_in, U4, f35_in, U5, f37_in, f41_in, U6, U7, f19_in, U8

Defined Pair Symbols:

F41_IN, F49_IN, F1_IN, F60_IN, F50_IN, F35_IN, U6', F19_IN

Compound Symbols:

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

(7) 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.

F1_IN(z0, z1) → c1(F19_IN(z0, z1))

We considered the (Usable) Rules:

f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)

And the Tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

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

POL(0) = 0   
POL(F19_IN(x₁, x₂)) = x₂² + x₁²   
POL(F1_IN(x₁, x₂)) = [1] + x₂² + x₁²   
POL(F35_IN(x₁, x₂)) = x₂² + x₁²   
POL(F41_IN(x₁, x₂)) = x₂² + x₁²   
POL(F49_IN(x₁, x₂)) = 0   
POL(F50_IN(x₁, x₂)) = [1] + x₂² + x₁²   
POL(F60_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3(x₁, x₂, x₃)) = [1] + x₁   
POL(U6'(x₁, x₂, x₃)) = x₃² + x₁²   
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(f49_in(x₁, x₂)) = x₁   
POL(f49_out1(x₁)) = [1] + x₁   
POL(f60_in(x₁, x₂)) = x₁   
POL(f60_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f19_in(z0, z1), z0, z1)
U1(f19_out1, z0, z1) → f1_out1
U1(f19_out2, z0, z1) → f1_out1
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)
f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
f50_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f50_out1
f35_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → f35_out1
f37_in(z0, z1) → f37_out1
f41_in(z0, z1) → U6(f49_in(z0, z1), z0, z1)
U6(f49_out1(z0), z1, z2) → U7(f50_in(z0, z2), z1, z2, z0)
U7(f50_out1, z0, z1, z2) → f41_out1(z2)
f19_in(z0, z1) → U8(f35_in(z0, z1), f37_in(z0, z1), z0, z1)
U8(f35_out1, z0, z1, z2) → f19_out1
U8(z0, f37_out1, z1, z2) → f19_out2

Tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

S tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

K tuples:

F1_IN(z0, z1) → c1(F19_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f60_in, U2, f49_in, U3, f50_in, U4, f35_in, U5, f37_in, f41_in, U6, U7, f19_in, U8

Defined Pair Symbols:

F41_IN, F49_IN, F1_IN, F60_IN, F50_IN, F35_IN, U6', F19_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:

F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F49_IN(s(z0), s(z1)) → c
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(z0, z1) → U1(f19_in(z0, z1), z0, z1)
U1(f19_out1, z0, z1) → f1_out1
U1(f19_out2, z0, z1) → f1_out1
f60_in(z0, 0) → f60_out1(z0)
f60_in(s(z0), s(z1)) → U2(f60_in(z0, z1), s(z0), s(z1))
U2(f60_out1(z0), s(z1), s(z2)) → f60_out1(z0)
f49_in(s(z0), s(z1)) → U3(f60_in(z0, z1), s(z0), s(z1))
U3(f60_out1(z0), s(z1), s(z2)) → f49_out1(z0)
f50_in(z0, z1) → U4(f1_in(z0, z1), z0, z1)
U4(f1_out1, z0, z1) → f50_out1
f35_in(z0, z1) → U5(f41_in(z0, z1), z0, z1)
U5(f41_out1(z0), z1, z2) → f35_out1
f37_in(z0, z1) → f37_out1
f41_in(z0, z1) → U6(f49_in(z0, z1), z0, z1)
U6(f49_out1(z0), z1, z2) → U7(f50_in(z0, z2), z1, z2, z0)
U7(f50_out1, z0, z1, z2) → f41_out1(z2)
f19_in(z0, z1) → U8(f35_in(z0, z1), f37_in(z0, z1), z0, z1)
U8(f35_out1, z0, z1, z2) → f19_out1
U8(z0, f37_out1, z1, z2) → f19_out2

Tuples:

F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))
F50_IN(z0, z1) → c9(F1_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c

S tuples:

F60_IN(s(z0), s(z1)) → c5(F60_IN(z0, z1))

K tuples:

F1_IN(z0, z1) → c1(F19_IN(z0, z1))
F19_IN(z0, z1) → c17(F35_IN(z0, z1))
F35_IN(z0, z1) → c11(F41_IN(z0, z1))
F41_IN(z0, z1) → c14(U6'(f49_in(z0, z1), z0, z1), F49_IN(z0, z1))
F49_IN(s(z0), s(z1)) → c(F60_IN(z0, z1))
U6'(f49_out1(z0), z1, z2) → c15(F50_IN(z0, z2))
F49_IN(s(z0), s(z1)) → c
F50_IN(z0, z1) → c9(F1_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f60_in, U2, f49_in, U3, f50_in, U4, f35_in, U5, f37_in, f41_in, U6, U7, f19_in, U8

Defined Pair Symbols:

F41_IN, F49_IN, F1_IN, F60_IN, F50_IN, F35_IN, U6', F19_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, failure(a)) | 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];
8 [label="8: 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\nX10 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: failure(a), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(!_1, ','(eq(T17, 0), eq(T18, 0))) | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT14 is ground\nX10 is free\nX11 is free\nX12 is free\ndiv(0, T14, T3, T4) !=? div(X10, 0, X11, X12)", 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\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(eq(T17, 0), eq(T18, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(eq(T17, 0), eq(T18, 0)), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: eq(T22, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: eq(T22, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))) | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)\ndiv(T1, T2, T3, T4) !=? div(X29, X30, s(X31), X32)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 4 | ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(minus(T45, T46, X54), ','(!_1, div(X54, s(T46), T34, T35)))\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: minus(T45, T46, X54)\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(!_1, div(T49, s(T46), T34, T35))\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 4 | minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true | minus(T54, 0, X54), SCOPE: 6, CLAUSE: 5\n\nT54 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free\nX61 is free\nminus(T45, T46, X54) !=? minus(X61, 0, X61)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minus(T54, 0, X54), SCOPE: 6, CLAUSE: 5\n\nT54 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: minus(T60, T61, X76)\n\nT60 is ground\nT61 is ground\nX76 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: div(T49, s(T46), T34, T35)\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 78 [arrowhead = none ];
78 -> 4;

79 [label="EVAL with clause\ndiv(X10, 0, X11, X12) :- ','(!_1, failure(a)).\nand substitutionT1 -> T8,\nX10 -> T8,\nT2 -> 0,\nT3 -> T9,\nX11 -> T9,\nT4 -> T10,\nX12 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 79 [arrowhead = none ];
79 -> 6;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 80 [arrowhead = none ];
80 -> 8;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 81 [arrowhead = none ];
81 -> 9;

82 [label="EVAL with clause\ndiv(0, X16, X17, X18) :- ','(!_1, ','(eq(X17, 0), eq(X18, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX16 -> T14,\nT3 -> T17,\nX17 -> T17,\nT4 -> T18,\nX18 -> T18,\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 82 [arrowhead = none ];
82 -> 15;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 83 [arrowhead = none ];
83 -> 17;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 84 [arrowhead = none ];
84 -> 11;

85 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 85 [arrowhead = none ];
85 -> 13;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="EVAL with clause\ndiv(X29, X30, s(X31), X32) :- ','(minus(X29, X30, X33), ','(!_1, div(X33, X30, X31, X32))).\nand substitutionT1 -> T30,\nX29 -> T30,\nT2 -> T31,\nX30 -> T31,\nX31 -> T34,\nT3 -> s(T34),\nT4 -> T35,\nX32 -> T35,\nT32 -> T34,\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 87 [arrowhead = none ];
87 -> 36;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 88 [arrowhead = none ];
88 -> 38;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 89 [arrowhead = none ];
89 -> 20;

90 [label="EVAL with clause\neq(X21, X21).\nand substitutionT17 -> 0,\nX21 -> 0,\nT21 -> 0,\nT18 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 90 [arrowhead = none ];
90 -> 23;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 91 [arrowhead = none ];
91 -> 24;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 92 [arrowhead = none ];
92 -> 26;

93 [label="EVAL with clause\neq(X24, X24).\nand substitutionT22 -> 0,\nX24 -> 0,\nT25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 93 [arrowhead = none ];
93 -> 29;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 96 [arrowhead = none ];
96 -> 39;

97 [label="EVAL with clause\ndiv(X94, X95, X96, X94).\nand substitutionT1 -> T77,\nX94 -> T77,\nT2 -> T78,\nX95 -> T78,\nT3 -> T79,\nX96 -> T79,\nT4 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 97 [arrowhead = none ];
97 -> 75;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 98 [arrowhead = none ];
98 -> 76;

99 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T30, T31, T3, T4), div(X10, 0, X11, X12))", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 99 [arrowhead = none ];
99 -> 40;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 100 [arrowhead = none ];
100 -> 42;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 101 [arrowhead = none ];
101 -> 43;

102 [label="EVAL with clause\nminus(s(X51), s(X52), X53) :- minus(X51, X52, X53).\nand substitutionX51 -> T45,\nT30 -> s(T45),\nX52 -> T46,\nT31 -> s(T46),\nX33 -> X54,\nX53 -> X54", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 102 [arrowhead = none ];
102 -> 45;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 103 [arrowhead = none ];
103 -> 46;

104 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 104 [arrowhead = none ];
104 -> 67;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 105 [arrowhead = none ];
105 -> 47;

106 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X54 -> T49", fontsize=14, style = filled, fillcolor = violet];
45 -> 106 [arrowhead = none ];
106 -> 48;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 107 [arrowhead = none ];
107 -> 51;

108 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 108 [arrowhead = none ];
108 -> 62;

109 [label="EVAL with clause\nminus(X61, 0, X61).\nand substitutionT45 -> T54,\nX61 -> T54,\nT46 -> 0,\nX54 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 109 [arrowhead = none ];
109 -> 52;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 110 [arrowhead = none ];
110 -> 53;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 111 [arrowhead = none ];
111 -> 54;

112 [label="EVAL with clause\nminus(s(X73), s(X74), X75) :- minus(X73, X74, X75).\nand substitutionX73 -> T60,\nT45 -> s(T60),\nX74 -> T61,\nT46 -> s(T61),\nX54 -> X76,\nX75 -> X76", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 112 [arrowhead = none ];
112 -> 58;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 113 [arrowhead = none ];
113 -> 59;

114 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 114 [arrowhead = none ];
114 -> 56;

115 [label="INSTANCE with matching:\nT45 -> T60\nT46 -> T61\nX54 -> X76", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 115 [arrowhead = none , style=dashed];
115 -> 47[style=dashed];

116 [label="INSTANCE with matching:\nT1 -> T49\nT2 -> s(T46)\nT3 -> T34\nT4 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 116 [arrowhead = none , style=dashed];
116 -> 2[style=dashed];

117 [label="EVAL with clause\ndiv(X88, X89, X90, X88).\nand substitutionT30 -> T71,\nX88 -> T71,\nT31 -> T72,\nX89 -> T72,\nT3 -> T73,\nX90 -> T73,\nT4 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 117 [arrowhead = none ];
117 -> 71;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 118 [arrowhead = none ];
118 -> 72;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 119 [arrowhead = none ];
119 -> 73;

120 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 120 [arrowhead = none ];
120 -> 77;

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f40_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f40_out1(z0), z1, z2) → f2_out1
U1(f40_out3, z0, z1) → f2_out1
f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)
f48_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f48_out1
f42_in(s(z0), s(z1)) → U4(f45_in(z0, z1), s(z0), s(z1))
U4(f45_out1(z0), s(z1), s(z2)) → f42_out1(z0)
f43_in(z0, z1) → f43_out2
f45_in(z0, z1) → U5(f47_in(z0, z1), z0, z1)
U5(f47_out1(z0), z1, z2) → U6(f48_in(z0, z2), z1, z2, z0)
U6(f48_out1, z0, z1, z2) → f45_out1(z2)
f40_in(z0, z1) → U7(f42_in(z0, z1), f43_in(z0, z1), z0, z1)
U7(f42_out1(z0), z1, z2, z3) → f40_out1(z0)
U7(z0, f43_out2, z1, z2) → f40_out3

Tuples:

F2_IN(z0, z1) → c1(U1'(f40_in(z0, z1), z0, z1), F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(U2'(f47_in(z0, z1), s(z0), s(z1)), F47_IN(z0, z1))
F48_IN(z0, z1) → c8(U3'(f2_in(z0, s(z1)), z0, z1), F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(U4'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(U6'(f48_in(z0, z2), z1, z2, z0), F48_IN(z0, z2))
F40_IN(z0, z1) → c16(U7'(f42_in(z0, z1), f43_in(z0, z1), z0, z1), F42_IN(z0, z1), F43_IN(z0, z1))

S tuples:

F2_IN(z0, z1) → c1(U1'(f40_in(z0, z1), z0, z1), F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(U2'(f47_in(z0, z1), s(z0), s(z1)), F47_IN(z0, z1))
F48_IN(z0, z1) → c8(U3'(f2_in(z0, s(z1)), z0, z1), F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(U4'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(U6'(f48_in(z0, z2), z1, z2, z0), F48_IN(z0, z2))
F40_IN(z0, z1) → c16(U7'(f42_in(z0, z1), f43_in(z0, z1), z0, z1), F42_IN(z0, z1), F43_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f47_in, U2, f48_in, U3, f42_in, U4, f43_in, f45_in, U5, U6, f40_in, U7

Defined Pair Symbols:

F2_IN, F47_IN, F48_IN, F42_IN, F45_IN, U5', F40_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(f40_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f40_out1(z0), z1, z2) → f2_out1
U1(f40_out3, z0, z1) → f2_out1
f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)
f48_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f48_out1
f42_in(s(z0), s(z1)) → U4(f45_in(z0, z1), s(z0), s(z1))
U4(f45_out1(z0), s(z1), s(z2)) → f42_out1(z0)
f43_in(z0, z1) → f43_out2
f45_in(z0, z1) → U5(f47_in(z0, z1), z0, z1)
U5(f47_out1(z0), z1, z2) → U6(f48_in(z0, z2), z1, z2, z0)
U6(f48_out1, z0, z1, z2) → f45_out1(z2)
f40_in(z0, z1) → U7(f42_in(z0, z1), f43_in(z0, z1), z0, z1)
U7(f42_out1(z0), z1, z2, z3) → f40_out1(z0)
U7(z0, f43_out2, z1, z2) → f40_out3

Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

S tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f47_in, U2, f48_in, U3, f42_in, U4, f43_in, f45_in, U5, U6, f40_in, U7

Defined Pair Symbols:

F45_IN, F2_IN, F47_IN, F48_IN, F42_IN, U5', F40_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.

F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))

We considered the (Usable) Rules:

f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)

And the Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

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

POL(0) = 0   
POL(F2_IN(x₁, x₂)) = x₁ + x₂   
POL(F40_IN(x₁, x₂)) = x₁ + x₂   
POL(F42_IN(x₁, x₂)) = x₁ + x₂   
POL(F45_IN(x₁, x₂)) = [1] + x₁ + x₂   
POL(F47_IN(x₁, x₂)) = 0   
POL(F48_IN(x₁, x₂)) = [1] + x₁ + x₂   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U5'(x₁, x₂, x₃)) = x₁ + x₃   
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(f47_in(x₁, x₂)) = [1] + x₁   
POL(f47_out1(x₁)) = [1] + x₁   
POL(s(x₁)) = [1] + x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f40_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f40_out1(z0), z1, z2) → f2_out1
U1(f40_out3, z0, z1) → f2_out1
f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)
f48_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f48_out1
f42_in(s(z0), s(z1)) → U4(f45_in(z0, z1), s(z0), s(z1))
U4(f45_out1(z0), s(z1), s(z2)) → f42_out1(z0)
f43_in(z0, z1) → f43_out2
f45_in(z0, z1) → U5(f47_in(z0, z1), z0, z1)
U5(f47_out1(z0), z1, z2) → U6(f48_in(z0, z2), z1, z2, z0)
U6(f48_out1, z0, z1, z2) → f45_out1(z2)
f40_in(z0, z1) → U7(f42_in(z0, z1), f43_in(z0, z1), z0, z1)
U7(f42_out1(z0), z1, z2, z3) → f40_out1(z0)
U7(z0, f43_out2, z1, z2) → f40_out3

Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

S tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

K tuples:

F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f47_in, U2, f48_in, U3, f42_in, U4, f43_in, f45_in, U5, U6, f40_in, U7

Defined Pair Symbols:

F45_IN, F2_IN, F47_IN, F48_IN, F42_IN, U5', F40_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:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f40_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f40_out1(z0), z1, z2) → f2_out1
U1(f40_out3, z0, z1) → f2_out1
f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)
f48_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f48_out1
f42_in(s(z0), s(z1)) → U4(f45_in(z0, z1), s(z0), s(z1))
U4(f45_out1(z0), s(z1), s(z2)) → f42_out1(z0)
f43_in(z0, z1) → f43_out2
f45_in(z0, z1) → U5(f47_in(z0, z1), z0, z1)
U5(f47_out1(z0), z1, z2) → U6(f48_in(z0, z2), z1, z2, z0)
U6(f48_out1, z0, z1, z2) → f45_out1(z2)
f40_in(z0, z1) → U7(f42_in(z0, z1), f43_in(z0, z1), z0, z1)
U7(f42_out1(z0), z1, z2, z3) → f40_out1(z0)
U7(z0, f43_out2, z1, z2) → f40_out3

Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

S tuples:

F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))

K tuples:

F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f47_in, U2, f48_in, U3, f42_in, U4, f43_in, f45_in, U5, U6, f40_in, U7

Defined Pair Symbols:

F45_IN, F2_IN, F47_IN, F48_IN, F42_IN, U5', F40_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.

F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))

We considered the (Usable) Rules:

f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)

And the Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_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(F40_IN(x₁, x₂)) = x₁²   
POL(F42_IN(x₁, x₂)) = x₁²   
POL(F45_IN(x₁, x₂)) = [1] + x₁ + x₁²   
POL(F47_IN(x₁, x₂)) = x₁   
POL(F48_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(f47_in(x₁, x₂)) = x₁   
POL(f47_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(f40_in(z0, z1), z0, z1)
f2_in(z0, z1) → f2_out1
U1(f40_out1(z0), z1, z2) → f2_out1
U1(f40_out3, z0, z1) → f2_out1
f47_in(z0, 0) → f47_out1(z0)
f47_in(s(z0), s(z1)) → U2(f47_in(z0, z1), s(z0), s(z1))
U2(f47_out1(z0), s(z1), s(z2)) → f47_out1(z0)
f48_in(z0, z1) → U3(f2_in(z0, s(z1)), z0, z1)
U3(f2_out1, z0, z1) → f48_out1
f42_in(s(z0), s(z1)) → U4(f45_in(z0, z1), s(z0), s(z1))
U4(f45_out1(z0), s(z1), s(z2)) → f42_out1(z0)
f43_in(z0, z1) → f43_out2
f45_in(z0, z1) → U5(f47_in(z0, z1), z0, z1)
U5(f47_out1(z0), z1, z2) → U6(f48_in(z0, z2), z1, z2, z0)
U6(f48_out1, z0, z1, z2) → f45_out1(z2)
f40_in(z0, z1) → U7(f42_in(z0, z1), f43_in(z0, z1), z0, z1)
U7(f42_out1(z0), z1, z2, z3) → f40_out1(z0)
U7(z0, f43_out2, z1, z2) → f40_out3

Tuples:

F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))

S tuples:none
K tuples:

F42_IN(s(z0), s(z1)) → c10(F45_IN(z0, z1))
F45_IN(z0, z1) → c13(U5'(f47_in(z0, z1), z0, z1), F47_IN(z0, z1))
U5'(f47_out1(z0), z1, z2) → c14(F48_IN(z0, z2))
F48_IN(z0, z1) → c8(F2_IN(z0, s(z1)))
F2_IN(z0, z1) → c1(F40_IN(z0, z1))
F40_IN(z0, z1) → c16(F42_IN(z0, z1))
F47_IN(s(z0), s(z1)) → c6(F47_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f47_in, U2, f48_in, U3, f42_in, U4, f43_in, f45_in, U5, U6, f40_in, U7

Defined Pair Symbols:

F45_IN, F2_IN, F47_IN, F48_IN, F42_IN, U5', F40_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))