Runtime Complexity of the query pattern
div(g,g,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) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =(Z, 0)).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
=(X, X).

Query: div(g,g,a)

(1) BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

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

Query: div(g,g,a)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, fail) | div(T8, 0, T3), SCOPE: 1, CLAUSE: 1 | div(T8, 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX5 is free\nX6 is free\ndiv(T1, T2, T3) !=? div(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, user_defined_=(T16, 0)) | div(0, T14, T3), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nX5 is free\nX6 is free\ndiv(0, T14, T3) !=? div(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3) !=? div(X5, 0, X6)\ndiv(T1, T2, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: user_defined_=(T16, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: user_defined_=(T16, 0), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(minus(T26, T27, X26), div(X26, T27, T29))\n\nT26 is ground\nT27 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\nX26 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: minus(T26, T27, X26)\n\nT26 is ground\nT27 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\nX26 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: div(T36, T27, T29)\n\nT26 is ground\nT27 is ground\nT36 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: minus(T26, T27, X26), SCOPE: 3, CLAUSE: 3 | minus(T26, T27, X26), SCOPE: 3, CLAUSE: 4 | minus(T26, T27, X26), SCOPE: 3, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\nX26 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: minus(T26, T27, X26), SCOPE: 3, CLAUSE: 4 | minus(T26, T27, X26), SCOPE: 3, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\nX26 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: minus(T26, T27, X26), SCOPE: 3, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX5 is free\nX6 is free\nX11 is free\nX12 is free\nX26 is free\ndiv(T26, T27, T3) !=? div(X5, 0, X6)\ndiv(T26, T27, T3) !=? div(0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: minus(T49, T50, X56)\n\nT49 is ground\nT50 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: minus(T49, T50, X56), SCOPE: 4, CLAUSE: 3 | minus(T49, T50, X56), SCOPE: 4, CLAUSE: 4 | minus(T49, T50, X56), SCOPE: 4, CLAUSE: 5\n\nT49 is ground\nT50 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true | minus(0, T55, X56), SCOPE: 4, CLAUSE: 4 | minus(0, T55, X56), SCOPE: 4, CLAUSE: 5\n\nT55 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: minus(T49, T50, X56), SCOPE: 4, CLAUSE: 4 | minus(T49, T50, X56), SCOPE: 4, CLAUSE: 5\n\nT49 is ground\nT50 is ground\nX56 is free\nX61 is free\nminus(T49, T50, X56) !=? minus(0, X61, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: minus(0, T55, X56), SCOPE: 4, CLAUSE: 4 | minus(0, T55, X56), SCOPE: 4, CLAUSE: 5\n\nT55 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true | minus(0, 0, X56), SCOPE: 4, CLAUSE: 5\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: minus(0, T55, X56), SCOPE: 4, CLAUSE: 5\n\nT55 is ground\nX56 is free\nX64 is free\nminus(0, T55, X56) !=? minus(X64, 0, X64)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: minus(0, 0, X56), SCOPE: 4, CLAUSE: 5\n\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true | minus(T58, 0, X56), SCOPE: 4, CLAUSE: 5\n\nT58 is ground\nX56 is free\nX61 is free\nminus(T58, 0, X56) !=? minus(0, X61, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: minus(T49, T50, X56), SCOPE: 4, CLAUSE: 5\n\nT49 is ground\nT50 is ground\nX56 is free\nX61 is free\nX73 is free\nminus(T49, T50, X56) !=? minus(0, X61, 0)\nminus(T49, T50, X56) !=? minus(X73, 0, X73)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: minus(T58, 0, X56), SCOPE: 4, CLAUSE: 5\n\nT58 is ground\nX56 is free\nX61 is free\nminus(T58, 0, X56) !=? minus(0, X61, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: minus(T64, T65, X88)\n\nT64 is ground\nT65 is ground\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 73 [arrowhead = none ];
73 -> 4;

74 [label="EVAL with clause\ndiv(X5, 0, X6) :- ','(!_1, fail).\nand substitutionT1 -> T8,\nX5 -> T8,\nT2 -> 0,\nT3 -> T9,\nX6 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 74 [arrowhead = none ];
74 -> 6;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 76 [arrowhead = none ];
76 -> 9;

77 [label="EVAL with clause\ndiv(0, X11, X12) :- ','(!_1, user_defined_=(X12, 0)).\nand substitutionT1 -> 0,\nT2 -> T14,\nX11 -> T14,\nT3 -> T16,\nX12 -> T16,\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 77 [arrowhead = none ];
77 -> 13;

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

79 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 79 [arrowhead = none ];
79 -> 12;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 80 [arrowhead = none ];
80 -> 15;

81 [label="EVAL with clause\ndiv(X23, X24, s(X25)) :- ','(minus(X23, X24, X26), div(X26, X24, X25)).\nand substitutionT1 -> T26,\nX23 -> T26,\nT2 -> T27,\nX24 -> T27,\nX25 -> T29,\nT3 -> s(T29),\nT28 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 81 [arrowhead = none ];
81 -> 27;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 83 [arrowhead = none ];
83 -> 16;

84 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT16 -> 0,\nX15 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 85 [arrowhead = none ];
85 -> 18;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 86 [arrowhead = none ];
86 -> 19;

87 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nT27 is ground\nT36 is ground\nreplacements:X26 -> T36", fontsize=14, style = filled, fillcolor = violet];
27 -> 88 [arrowhead = none ];
88 -> 35;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 89 [arrowhead = none ];
89 -> 38;

90 [label="INSTANCE with matching:\nT1 -> T36\nT2 -> T27\nT3 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 90 [arrowhead = none , style=dashed];
90 -> 2[style=dashed];

91 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (div(T26, T27, T3), div(0, X11, X12))", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 91 [arrowhead = none ];
91 -> 41;

92 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T26, T27, T3), div(X5, 0, X6))", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 92 [arrowhead = none ];
92 -> 42;

93 [label="EVAL with clause\nminus(s(X53), s(X54), X55) :- minus(X53, X54, X55).\nand substitutionX53 -> T49,\nT26 -> s(T49),\nX54 -> T50,\nT27 -> s(T50),\nX26 -> X56,\nX55 -> X56", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 93 [arrowhead = none ];
93 -> 48;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 94 [arrowhead = none ];
94 -> 49;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 95 [arrowhead = none ];
95 -> 53;

96 [label="EVAL with clause\nminus(0, X61, 0).\nand substitutionT49 -> 0,\nT50 -> T55,\nX61 -> T55,\nX56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 96 [arrowhead = none ];
96 -> 55;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 97 [arrowhead = none ];
97 -> 56;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 98 [arrowhead = none ];
98 -> 57;

99 [label="EVAL with clause\nminus(X73, 0, X73).\nand substitutionT49 -> T58,\nX73 -> T58,\nT50 -> 0,\nX56 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 99 [arrowhead = none ];
99 -> 65;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 100 [arrowhead = none ];
100 -> 66;

101 [label="EVAL with clause\nminus(X64, 0, X64).\nand substitutionX64 -> 0,\nT55 -> 0,\nX56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 101 [arrowhead = none ];
101 -> 59;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 102 [arrowhead = none ];
102 -> 60;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 103 [arrowhead = none ];
103 -> 61;

104 [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 -> 104 [arrowhead = none ];
104 -> 64;

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];
61 -> 105 [arrowhead = none ];
105 -> 63;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 106 [arrowhead = none ];
106 -> 67;

107 [label="EVAL with clause\nminus(s(X85), s(X86), X87) :- minus(X85, X86, X87).\nand substitutionX85 -> T64,\nT49 -> s(T64),\nX86 -> T65,\nT50 -> s(T65),\nX56 -> X88,\nX87 -> X88", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 107 [arrowhead = none ];
107 -> 70;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 108 [arrowhead = none ];
108 -> 72;

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

110 [label="INSTANCE with matching:\nT49 -> T64\nT50 -> T65\nX56 -> X88", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 110 [arrowhead = none , style=dashed];
110 -> 48[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F2_IN(z0, z1) → c1(U1'(f27_in(z0, z1), z0, z1), F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(U2'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c8(U3'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(U5'(f2_in(z0, z2), z1, z2, z0), F2_IN(z0, z2))

S tuples:

F2_IN(z0, z1) → c1(U1'(f27_in(z0, z1), z0, z1), F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(U2'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c8(U3'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(U5'(f2_in(z0, z2), z1, z2, z0), F2_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F2_IN, F48_IN, F34_IN, F27_IN, U4'

Compound Symbols:

c1, c6, c8, c10, c11

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F2_IN(z0, z1) → c1(U1'(f27_in(z0, z1), z0, z1), F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(U2'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(U5'(f2_in(z0, z2), z1, z2, z0), F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c(U3'(f48_in(z0, z1), s(z0), s(z1)))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))

S tuples:

F2_IN(z0, z1) → c1(U1'(f27_in(z0, z1), z0, z1), F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(U2'(f48_in(z0, z1), s(z0), s(z1)), F48_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(U5'(f2_in(z0, z2), z1, z2, z0), F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c(U3'(f48_in(z0, z1), s(z0), s(z1)))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F2_IN, F48_IN, F27_IN, U4', F34_IN

Compound Symbols:

c1, c6, c10, c11, c

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

S tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F27_IN, F34_IN, F2_IN, F48_IN, U4'

Compound Symbols:

c10, c, c1, c6, c11, c

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

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

F34_IN(s(z0), s(z1)) → c

We considered the (Usable) Rules:

f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)

And the Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

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

POL(0) = 0   
POL(F27_IN(x₁, x₂)) = [2] + x₁   
POL(F2_IN(x₁, x₂)) = [2] + x₁   
POL(F34_IN(x₁, x₂)) = [2]   
POL(F48_IN(x₁, x₂)) = [2]   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3(x₁, x₂, x₃)) = [1] + x₁   
POL(U4'(x₁, x₂, x₃)) = x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(f34_in(x₁, x₂)) = x₁   
POL(f34_out1(x₁)) = [2] + x₁   
POL(f48_in(x₁, x₂)) = [2] + x₁   
POL(f48_out1(x₁)) = [1] + x₁   
POL(s(x₁)) = [3] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

S tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))

K tuples:

F34_IN(s(z0), s(z1)) → c

Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F27_IN, F34_IN, F2_IN, F48_IN, U4'

Compound Symbols:

c10, c, c1, c6, c11, c

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

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

U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))

We considered the (Usable) Rules:

f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)

And the Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

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

POL(0) = 0   
POL(F27_IN(x₁, x₂)) = x₁   
POL(F2_IN(x₁, x₂)) = x₁   
POL(F34_IN(x₁, x₂)) = 0   
POL(F48_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3(x₁, x₂, x₃)) = [1] + x₁   
POL(U4'(x₁, x₂, x₃)) = x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(f34_in(x₁, x₂)) = x₁   
POL(f34_out1(x₁)) = [1] + x₁   
POL(f48_in(x₁, x₂)) = x₁   
POL(f48_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

S tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))

K tuples:

F34_IN(s(z0), s(z1)) → c
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F27_IN, F34_IN, F2_IN, F48_IN, U4'

Compound Symbols:

c10, c, c1, c6, c11, c

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1
f2_in(z0, z1) → U1(f27_in(z0, z1), z0, z1)
U1(f27_out1(z0), z1, z2) → f2_out1
f48_in(0, z0) → f48_out1(0)
f48_in(0, 0) → f48_out1(0)
f48_in(z0, 0) → f48_out1(z0)
f48_in(s(z0), s(z1)) → U2(f48_in(z0, z1), s(z0), s(z1))
U2(f48_out1(z0), s(z1), s(z2)) → f48_out1(z0)
f34_in(s(z0), s(z1)) → U3(f48_in(z0, z1), s(z0), s(z1))
U3(f48_out1(z0), s(z1), s(z2)) → f34_out1(z0)
f27_in(z0, z1) → U4(f34_in(z0, z1), z0, z1)
U4(f34_out1(z0), z1, z2) → U5(f2_in(z0, z2), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f27_out1(z2)

Tuples:

F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F34_IN(s(z0), s(z1)) → c

S tuples:

F48_IN(s(z0), s(z1)) → c6(F48_IN(z0, z1))

K tuples:

F34_IN(s(z0), s(z1)) → c
U4'(f34_out1(z0), z1, z2) → c11(F2_IN(z0, z2))
F2_IN(z0, z1) → c1(F27_IN(z0, z1))
F27_IN(z0, z1) → c10(U4'(f34_in(z0, z1), z0, z1), F34_IN(z0, z1))
F34_IN(s(z0), s(z1)) → c(F48_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f48_in, U2, f34_in, U3, f27_in, U4, U5

Defined Pair Symbols:

F27_IN, F34_IN, F2_IN, F48_IN, U4'

Compound Symbols:

c10, c, c1, c6, c11, c

(15) 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)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, fail) | div(T6, 0, T3), SCOPE: 1, CLAUSE: 1 | div(T6, 0, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\ndiv(T1, T2, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_1, user_defined_=(T12, 0)) | div(0, T10, T3), SCOPE: 1, CLAUSE: 2\n\nT10 is ground\nX3 is free\nX4 is free\ndiv(0, T10, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\ndiv(T1, T2, T3) !=? div(X3, 0, X4)\ndiv(T1, T2, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: user_defined_=(T12, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: user_defined_=(T12, 0), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(minus(T19, T20, X18), div(X18, T20, T22))\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 3 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 4 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 4 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(minus(T29, T30, X32), div(X32, s(T30), T22))\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: minus(T29, T30, X32)\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: div(T33, s(T30), T22)\n\nT30 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 3 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 4 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true | minus(0, T38, X32), SCOPE: 4, CLAUSE: 4 | minus(0, T38, X32), SCOPE: 4, CLAUSE: 5\n\nT38 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 4 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free\nX39 is free\nminus(T29, T30, X32) !=? minus(0, X39, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: minus(0, T38, X32), SCOPE: 4, CLAUSE: 4 | minus(0, T38, X32), SCOPE: 4, CLAUSE: 5\n\nT38 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true | minus(0, 0, X32), SCOPE: 4, CLAUSE: 5\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: minus(0, T38, X32), SCOPE: 4, CLAUSE: 5\n\nT38 is ground\nX32 is free\nX42 is free\nminus(0, T38, X32) !=? minus(X42, 0, X42)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(0, 0, X32), SCOPE: 4, CLAUSE: 5\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true | minus(T41, 0, X32), SCOPE: 4, CLAUSE: 5\n\nT41 is ground\nX32 is free\nX39 is free\nminus(T41, 0, X32) !=? minus(0, X39, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free\nX39 is free\nX51 is free\nminus(T29, T30, X32) !=? minus(0, X39, 0)\nminus(T29, T30, X32) !=? minus(X51, 0, X51)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: minus(T41, 0, X32), SCOPE: 4, CLAUSE: 5\n\nT41 is ground\nX32 is free\nX39 is free\nminus(T41, 0, X32) !=? minus(0, X39, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: minus(T47, T48, X66)\n\nT47 is ground\nT48 is ground\nX66 is free", 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(X3, 0, X4) :- ','(!_1, fail).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T7,\nX4 -> T7", 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 -> 10;

79 [label="EVAL with clause\ndiv(0, X7, X8) :- ','(!_1, user_defined_=(X8, 0)).\nand substitutionT1 -> 0,\nT2 -> T10,\nX7 -> T10,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 79 [arrowhead = none ];
79 -> 20;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 80 [arrowhead = none ];
80 -> 21;

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

82 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 82 [arrowhead = none ];
82 -> 22;

83 [label="EVAL with clause\ndiv(X15, X16, s(X17)) :- ','(minus(X15, X16, X18), div(X18, X16, X17)).\nand substitutionT1 -> T19,\nX15 -> T19,\nT2 -> T20,\nX16 -> T20,\nX17 -> T22,\nT3 -> s(T22),\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="EVAL with clause\nuser_defined_=(X11, X11).\nand substitutionT12 -> 0,\nX11 -> 0,\nT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 86 [arrowhead = none ];
86 -> 24;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 89 [arrowhead = none ];
89 -> 31;

90 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (div(T19, T20, T3), div(0, X7, X8))", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 90 [arrowhead = none ];
90 -> 32;

91 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T19, T20, T3), div(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 91 [arrowhead = none ];
91 -> 33;

92 [label="EVAL with clause\nminus(s(X29), s(X30), X31) :- minus(X29, X30, X31).\nand substitutionX29 -> T29,\nT19 -> s(T29),\nX30 -> T30,\nT20 -> s(T30),\nX18 -> X32,\nX31 -> X32", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 92 [arrowhead = none ];
92 -> 36;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nT33 is ground\nreplacements:X32 -> T33", fontsize=14, style = filled, fillcolor = violet];
36 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 96 [arrowhead = none ];
96 -> 43;

97 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> s(T30)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 97 [arrowhead = none , style=dashed];
97 -> 1[style=dashed];

98 [label="EVAL with clause\nminus(0, X39, 0).\nand substitutionT29 -> 0,\nT30 -> T38,\nX39 -> T38,\nX32 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 98 [arrowhead = none ];
98 -> 44;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="EVAL with clause\nminus(X51, 0, X51).\nand substitutionT29 -> T41,\nX51 -> T41,\nT30 -> 0,\nX32 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 101 [arrowhead = none ];
101 -> 58;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 102 [arrowhead = none ];
102 -> 62;

103 [label="EVAL with clause\nminus(X42, 0, X42).\nand substitutionX42 -> 0,\nT38 -> 0,\nX32 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 104 [arrowhead = none ];
104 -> 50;

105 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 105 [arrowhead = none ];
105 -> 51;

106 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 106 [arrowhead = none ];
106 -> 54;

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];
51 -> 107 [arrowhead = none ];
107 -> 52;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 108 [arrowhead = none ];
108 -> 69;

109 [label="EVAL with clause\nminus(s(X63), s(X64), X65) :- minus(X63, X64, X65).\nand substitutionX63 -> T47,\nT29 -> s(T47),\nX64 -> T48,\nT30 -> s(T48),\nX32 -> X66,\nX65 -> X66", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 109 [arrowhead = none ];
109 -> 73;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 110 [arrowhead = none ];
110 -> 74;

111 [label="BACKTRACK\nfor clause: minus(s(X), s(Y), Z) :- minus(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 111 [arrowhead = none ];
111 -> 71;

112 [label="INSTANCE with matching:\nT29 -> T47\nT30 -> T48\nX32 -> X66", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 112 [arrowhead = none , style=dashed];
112 -> 39[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(s(z0), s(z1)) → U1(f36_in(z0, z1), s(z0), s(z1))
U1(f36_out1(z0), s(z1), s(z2)) → f1_out1
f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)
f36_in(z0, z1) → U3(f39_in(z0, z1), z0, z1)
U3(f39_out1(z0), z1, z2) → U4(f1_in(z0, s(z2)), z1, z2, z0)
U4(f1_out1, z0, z1, z2) → f36_out1(z2)

Tuples:

F1_IN(s(z0), s(z1)) → c1(U1'(f36_in(z0, z1), s(z0), s(z1)), F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(U2'(f39_in(z0, z1), s(z0), s(z1)), F39_IN(z0, z1))
F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(U4'(f1_in(z0, s(z2)), z1, z2, z0), F1_IN(z0, s(z2)))

S tuples:

F1_IN(s(z0), s(z1)) → c1(U1'(f36_in(z0, z1), s(z0), s(z1)), F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(U2'(f39_in(z0, z1), s(z0), s(z1)), F39_IN(z0, z1))
F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(U4'(f1_in(z0, s(z2)), z1, z2, z0), F1_IN(z0, s(z2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f39_in, U2, f36_in, U3, U4

Defined Pair Symbols:

F1_IN, F39_IN, F36_IN, U3'

Compound Symbols:

c1, c6, c8, c9

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

Removed 3 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(s(z0), s(z1)) → U1(f36_in(z0, z1), s(z0), s(z1))
U1(f36_out1(z0), s(z1), s(z2)) → f1_out1
f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)
f36_in(z0, z1) → U3(f39_in(z0, z1), z0, z1)
U3(f39_out1(z0), z1, z2) → U4(f1_in(z0, s(z2)), z1, z2, z0)
U4(f1_out1, z0, z1, z2) → f36_out1(z2)

Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

S tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f39_in, U2, f36_in, U3, U4

Defined Pair Symbols:

F36_IN, F1_IN, F39_IN, U3'

Compound Symbols:

c8, c1, c6, c9

(19) 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.

F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

We considered the (Usable) Rules:

f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)

And the Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

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

POL(0) = [1]   
POL(F1_IN(x₁, x₂)) = [2]x₁   
POL(F36_IN(x₁, x₂)) = [1] + [2]x₁   
POL(F39_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃)) = [1] + [2]x₁   
POL(c1(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f39_in(x₁, x₂)) = x₁   
POL(f39_out1(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(s(z0), s(z1)) → U1(f36_in(z0, z1), s(z0), s(z1))
U1(f36_out1(z0), s(z1), s(z2)) → f1_out1
f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)
f36_in(z0, z1) → U3(f39_in(z0, z1), z0, z1)
U3(f39_out1(z0), z1, z2) → U4(f1_in(z0, s(z2)), z1, z2, z0)
U4(f1_out1, z0, z1, z2) → f36_out1(z2)

Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

S tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))

K tuples:

F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f36_in, U3, U4

Defined Pair Symbols:

F36_IN, F1_IN, F39_IN, U3'

Compound Symbols:

c8, c1, c6, c9

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(s(z0), s(z1)) → U1(f36_in(z0, z1), s(z0), s(z1))
U1(f36_out1(z0), s(z1), s(z2)) → f1_out1
f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)
f36_in(z0, z1) → U3(f39_in(z0, z1), z0, z1)
U3(f39_out1(z0), z1, z2) → U4(f1_in(z0, s(z2)), z1, z2, z0)
U4(f1_out1, z0, z1, z2) → f36_out1(z2)

Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

S tuples:

F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))

K tuples:

F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))
F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f36_in, U3, U4

Defined Pair Symbols:

F36_IN, F1_IN, F39_IN, U3'

Compound Symbols:

c8, c1, c6, c9

(23) 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.

F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))

We considered the (Usable) Rules:

f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)

And the Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

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

POL(0) = 0   
POL(F1_IN(x₁, x₂)) = x₁²   
POL(F36_IN(x₁, x₂)) = [1] + x₁ + x₁²   
POL(F39_IN(x₁, x₂)) = x₁   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃)) = x₁²   
POL(c1(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f39_in(x₁, x₂)) = x₁   
POL(f39_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0, z0) → f1_out1
f1_in(s(z0), s(z1)) → U1(f36_in(z0, z1), s(z0), s(z1))
U1(f36_out1(z0), s(z1), s(z2)) → f1_out1
f39_in(0, z0) → f39_out1(0)
f39_in(0, 0) → f39_out1(0)
f39_in(z0, 0) → f39_out1(z0)
f39_in(s(z0), s(z1)) → U2(f39_in(z0, z1), s(z0), s(z1))
U2(f39_out1(z0), s(z1), s(z2)) → f39_out1(z0)
f36_in(z0, z1) → U3(f39_in(z0, z1), z0, z1)
U3(f39_out1(z0), z1, z2) → U4(f1_in(z0, s(z2)), z1, z2, z0)
U4(f1_out1, z0, z1, z2) → f36_out1(z2)

Tuples:

F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))

S tuples:none
K tuples:

F1_IN(s(z0), s(z1)) → c1(F36_IN(z0, z1))
U3'(f39_out1(z0), z1, z2) → c9(F1_IN(z0, s(z2)))
F36_IN(z0, z1) → c8(U3'(f39_in(z0, z1), z0, z1), F39_IN(z0, z1))
F39_IN(s(z0), s(z1)) → c6(F39_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f39_in, U2, f36_in, U3, U4

Defined Pair Symbols:

F36_IN, F1_IN, F39_IN, U3'

Compound Symbols:

c8, c1, c6, c9

(25) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(26) BOUNDS(O(1), O(1))