Runtime Complexity proof of /tmp/tmpA5WCXE/duplicate2.pl

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

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 65 ms)

↳2 CdtProblem

    ↳3 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 97 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 78 ms)

↳10 CdtProblem

    ↳11 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 68 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

duplicate([], []).
duplicate(X, .(H, .(H, Z))) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), duplicate(T, Z)))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Query: duplicate(g,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: duplicate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0 | duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nduplicate(T1, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(call(empty([])), ','(!_2, failure(a))), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))) | ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10))))) | ?_3 | ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(','(!_2, failure(a)), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(failure(a), ','(head([], T9), ','(tail([], X15), duplicate(X15, T10)))), SCOPE: 5, CLAUSE: 9\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(no(empty(T17)), ','(head(T17, T20), ','(tail(T17, X29), duplicate(X29, T21))))\n\nT17 is ground\nX29 is free\nduplicate(T17, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(no(empty(T17)), ','(head(T17, T20), ','(tail(T17, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T17)), ','(head(T17, T20), ','(tail(T17, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT17 is ground\nX29 is free\nduplicate(T17, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(','(call(empty(T24)), ','(!_6, failure(a))), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))) | ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(empty(T24), ','(','(!_6, failure(a)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21))))) | ?_7 | ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(empty(T24), ','(','(!_6, failure(a)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(','(!_6, failure(a)), ','(head([], T20), ','(tail([], X29), duplicate(X29, T21)))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], T20), ','(tail([], X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ?_8 | ?_7 | ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])\nempty(T24) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(failure(a), ','(head([], T20), ','(tail([], X29), duplicate(X29, T21))))\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(head([], T20), ','(tail([], X29), duplicate(X29, T21)))), SCOPE: 9, CLAUSE: 9\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ?_7 | ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])\nempty(T24) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(empty(T24)), ','(head(T24, T20), ','(tail(T24, X29), duplicate(X29, T21)))), SCOPE: 6, CLAUSE: 8\n\nT24 is ground\nX29 is free\nduplicate(T24, T2) !=? duplicate([], [])\nempty(T24) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(head(T27, T20), ','(tail(T27, X29), duplicate(X29, T21)))\n\nT27 is ground\nX29 is free\nduplicate(T27, T2) !=? duplicate([], [])\nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(T27, T20), ','(tail(T27, X29), duplicate(X29, T21))), SCOPE: 10, CLAUSE: 2 | ','(head(T27, T20), ','(tail(T27, X29), duplicate(X29, T21))), SCOPE: 10, CLAUSE: 3\n\nT27 is ground\nX29 is free\nduplicate(T27, T2) !=? duplicate([], [])\nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(head(T27, T20), ','(tail(T27, X29), duplicate(X29, T21))), SCOPE: 10, CLAUSE: 3\n\nT27 is ground\nX29 is free\nduplicate(T27, T2) !=? duplicate([], [])\nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(tail(.(T33, T34), X29), duplicate(X29, T35))\n\nT33 is ground\nT34 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(tail(.(T33, T34), X29), duplicate(X29, T35)), SCOPE: 11, CLAUSE: 4 | ','(tail(.(T33, T34), X29), duplicate(X29, T35)), SCOPE: 11, CLAUSE: 5\n\nT33 is ground\nT34 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(T33, T34), X29), duplicate(X29, T35)), SCOPE: 11, CLAUSE: 5\n\nT33 is ground\nT34 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: duplicate(T41, T35)\n\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 73 [arrowhead = none ];
73 -> 4;

74 [label="EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 74 [arrowhead = none ];
74 -> 5;

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

76 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 76 [arrowhead = none ];
76 -> 9;

77 [label="EVAL with clause\nduplicate(X26, .(X27, .(X27, X28))) :- ','(no(empty(X26)), ','(head(X26, X27), ','(tail(X26, X29), duplicate(X29, X28)))).\nand substitutionT1 -> T17,\nX26 -> T17,\nX27 -> T20,\nX28 -> T21,\nT2 -> .(T20, .(T20, T21)),\nT18 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 77 [arrowhead = none ];
77 -> 33;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 78 [arrowhead = none ];
78 -> 35;

79 [label="EVAL with clause\nduplicate(X12, .(X13, .(X13, X14))) :- ','(no(empty(X12)), ','(head(X12, X13), ','(tail(X12, X15), duplicate(X15, X14)))).\nand substitutionX12 -> [],\nX13 -> T9,\nX14 -> T10,\nT2 -> .(T9, .(T9, T10)),\nT7 -> T9,\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 80 [arrowhead = none ];
80 -> 15;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 81 [arrowhead = none ];
81 -> 16;

82 [label="ONLY EVAL with clause\nno(X18) :- ','(call(X18), ','(!_2, failure(a))).\nand substitutionX18 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 82 [arrowhead = none ];
82 -> 18;

83 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 87 [arrowhead = none ];
87 -> 28;

88 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 88 [arrowhead = none ];
88 -> 30;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 89 [arrowhead = none ];
89 -> 36;

90 [label="ONLY EVAL with clause\nno(X32) :- ','(call(X32), ','(!_6, failure(a))).\nand substitutionT17 -> T24,\nX32 -> empty(T24)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 90 [arrowhead = none ];
90 -> 40;

91 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 91 [arrowhead = none ];
91 -> 41;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 92 [arrowhead = none ];
92 -> 44;

93 [label="EVAL with clause\nempty([]).\nand substitutionT24 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 93 [arrowhead = none ];
93 -> 46;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 94 [arrowhead = none ];
94 -> 49;

95 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 95 [arrowhead = none ];
95 -> 51;

96 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 96 [arrowhead = none ];
96 -> 55;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 97 [arrowhead = none ];
97 -> 52;

98 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 98 [arrowhead = none ];
98 -> 53;

99 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 99 [arrowhead = none ];
99 -> 56;

100 [label="ONLY EVAL with clause\nno(X35).\nand substitutionT24 -> T27,\nX35 -> empty(T27)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 100 [arrowhead = none ];
100 -> 58;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 101 [arrowhead = none ];
101 -> 61;

102 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T27), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 102 [arrowhead = none ];
102 -> 62;

103 [label="EVAL with clause\nhead(.(X41, X42), X41).\nand substitutionX41 -> T33,\nX42 -> T34,\nT27 -> .(T33, T34),\nT20 -> T33,\nT21 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 103 [arrowhead = none ];
103 -> 66;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 104 [arrowhead = none ];
104 -> 68;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 105 [arrowhead = none ];
105 -> 69;

106 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 106 [arrowhead = none ];
106 -> 70;

107 [label="ONLY EVAL with clause\ntail(.(X47, X48), X48).\nand substitutionT33 -> T40,\nX47 -> T40,\nT34 -> T41,\nX48 -> T41,\nX29 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 107 [arrowhead = none ];
107 -> 72;

108 [label="INSTANCE with matching:\nT1 -> T41\nT2 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 108 [arrowhead = none , style=dashed];
108 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(5) 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(.(z0, z1)) → c1(F1_IN(z1))

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

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))

(9) 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: duplicate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0 | duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nduplicate(T1, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(','(call(empty([])), ','(!_2, failure(a))), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))) | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))) | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 5, CLAUSE: 9\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16))))\n\nT12 is ground\nX21 is free\nduplicate(T12, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT12 is ground\nX21 is free\nduplicate(T12, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(','(call(empty(T19)), ','(!_6, failure(a))), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))) | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(empty(T19), ','(','(!_6, failure(a)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16))))) | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(empty(T19), ','(','(!_6, failure(a)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(','(!_6, failure(a)), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ?_8 | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(failure(a), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16))))\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(failure(a), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))), SCOPE: 9, CLAUSE: 9\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16)))\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 2 | ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 3\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 3\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T28, T29), X21), duplicate(X21, T30))\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 4 | ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 5\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 5\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: duplicate(T36, T30)\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 72 [arrowhead = none ];
72 -> 3;

73 [label="EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="EVAL with clause\nduplicate(X18, .(X19, .(X19, X20))) :- ','(no(empty(X18)), ','(head(X18, X19), ','(tail(X18, X21), duplicate(X21, X20)))).\nand substitutionT1 -> T12,\nX18 -> T12,\nX19 -> T15,\nX20 -> T16,\nT2 -> .(T15, .(T15, T16)),\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 76 [arrowhead = none ];
76 -> 31;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 77 [arrowhead = none ];
77 -> 32;

78 [label="EVAL with clause\nduplicate(X8, .(X9, .(X9, X10))) :- ','(no(empty(X8)), ','(head(X8, X9), ','(tail(X8, X11), duplicate(X11, X10)))).\nand substitutionX8 -> [],\nX9 -> T7,\nX10 -> T8,\nT2 -> .(T7, .(T7, T8)),\nT5 -> T7,\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 78 [arrowhead = none ];
78 -> 11;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 79 [arrowhead = none ];
79 -> 12;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 80 [arrowhead = none ];
80 -> 14;

81 [label="ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 82 [arrowhead = none ];
82 -> 19;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 84 [arrowhead = none ];
84 -> 24;

85 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 85 [arrowhead = none ];
85 -> 26;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 86 [arrowhead = none ];
86 -> 27;

87 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 87 [arrowhead = none ];
87 -> 29;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 88 [arrowhead = none ];
88 -> 34;

89 [label="ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_6, failure(a))).\nand substitutionT12 -> T19,\nX24 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 89 [arrowhead = none ];
89 -> 37;

90 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 91 [arrowhead = none ];
91 -> 39;

92 [label="EVAL with clause\nempty([]).\nand substitutionT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 92 [arrowhead = none ];
92 -> 42;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 93 [arrowhead = none ];
93 -> 43;

94 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 94 [arrowhead = none ];
94 -> 45;

95 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 95 [arrowhead = none ];
95 -> 50;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 96 [arrowhead = none ];
96 -> 47;

97 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 97 [arrowhead = none ];
97 -> 48;

98 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 98 [arrowhead = none ];
98 -> 54;

99 [label="ONLY EVAL with clause\nno(X27).\nand substitutionT19 -> T22,\nX27 -> empty(T22)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 99 [arrowhead = none ];
99 -> 57;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 100 [arrowhead = none ];
100 -> 59;

101 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T22), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 101 [arrowhead = none ];
101 -> 60;

102 [label="EVAL with clause\nhead(.(X33, X34), X33).\nand substitutionX33 -> T28,\nX34 -> T29,\nT22 -> .(T28, T29),\nT15 -> T28,\nT16 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 102 [arrowhead = none ];
102 -> 63;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 103 [arrowhead = none ];
103 -> 64;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 104 [arrowhead = none ];
104 -> 65;

105 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 105 [arrowhead = none ];
105 -> 67;

106 [label="ONLY EVAL with clause\ntail(.(X39, X40), X40).\nand substitutionT28 -> T35,\nX39 -> T35,\nT29 -> T36,\nX40 -> T36,\nX21 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 106 [arrowhead = none ];
106 -> 71;

107 [label="INSTANCE with matching:\nT1 -> T36\nT2 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 107 [arrowhead = none , style=dashed];
107 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

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

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

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

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

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F2_IN(x₁)) = x₁
POL(c1(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1