Runtime Complexity proof of /tmp/tmpMOKjW8/len2.pl

Runtime Complexity of the query pattern
len(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), 60 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 100 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 77 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 86 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

len([], 0).
len(Xs, s(N)) :- ','(no(empty(Xs)), ','(tail(Xs, Ys), len(Ys, N))).
tail([], []).
tail(.(X, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: len(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];
2 [label="2: len(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: len(T1, T2), SCOPE: 1, CLAUSE: 0 | len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlen(T1, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(empty([])), ','(tail([], X9), len(X9, T6)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 5 | ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(call(empty([])), ','(!_2, failure(a))), ','(tail([], X9), len(X9, T6))) | ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X9), len(X9, T6)))) | ?_3 | ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X9), len(X9, T6)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(','(!_2, failure(a)), ','(tail([], X9), len(X9, T6))) | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X9), len(X9, T6))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), ','(tail([], X9), len(X9, T6)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(failure(a), ','(tail([], X9), len(X9, T6))), SCOPE: 5, CLAUSE: 7\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(no(empty(T11)), ','(tail(T11, X20), len(X20, T13)))\n\nT11 is ground\nX20 is free\nlen(T11, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(no(empty(T11)), ','(tail(T11, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 5 | ','(no(empty(T11)), ','(tail(T11, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT11 is ground\nX20 is free\nlen(T11, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(','(call(empty(T16)), ','(!_6, failure(a))), ','(tail(T16, X20), len(X20, T13))) | ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(empty(T16), ','(','(!_6, failure(a)), ','(tail(T16, X20), len(X20, T13)))) | ?_7 | ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(empty(T16), ','(','(!_6, failure(a)), ','(tail(T16, X20), len(X20, T13)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(','(!_6, failure(a)), ','(tail([], X20), len(X20, T13))) | ?_8 | ?_7 | ','(no(empty([])), ','(tail([], X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ?_8 | ?_7 | ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(tail([], X20), len(X20, T13)))\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(failure(a), ','(tail([], X20), len(X20, T13))), SCOPE: 9, CLAUSE: 7\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ?_7 | ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(no(empty(T16)), ','(tail(T16, X20), len(X20, T13))), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX20 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail(T19, X20), len(X20, T13))\n\nT19 is ground\nX20 is free\nlen(T19, T2) !=? len([], 0)\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(T19, X20), len(X20, T13)), SCOPE: 10, CLAUSE: 2 | ','(tail(T19, X20), len(X20, T13)), SCOPE: 10, CLAUSE: 3\n\nT19 is ground\nX20 is free\nlen(T19, T2) !=? len([], 0)\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(T19, X20), len(X20, T13)), SCOPE: 10, CLAUSE: 3\n\nT19 is ground\nX20 is free\nlen(T19, T2) !=? len([], 0)\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: len(T25, T13)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 67 [arrowhead = none ];
67 -> 3;

68 [label="EVAL with clause\nlen([], 0).\nand substitutionT1 -> [],\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 68 [arrowhead = none ];
68 -> 8;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 69 [arrowhead = none ];
69 -> 9;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 70 [arrowhead = none ];
70 -> 10;

71 [label="EVAL with clause\nlen(X18, s(X19)) :- ','(no(empty(X18)), ','(tail(X18, X20), len(X20, X19))).\nand substitutionT1 -> T11,\nX18 -> T11,\nX19 -> T13,\nT2 -> s(T13),\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 71 [arrowhead = none ];
71 -> 36;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 72 [arrowhead = none ];
72 -> 38;

73 [label="EVAL with clause\nlen(X7, s(X8)) :- ','(no(empty(X7)), ','(tail(X7, X9), len(X9, X8))).\nand substitutionX7 -> [],\nX8 -> T6,\nT2 -> s(T6),\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 74 [arrowhead = none ];
74 -> 15;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 75 [arrowhead = none ];
75 -> 17;

76 [label="ONLY EVAL with clause\nno(X12) :- ','(call(X12), ','(!_2, failure(a))).\nand substitutionX12 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 77 [arrowhead = none ];
77 -> 20;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 78 [arrowhead = none ];
78 -> 22;

79 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 79 [arrowhead = none ];
79 -> 24;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 80 [arrowhead = none ];
80 -> 26;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 82 [arrowhead = none ];
82 -> 30;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 83 [arrowhead = none ];
83 -> 40;

84 [label="ONLY EVAL with clause\nno(X23) :- ','(call(X23), ','(!_6, failure(a))).\nand substitutionT11 -> T16,\nX23 -> empty(T16)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 84 [arrowhead = none ];
84 -> 46;

85 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 85 [arrowhead = none ];
85 -> 47;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 86 [arrowhead = none ];
86 -> 49;

87 [label="EVAL with clause\nempty([]).\nand substitutionT16 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 87 [arrowhead = none ];
87 -> 50;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 88 [arrowhead = none ];
88 -> 51;

89 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 89 [arrowhead = none ];
89 -> 52;

90 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 90 [arrowhead = none ];
90 -> 57;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 91 [arrowhead = none ];
91 -> 54;

92 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 92 [arrowhead = none ];
92 -> 55;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 93 [arrowhead = none ];
93 -> 59;

94 [label="ONLY EVAL with clause\nno(X26).\nand substitutionT16 -> T19,\nX26 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 94 [arrowhead = none ];
94 -> 61;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 95 [arrowhead = none ];
95 -> 63;

96 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T19), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 96 [arrowhead = none ];
96 -> 64;

97 [label="EVAL with clause\ntail(.(X31, X32), X32).\nand substitutionX31 -> T24,\nX32 -> T25,\nT19 -> .(T24, T25),\nX20 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 97 [arrowhead = none ];
97 -> 65;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 98 [arrowhead = none ];
98 -> 66;

99 [label="INSTANCE with matching:\nT1 -> T25\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 99 [arrowhead = none , style=dashed];
99 -> 2[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1(0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1(z0), .(z1, z2)) → f2_out1(s(z0))

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

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1(0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1(z0), .(z1, z2)) → f2_out1(s(z0))

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

(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.

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1(0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1(z0), .(z1, z2)) → f2_out1(s(z0))

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

(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];
1 [label="1: len(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: len(T1, T2), SCOPE: 1, CLAUSE: 0 | len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlen(T1, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(no(empty([])), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 5 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(call(empty([])), ','(!_2, failure(a))), ','(tail([], X6), len(X6, T5))) | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5)))) | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5))) | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), ','(tail([], X6), len(X6, T5))), SCOPE: 5, CLAUSE: 7\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10)))\n\nT8 is ground\nX14 is free\nlen(T8, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 5 | ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT8 is ground\nX14 is free\nlen(T8, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(','(call(empty(T13)), ','(!_6, failure(a))), ','(tail(T13, X14), len(X14, T10))) | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(empty(T13), ','(','(!_6, failure(a)), ','(tail(T13, X14), len(X14, T10)))) | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(empty(T13), ','(','(!_6, failure(a)), ','(tail(T13, X14), len(X14, T10)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(','(!_6, failure(a)), ','(tail([], X14), len(X14, T10))) | ?_8 | ?_7 | ','(no(empty([])), ','(tail([], X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_8 | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(failure(a), ','(tail([], X14), len(X14, T10)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(failure(a), ','(tail([], X14), len(X14, T10))), SCOPE: 9, CLAUSE: 7\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(tail(T16, X14), len(X14, T10))\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 2 | ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 3\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 3\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: len(T22, T10)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 63 [arrowhead = none ];
63 -> 4;

64 [label="EVAL with clause\nlen([], 0).\nand substitutionT1 -> [],\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 64 [arrowhead = none ];
64 -> 5;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 65 [arrowhead = none ];
65 -> 6;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 66 [arrowhead = none ];
66 -> 7;

67 [label="EVAL with clause\nlen(X12, s(X13)) :- ','(no(empty(X12)), ','(tail(X12, X14), len(X14, X13))).\nand substitutionT1 -> T8,\nX12 -> T8,\nX13 -> T10,\nT2 -> s(T10),\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL with clause\nlen(X4, s(X5)) :- ','(no(empty(X4)), ','(tail(X4, X6), len(X6, X5))).\nand substitutionX4 -> [],\nX5 -> T5,\nT2 -> s(T5),\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 69 [arrowhead = none ];
69 -> 11;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 70 [arrowhead = none ];
70 -> 12;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 71 [arrowhead = none ];
71 -> 13;

72 [label="ONLY EVAL with clause\nno(X9) :- ','(call(X9), ','(!_2, failure(a))).\nand substitutionX9 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 72 [arrowhead = none ];
72 -> 16;

73 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 73 [arrowhead = none ];
73 -> 18;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 75 [arrowhead = none ];
75 -> 23;

76 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 77 [arrowhead = none ];
77 -> 27;

78 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 78 [arrowhead = none ];
78 -> 29;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 79 [arrowhead = none ];
79 -> 33;

80 [label="ONLY EVAL with clause\nno(X17) :- ','(call(X17), ','(!_6, failure(a))).\nand substitutionT8 -> T13,\nX17 -> empty(T13)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 80 [arrowhead = none ];
80 -> 34;

81 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 81 [arrowhead = none ];
81 -> 35;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 82 [arrowhead = none ];
82 -> 37;

83 [label="EVAL with clause\nempty([]).\nand substitutionT13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 83 [arrowhead = none ];
83 -> 39;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 84 [arrowhead = none ];
84 -> 41;

85 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 85 [arrowhead = none ];
85 -> 42;

86 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 86 [arrowhead = none ];
86 -> 45;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 87 [arrowhead = none ];
87 -> 43;

88 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 88 [arrowhead = none ];
88 -> 44;

89 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 89 [arrowhead = none ];
89 -> 48;

90 [label="ONLY EVAL with clause\nno(X20).\nand substitutionT13 -> T16,\nX20 -> empty(T16)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 90 [arrowhead = none ];
90 -> 53;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 91 [arrowhead = none ];
91 -> 56;

92 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T16), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 92 [arrowhead = none ];
92 -> 58;

93 [label="EVAL with clause\ntail(.(X25, X26), X26).\nand substitutionX25 -> T21,\nX26 -> T22,\nT16 -> .(T21, T22),\nX14 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 93 [arrowhead = none ];
93 -> 60;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 94 [arrowhead = none ];
94 -> 62;

95 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 95 [arrowhead = none , style=dashed];
95 -> 1[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1(0)
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1(z0), .(z1, z2)) → f1_out1(s(z0))

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

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1(0)
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1(z0), .(z1, z2)) → f1_out1(s(z0))

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

(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.

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1(0)
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1(z0), .(z1, z2)) → f1_out1(s(z0))

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