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

(0) Obligation:

Clauses:

fl([], [], 0).
fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Query: fl(g,g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

fl([], [], 0).
append([], X, X).
fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Query: fl(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: fl(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fl(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | fl([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nfl(T1, T2, T3) !=? fl([], [], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: fl([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(append(T12, X18, T14), fl(T13, X18, T16))\n\nT12 is ground\nT13 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append(T12, X18, T14)\n\nT12 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: fl(T13, T19, T16)\n\nT13 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T12, X18, T14), SCOPE: 2, CLAUSE: 1 | append(T12, X18, T14), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | append([], X18, T24), SCOPE: 2, CLAUSE: 3\n\nT24 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append(T12, X18, T14), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nT14 is ground\nX18 is free\nX31 is free\nappend(T12, X18, T14) !=? append([], X31, X31)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append([], X18, T24), SCOPE: 2, CLAUSE: 3\n\nT24 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append(T32, X51, T33)\n\nT32 is ground\nT33 is ground\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 34 [arrowhead = none ];
34 -> 4;

35 [label="EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 36 [arrowhead = none ];
36 -> 7;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="EVAL with clause\nfl(.(X14, X15), X16, s(X17)) :- ','(append(X14, X18, X16), fl(X15, X18, X17)).\nand substitutionX14 -> T12,\nX15 -> T13,\nT1 -> .(T12, T13),\nT2 -> T14,\nX16 -> T14,\nX17 -> T16,\nT3 -> s(T16),\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 38 [arrowhead = none ];
38 -> 18;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 39 [arrowhead = none ];
39 -> 19;

40 [label="BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 40 [arrowhead = none ];
40 -> 15;

41 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 41 [arrowhead = none ];
41 -> 22;

42 [label="SPLIT 2\nnew knowledge:\nT12 is ground\nT19 is ground\nT14 is ground\nreplacements:X18 -> T19", fontsize=14, style = filled, fillcolor = violet];
18 -> 42 [arrowhead = none ];
42 -> 23;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 43 [arrowhead = none ];
43 -> 25;

44 [label="INSTANCE with matching:\nT1 -> T13\nT2 -> T19\nT3 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 44 [arrowhead = none , style=dashed];
44 -> 2[style=dashed];

45 [label="EVAL with clause\nappend([], X31, X31).\nand substitutionT12 -> [],\nX18 -> T24,\nX31 -> T24,\nT14 -> T24,\nX32 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 45 [arrowhead = none ];
45 -> 26;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 47 [arrowhead = none ];
47 -> 28;

48 [label="EVAL with clause\nappend(.(X47, X48), X49, .(X47, X50)) :- append(X48, X49, X50).\nand substitutionX47 -> T31,\nX48 -> T32,\nT12 -> .(T31, T32),\nX18 -> X51,\nX49 -> X51,\nX50 -> T33,\nT14 -> .(T31, T33)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 48 [arrowhead = none ];
48 -> 32;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 49 [arrowhead = none ];
49 -> 33;

50 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 50 [arrowhead = none ];
50 -> 29;

51 [label="INSTANCE with matching:\nT12 -> T32\nX18 -> X51\nT14 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 51 [arrowhead = none , style=dashed];
51 -> 22[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1(0)
f2_in(.(z0, z1), z2) → U1(f18_in(z0, z2, z1), .(z0, z1), z2)
U1(f18_out1(z0, z1), .(z2, z3), z4) → f2_out1(s(z1))
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f18_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f2_in(z3, z0), z1, z2, z3, z0)
U4(f2_out1(z0), z1, z2, z3, z4) → f18_out1(z4, z0)

Tuples:

F2_IN(.(z0, z1), z2) → c1(U1'(f18_in(z0, z2, z1), .(z0, z1), z2), F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f22_in(z1, z2), .(z0, z1), .(z0, z2)), F22_IN(z1, z2))
F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(U4'(f2_in(z3, z0), z1, z2, z3, z0), F2_IN(z3, z0))

S tuples:

F2_IN(.(z0, z1), z2) → c1(U1'(f18_in(z0, z2, z1), .(z0, z1), z2), F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f22_in(z1, z2), .(z0, z1), .(z0, z2)), F22_IN(z1, z2))
F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(U4'(f2_in(z3, z0), z1, z2, z3, z0), F2_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f22_in, U2, f18_in, U3, U4

Defined Pair Symbols:

F2_IN, F22_IN, F18_IN, U3'

Compound Symbols:

c1, c4, c6, c7

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1(0)
f2_in(.(z0, z1), z2) → U1(f18_in(z0, z2, z1), .(z0, z1), z2)
U1(f18_out1(z0, z1), .(z2, z3), z4) → f2_out1(s(z1))
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f18_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f2_in(z3, z0), z1, z2, z3, z0)
U4(f2_out1(z0), z1, z2, z3, z4) → f18_out1(z4, z0)

Tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

S tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f22_in, U2, f18_in, U3, U4

Defined Pair Symbols:

F18_IN, F2_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

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

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

F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))

We considered the (Usable) Rules:

f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)

And the Tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F18_IN(x₁, x₂, x₃)) = [2]x₃   
POL(F22_IN(x₁, x₂)) = 0   
POL(F2_IN(x₁, x₂)) = [2]x₁   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄)) = [2]x₄   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁, x₂)) = 0   
POL(f22_out1(x₁)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1(0)
f2_in(.(z0, z1), z2) → U1(f18_in(z0, z2, z1), .(z0, z1), z2)
U1(f18_out1(z0, z1), .(z2, z3), z4) → f2_out1(s(z1))
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f18_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f2_in(z3, z0), z1, z2, z3, z0)
U4(f2_out1(z0), z1, z2, z3, z4) → f18_out1(z4, z0)

Tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

S tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

K tuples:

F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))

Defined Rule Symbols:

f2_in, U1, f22_in, U2, f18_in, U3, U4

Defined Pair Symbols:

F18_IN, F2_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1(0)
f2_in(.(z0, z1), z2) → U1(f18_in(z0, z2, z1), .(z0, z1), z2)
U1(f18_out1(z0, z1), .(z2, z3), z4) → f2_out1(s(z1))
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f18_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f2_in(z3, z0), z1, z2, z3, z0)
U4(f2_out1(z0), z1, z2, z3, z4) → f18_out1(z4, z0)

Tuples:

F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

S tuples:

F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))

K tuples:

F2_IN(.(z0, z1), z2) → c1(F18_IN(z0, z2, z1))
F18_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(F2_IN(z3, z0))

Defined Rule Symbols:

f2_in, U1, f22_in, U2, f18_in, U3, U4

Defined Pair Symbols:

F18_IN, F2_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(11) 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: fl(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fl(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | fl([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nfl(T1, T2, T3) !=? fl([], [], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: fl([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(append(T8, X13, T10), fl(T9, X13, T12))\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 1 | ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: fl(T9, T15, T12) | ','(append([], X13, T15), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT15 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free\nX18 is free\nappend(T8, X13, T10) !=? append([], X18, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: fl(T9, T15, T12)\n\nT9 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(append([], X13, T15), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT15 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(append(T23, X38, T24), fl(T9, X38, T12))\n\nT9 is ground\nT23 is ground\nT24 is ground\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 32 [arrowhead = none ];
32 -> 3;

33 [label="EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 33 [arrowhead = none ];
33 -> 6;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="EVAL with clause\nfl(.(X9, X10), X11, s(X12)) :- ','(append(X9, X13, X11), fl(X10, X13, X12)).\nand substitutionX9 -> T8,\nX10 -> T9,\nT1 -> .(T8, T9),\nT2 -> T10,\nX11 -> T10,\nX12 -> T12,\nT3 -> s(T12),\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 36 [arrowhead = none ];
36 -> 11;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 37 [arrowhead = none ];
37 -> 12;

38 [label="BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 10;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="EVAL with clause\nappend([], X18, X18).\nand substitutionT8 -> [],\nX13 -> T15,\nX18 -> T15,\nT10 -> T15,\nX19 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 41 [arrowhead = none ];
41 -> 17;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 42 [arrowhead = none ];
42 -> 20;

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 43 [arrowhead = none ];
43 -> 21;

44 [label="EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T22,\nX35 -> T23,\nT8 -> .(T22, T23),\nX13 -> X38,\nX36 -> X38,\nX37 -> T24,\nT10 -> .(T22, T24)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 44 [arrowhead = none ];
44 -> 30;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 45 [arrowhead = none ];
45 -> 31;

46 [label="INSTANCE with matching:\nT1 -> T9\nT2 -> T15\nT3 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 46 [arrowhead = none , style=dashed];
46 -> 1[style=dashed];

47 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 47 [arrowhead = none ];
47 -> 24;

48 [label="INSTANCE with matching:\nT8 -> T23\nX13 -> X38\nT10 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 48 [arrowhead = none , style=dashed];
48 -> 11[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1(0)
f1_in(.(z0, z1), z2) → U1(f11_in(z0, z2, z1), .(z0, z1), z2)
U1(f11_out1(z0, z1), .(z2, z3), z4) → f1_out1(s(z1))
f11_in([], z0, z1) → U2(f16_in(z1, z0), [], z0, z1)
f11_in(.(z0, z1), .(z0, z2), z3) → U3(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3)
U2(f16_out1(z0), [], z1, z2) → f11_out1(z1, z0)
U2(f16_out2(z0, z1), [], z2, z3) → f11_out1(z0, z1)
U3(f11_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f11_out1(z0, z1)
f16_in(z0, z1) → U4(f1_in(z0, z1), f21_in(z1, z0), z0, z1)
U4(f1_out1(z0), z1, z2, z3) → f16_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4) → f16_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), z2) → c1(U1'(f11_in(z0, z2, z1), .(z0, z1), z2), F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(U2'(f16_in(z1, z0), [], z0, z1), F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(U3'(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3), F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(U4'(f1_in(z0, z1), f21_in(z1, z0), z0, z1), F1_IN(z0, z1))

S tuples:

F1_IN(.(z0, z1), z2) → c1(U1'(f11_in(z0, z2, z1), .(z0, z1), z2), F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(U2'(f16_in(z1, z0), [], z0, z1), F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(U3'(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3), F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(U4'(f1_in(z0, z1), f21_in(z1, z0), z0, z1), F1_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f11_in, U2, U3, f16_in, U4

Defined Pair Symbols:

F1_IN, F11_IN, F16_IN

Compound Symbols:

c1, c3, c4, c8

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

Removed 4 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1(0)
f1_in(.(z0, z1), z2) → U1(f11_in(z0, z2, z1), .(z0, z1), z2)
U1(f11_out1(z0, z1), .(z2, z3), z4) → f1_out1(s(z1))
f11_in([], z0, z1) → U2(f16_in(z1, z0), [], z0, z1)
f11_in(.(z0, z1), .(z0, z2), z3) → U3(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3)
U2(f16_out1(z0), [], z1, z2) → f11_out1(z1, z0)
U2(f16_out2(z0, z1), [], z2, z3) → f11_out1(z0, z1)
U3(f11_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f11_out1(z0, z1)
f16_in(z0, z1) → U4(f1_in(z0, z1), f21_in(z1, z0), z0, z1)
U4(f1_out1(z0), z1, z2, z3) → f16_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4) → f16_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

S tuples:

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f11_in, U2, U3, f16_in, U4

Defined Pair Symbols:

F1_IN, F11_IN, F16_IN

Compound Symbols:

c1, c3, c4, c8

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

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

F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))

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

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F11_IN(x₁, x₂, x₃)) = [2]x₂   
POL(F16_IN(x₁, x₂)) = [2]x₂   
POL(F1_IN(x₁, x₂)) = [2]x₂   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1(0)
f1_in(.(z0, z1), z2) → U1(f11_in(z0, z2, z1), .(z0, z1), z2)
U1(f11_out1(z0, z1), .(z2, z3), z4) → f1_out1(s(z1))
f11_in([], z0, z1) → U2(f16_in(z1, z0), [], z0, z1)
f11_in(.(z0, z1), .(z0, z2), z3) → U3(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3)
U2(f16_out1(z0), [], z1, z2) → f11_out1(z1, z0)
U2(f16_out2(z0, z1), [], z2, z3) → f11_out1(z0, z1)
U3(f11_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f11_out1(z0, z1)
f16_in(z0, z1) → U4(f1_in(z0, z1), f21_in(z1, z0), z0, z1)
U4(f1_out1(z0), z1, z2, z3) → f16_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4) → f16_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

S tuples:

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

K tuples:

F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))

Defined Rule Symbols:

f1_in, U1, f11_in, U2, U3, f16_in, U4

Defined Pair Symbols:

F1_IN, F11_IN, F16_IN

Compound Symbols:

c1, c3, c4, c8

(17) 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), z2) → c1(F11_IN(z0, z2, z1))

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

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F11_IN(x₁, x₂, x₃)) = x₃   
POL(F16_IN(x₁, x₂)) = x₁   
POL(F1_IN(x₁, x₂)) = x₁   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1(0)
f1_in(.(z0, z1), z2) → U1(f11_in(z0, z2, z1), .(z0, z1), z2)
U1(f11_out1(z0, z1), .(z2, z3), z4) → f1_out1(s(z1))
f11_in([], z0, z1) → U2(f16_in(z1, z0), [], z0, z1)
f11_in(.(z0, z1), .(z0, z2), z3) → U3(f11_in(z1, z2, z3), .(z0, z1), .(z0, z2), z3)
U2(f16_out1(z0), [], z1, z2) → f11_out1(z1, z0)
U2(f16_out2(z0, z1), [], z2, z3) → f11_out1(z0, z1)
U3(f11_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f11_out1(z0, z1)
f16_in(z0, z1) → U4(f1_in(z0, z1), f21_in(z1, z0), z0, z1)
U4(f1_out1(z0), z1, z2, z3) → f16_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4) → f16_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))
F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

S tuples:

F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))

K tuples:

F11_IN(.(z0, z1), .(z0, z2), z3) → c4(F11_IN(z1, z2, z3))
F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))

Defined Rule Symbols:

f1_in, U1, f11_in, U2, U3, f16_in, U4

Defined Pair Symbols:

F1_IN, F11_IN, F16_IN

Compound Symbols:

c1, c3, c4, c8

(19) CdtKnowledgeProof (EQUIVALENT transformation)

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

F11_IN([], z0, z1) → c3(F16_IN(z1, z0))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))
F16_IN(z0, z1) → c8(F1_IN(z0, z1))
F1_IN(.(z0, z1), z2) → c1(F11_IN(z0, z2, z1))

Now S is empty

(20) BOUNDS(O(1), O(1))