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

(0) Obligation:

Clauses:

sum([], [], []).
sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3)).
add(0, X, X).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Query: sum(g,g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

sum([], [], []).
add(0, X, X).
sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3)).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Query: sum(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: sum(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | sum(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | sum([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nsum(T1, T2, T3) !=? sum([], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: sum([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(add(T16, T18, T22), sum(T17, T19, T23))\n\nT16 is ground\nT17 is ground\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: add(T16, T18, T22)\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: sum(T17, T19, T26)\n\nT17 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: add(T16, T18, T22), SCOPE: 2, CLAUSE: 1 | add(T16, T18, T22), SCOPE: 2, CLAUSE: 3\n\nT16 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | add(0, T31, T22), SCOPE: 2, CLAUSE: 3\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: add(T16, T18, T22), SCOPE: 2, CLAUSE: 3\n\nT16 is ground\nT18 is ground\nX34 is free\nadd(T16, T18, T22) !=? add(0, X34, X34)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: add(0, T31, T22), SCOPE: 2, CLAUSE: 3\n\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: add(T38, T39, T41)\n\nT38 is ground\nT39 is ground", 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\nsum([], [], []).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> []", 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 -> 8;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 37 [arrowhead = none ];
37 -> 9;

38 [label="EVAL with clause\nsum(.(X22, X23), .(X24, X25), .(X26, X27)) :- ','(add(X22, X24, X26), sum(X23, X25, X27)).\nand substitutionX22 -> T16,\nX23 -> T17,\nT1 -> .(T16, T17),\nX24 -> T18,\nX25 -> T19,\nT2 -> .(T18, T19),\nX26 -> T22,\nX27 -> T23,\nT3 -> .(T22, T23),\nT20 -> T22,\nT21 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 38 [arrowhead = none ];
38 -> 16;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 39 [arrowhead = none ];
39 -> 17;

40 [label="BACKTRACK\nfor clause: sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 40 [arrowhead = none ];
40 -> 12;

41 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 41 [arrowhead = none ];
41 -> 22;

42 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nT18 is ground\nT22 is ground\nreplacements:T23 -> T26", fontsize=14, style = filled, fillcolor = violet];
16 -> 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 -> T17\nT2 -> T19\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 44 [arrowhead = none , style=dashed];
44 -> 2[style=dashed];

45 [label="EVAL with clause\nadd(0, X34, X34).\nand substitutionT16 -> 0,\nT18 -> T31,\nX34 -> T31,\nT22 -> T31", 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 -> 28;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 47 [arrowhead = none ];
47 -> 29;

48 [label="EVAL with clause\nadd(s(X44), X45, s(X46)) :- add(X44, X45, X46).\nand substitutionX44 -> T38,\nT16 -> s(T38),\nT18 -> T39,\nX45 -> T39,\nX46 -> T41,\nT22 -> s(T41),\nT40 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 48 [arrowhead = none ];
48 -> 32;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 49 [arrowhead = none ];
49 -> 33;

50 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 50 [arrowhead = none ];
50 -> 31;

51 [label="INSTANCE with matching:\nT16 -> T38\nT18 -> T39\nT22 -> T41", 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([])
f2_in(.(z0, z1), .(z2, z3)) → U1(f16_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f16_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1(.(z0, z1))
f22_in(0, z0) → f22_out1(z0)
f22_in(s(z0), z1) → U2(f22_in(z0, z1), s(z0), z1)
U2(f22_out1(z0), s(z1), z2) → f22_out1(s(z0))
f16_in(z0, z1, z2, z3) → U3(f22_in(z0, z1), z0, z1, z2, z3)
U3(f22_out1(z0), z1, z2, z3, z4) → U4(f2_in(z3, z4), z1, z2, z3, z4, z0)
U4(f2_out1(z0), z1, z2, z3, z4, z5) → f16_out1(z5, z0)

Tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f16_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3)), F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(U2'(f22_in(z0, z1), s(z0), z1), F22_IN(z0, z1))
F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(U4'(f2_in(z3, z4), z1, z2, z3, z4, z0), F2_IN(z3, z4))

S tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f16_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3)), F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(U2'(f22_in(z0, z1), s(z0), z1), F22_IN(z0, z1))
F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(U4'(f2_in(z3, z4), z1, z2, z3, z4, z0), F2_IN(z3, z4))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f22_in, U2, f16_in, U3, U4

Defined Pair Symbols:

F2_IN, F22_IN, F16_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([])
f2_in(.(z0, z1), .(z2, z3)) → U1(f16_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f16_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1(.(z0, z1))
f22_in(0, z0) → f22_out1(z0)
f22_in(s(z0), z1) → U2(f22_in(z0, z1), s(z0), z1)
U2(f22_out1(z0), s(z1), z2) → f22_out1(s(z0))
f16_in(z0, z1, z2, z3) → U3(f22_in(z0, z1), z0, z1, z2, z3)
U3(f22_out1(z0), z1, z2, z3, z4) → U4(f2_in(z3, z4), z1, z2, z3, z4, z0)
U4(f2_out1(z0), z1, z2, z3, z4, z5) → f16_out1(z5, z0)

Tuples:

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))

S tuples:

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f22_in, U2, f16_in, U3, U4

Defined Pair Symbols:

F16_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.

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))

We considered the (Usable) Rules:

f22_in(0, z0) → f22_out1(z0)
f22_in(s(z0), z1) → U2(f22_in(z0, z1), s(z0), z1)
U2(f22_out1(z0), s(z1), z2) → f22_out1(s(z0))

And the Tuples:

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(0) = [1]   
POL(F16_IN(x₁, x₂, x₃, x₄)) = [1] + [2]x₁ + x₂ + [2]x₃ + x₄   
POL(F22_IN(x₁, x₂)) = x₁ + x₂   
POL(F2_IN(x₁, x₂)) = [2]x₁ + x₂   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄, x₅)) = [2]x₄ + x₅   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁, x₂)) = x₁   
POL(f22_out1(x₁)) = 0   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1([])
f2_in(.(z0, z1), .(z2, z3)) → U1(f16_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f16_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1(.(z0, z1))
f22_in(0, z0) → f22_out1(z0)
f22_in(s(z0), z1) → U2(f22_in(z0, z1), s(z0), z1)
U2(f22_out1(z0), s(z1), z2) → f22_out1(s(z0))
f16_in(z0, z1, z2, z3) → U3(f22_in(z0, z1), z0, z1, z2, z3)
U3(f22_out1(z0), z1, z2, z3, z4) → U4(f2_in(z3, z4), z1, z2, z3, z4, z0)
U4(f2_out1(z0), z1, z2, z3, z4, z5) → f16_out1(z5, z0)

Tuples:

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))

S tuples:

U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))

K tuples:

F16_IN(z0, z1, z2, z3) → c6(U3'(f22_in(z0, z1), z0, z1, z2, z3), F22_IN(z0, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))
F22_IN(s(z0), z1) → c4(F22_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f22_in, U2, f16_in, U3, U4

Defined Pair Symbols:

F16_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:

U3'(f22_out1(z0), z1, z2, z3, z4) → c7(F2_IN(z3, z4))
F2_IN(.(z0, z1), .(z2, z3)) → c1(F16_IN(z0, z2, z1, z3))

Now S is empty

(10) BOUNDS(O(1), O(1))

(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: sum(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | sum(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | sum([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nsum(T1, T2, T3) !=? sum([], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: sum([], [], T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(add(T10, T12, T16), sum(T11, T13, T17))\n\nT10 is ground\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(add(T10, T12, T16), sum(T11, T13, T17)), SCOPE: 2, CLAUSE: 1 | ','(add(T10, T12, T16), sum(T11, T13, T17)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: sum(T11, T13, T21) | ','(add(0, T20, T16), sum(T11, T13, T17)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT13 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(add(T10, T12, T16), sum(T11, T13, T17)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nT13 is ground\nX24 is free\nadd(T10, T12, T16) !=? add(0, X24, X24)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: sum(T11, T13, T21)\n\nT11 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(add(0, T20, T16), sum(T11, T13, T17)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT13 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(add(T28, T29, T31), sum(T11, T13, T32))\n\nT11 is ground\nT13 is ground\nT28 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL with clause\nsum([], [], []).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="EVAL with clause\nsum(.(X16, X17), .(X18, X19), .(X20, X21)) :- ','(add(X16, X18, X20), sum(X17, X19, X21)).\nand substitutionX16 -> T10,\nX17 -> T11,\nT1 -> .(T10, T11),\nX18 -> T12,\nX19 -> T13,\nT2 -> .(T12, T13),\nX20 -> T16,\nX21 -> T17,\nT3 -> .(T16, T17),\nT14 -> T16,\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 35 [arrowhead = none ];
35 -> 13;

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

37 [label="BACKTRACK\nfor clause: sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 38 [arrowhead = none ];
38 -> 15;

39 [label="EVAL with clause\nadd(0, X24, X24).\nand substitutionT10 -> 0,\nT12 -> T20,\nX24 -> T20,\nT16 -> T20,\nT17 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 39 [arrowhead = none ];
39 -> 18;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 40 [arrowhead = none ];
40 -> 19;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 41 [arrowhead = none ];
41 -> 20;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 42 [arrowhead = none ];
42 -> 21;

43 [label="EVAL with clause\nadd(s(X34), X35, s(X36)) :- add(X34, X35, X36).\nand substitutionX34 -> T28,\nT10 -> s(T28),\nT12 -> T29,\nX35 -> T29,\nX36 -> T31,\nT16 -> s(T31),\nT30 -> T31,\nT17 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 43 [arrowhead = none ];
43 -> 27;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 44 [arrowhead = none ];
44 -> 30;

45 [label="INSTANCE with matching:\nT1 -> T11\nT2 -> T13\nT3 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 45 [arrowhead = none , style=dashed];
45 -> 1[style=dashed];

46 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 46 [arrowhead = none ];
46 -> 24;

47 [label="INSTANCE with matching:\nT10 -> T28\nT12 -> T29\nT16 -> T31\nT17 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 47 [arrowhead = none , style=dashed];
47 -> 13[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1([])
f1_in(.(z0, z1), .(z2, z3)) → U1(f13_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f13_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1(.(z0, z1))
f13_in(0, z0, z1, z2) → U2(f18_in(z1, z2, z0), 0, z0, z1, z2)
f13_in(s(z0), z1, z2, z3) → U3(f13_in(z0, z1, z2, z3), s(z0), z1, z2, z3)
U2(f18_out1(z0), 0, z1, z2, z3) → f13_out1(z1, z0)
U2(f18_out2(z0, z1), 0, z2, z3, z4) → f13_out1(z0, z1)
U3(f13_out1(z0, z1), s(z2), z3, z4, z5) → f13_out1(s(z0), z1)
f18_in(z0, z1, z2) → U4(f1_in(z0, z1), f21_in(z2, z0, z1), z0, z1, z2)
U4(f1_out1(z0), z1, z2, z3, z4) → f18_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4, z5) → f18_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f13_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3)), F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(U2'(f18_in(z1, z2, z0), 0, z0, z1, z2), F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(U3'(f13_in(z0, z1, z2, z3), s(z0), z1, z2, z3), F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(U4'(f1_in(z0, z1), f21_in(z2, z0, z1), z0, z1, z2), F1_IN(z0, z1))

S tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f13_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3)), F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(U2'(f18_in(z1, z2, z0), 0, z0, z1, z2), F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(U3'(f13_in(z0, z1, z2, z3), s(z0), z1, z2, z3), F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(U4'(f1_in(z0, z1), f21_in(z2, z0, z1), z0, z1, z2), F1_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f13_in, U2, U3, f18_in, U4

Defined Pair Symbols:

F1_IN, F13_IN, F18_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([])
f1_in(.(z0, z1), .(z2, z3)) → U1(f13_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f13_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1(.(z0, z1))
f13_in(0, z0, z1, z2) → U2(f18_in(z1, z2, z0), 0, z0, z1, z2)
f13_in(s(z0), z1, z2, z3) → U3(f13_in(z0, z1, z2, z3), s(z0), z1, z2, z3)
U2(f18_out1(z0), 0, z1, z2, z3) → f13_out1(z1, z0)
U2(f18_out2(z0, z1), 0, z2, z3, z4) → f13_out1(z0, z1)
U3(f13_out1(z0, z1), s(z2), z3, z4, z5) → f13_out1(s(z0), z1)
f18_in(z0, z1, z2) → U4(f1_in(z0, z1), f21_in(z2, z0, z1), z0, z1, z2)
U4(f1_out1(z0), z1, z2, z3, z4) → f18_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4, z5) → f18_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

S tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f13_in, U2, U3, f18_in, U4

Defined Pair Symbols:

F1_IN, F13_IN, F18_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.

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

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

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(0) = [2]   
POL(F13_IN(x₁, x₂, x₃, x₄)) = [1] + x₁ + [2]x₃   
POL(F18_IN(x₁, x₂, x₃)) = [2] + [2]x₁   
POL(F1_IN(x₁, x₂)) = [1] + [2]x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(s(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1([])
f1_in(.(z0, z1), .(z2, z3)) → U1(f13_in(z0, z2, z1, z3), .(z0, z1), .(z2, z3))
U1(f13_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1(.(z0, z1))
f13_in(0, z0, z1, z2) → U2(f18_in(z1, z2, z0), 0, z0, z1, z2)
f13_in(s(z0), z1, z2, z3) → U3(f13_in(z0, z1, z2, z3), s(z0), z1, z2, z3)
U2(f18_out1(z0), 0, z1, z2, z3) → f13_out1(z1, z0)
U2(f18_out2(z0, z1), 0, z2, z3, z4) → f13_out1(z0, z1)
U3(f13_out1(z0, z1), s(z2), z3, z4, z5) → f13_out1(s(z0), z1)
f18_in(z0, z1, z2) → U4(f1_in(z0, z1), f21_in(z2, z0, z1), z0, z1, z2)
U4(f1_out1(z0), z1, z2, z3, z4) → f18_out1(z0)
U4(z0, f21_out1(z1, z2), z3, z4, z5) → f18_out2(z1, z2)

Tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F13_IN(s(z0), z1, z2, z3) → c4(F13_IN(z0, z1, z2, z3))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

S tuples:

F13_IN(s(z0), z1, z2, z3) → c4(F13_IN(z0, z1, z2, z3))

K tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(F13_IN(z0, z2, z1, z3))
F13_IN(0, z0, z1, z2) → c3(F18_IN(z1, z2, z0))
F18_IN(z0, z1, z2) → c8(F1_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f13_in, U2, U3, f18_in, U4

Defined Pair Symbols:

F1_IN, F13_IN, F18_IN

Compound Symbols:

c1, c3, c4, c8