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

(0) Obligation:

Clauses:

sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
sum(X, 0, X).

Query: sum(a,a,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).

Query: sum(a,a,g)

(3) 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\nT3 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: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | sum(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX3 is free\nsum(T1, T2, T3) !=? sum(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: sum(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: sum(T16, T17, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: sum(T29, T30, T28)\n\nT28 is ground\nX3 is free\nsum(T1, T2, s(T28)) !=? sum(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 23 [arrowhead = none ];
23 -> 4;

24 [label="EVAL with clause\nsum(X3, 0, X3).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 24 [arrowhead = none ];
24 -> 5;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 25 [arrowhead = none ];
25 -> 7;

26 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="EVAL with clause\nsum(X21, s(X22), s(X23)) :- sum(X21, X22, X23).\nand substitutionT1 -> T29,\nX21 -> T29,\nX22 -> T30,\nT2 -> s(T30),\nX23 -> T28,\nT3 -> s(T28),\nT26 -> T29,\nT27 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 27 [arrowhead = none ];
27 -> 21;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 28 [arrowhead = none ];
28 -> 22;

29 [label="EVAL with clause\nsum(X10, s(X11), s(X12)) :- sum(X10, X11, X12).\nand substitutionT1 -> T16,\nX10 -> T16,\nX11 -> T17,\nT2 -> s(T17),\nX12 -> T15,\nT6 -> s(T15),\nT13 -> T16,\nT14 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 29 [arrowhead = none ];
29 -> 13;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 30 [arrowhead = none ];
30 -> 15;

31 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T17\nT3 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
13 -> 31 [arrowhead = none , style=dashed];
31 -> 1[style=dashed];

32 [label="INSTANCE with matching:\nT1 -> T29\nT2 -> T30\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 32 [arrowhead = none , style=dashed];
32 -> 1[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0, 0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))
U2(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))

Tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))
F1_IN(s(z0)) → c2(U2'(f1_in(z0), s(z0)), F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))
F1_IN(s(z0)) → c2(U2'(f1_in(z0), s(z0)), F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0, 0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))
U2(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

(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.

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

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

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

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

POL(F1_IN(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0, 0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))
U2(f1_out1(z0, z1), s(z2)) → f1_out1(z0, s(z1))

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

S tuples:none
K tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set 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];
2 [label="2: sum(T1, T2, T3)\n\nT3 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: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | sum(T1, T2, T5), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: sum(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX2 is free\nsum(T1, T2, T3) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: sum(T1, T2, T5), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: sum(T12, T13, T11)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: sum(T12, T13, T11), SCOPE: 2, CLAUSE: 0 | sum(T12, T13, T11), SCOPE: 2, CLAUSE: 1\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | sum(T12, T13, T16), SCOPE: 2, CLAUSE: 1\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: sum(T12, T13, T11), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nX11 is free\nsum(T12, T13, T11) !=? sum(X11, 0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: sum(T12, T13, T16), SCOPE: 2, CLAUSE: 1\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: sum(T26, T27, T25)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: sum(T39, T40, T38)\n\nT38 is ground\nX11 is free\nsum(T12, T13, s(T38)) !=? sum(X11, 0, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: sum(T49, T50, T48)\n\nT48 is ground\nX2 is free\nsum(T1, T2, s(T48)) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: sum(T49, T50, T48), SCOPE: 3, CLAUSE: 0 | sum(T49, T50, T48), SCOPE: 3, CLAUSE: 1\n\nT48 is ground\nX2 is free\nsum(T1, T2, s(T48)) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | sum(T49, T50, T53), SCOPE: 3, CLAUSE: 1\n\nT53 is ground\nX2 is free\nsum(T1, T2, s(T53)) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: sum(T49, T50, T48), SCOPE: 3, CLAUSE: 1\n\nT48 is ground\nX2 is free\nX42 is free\nsum(T1, T2, s(T48)) !=? sum(X2, 0, X2)\nsum(T49, T50, T48) !=? sum(X42, 0, X42)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: sum(T49, T50, T53), SCOPE: 3, CLAUSE: 1\n\nT53 is ground\nX2 is free\nsum(T1, T2, s(T53)) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: sum(T63, T64, T62)\n\nT62 is ground\nX2 is free\nsum(T1, T2, s(s(T62))) !=? sum(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: sum(T76, T77, T75)\n\nT75 is ground\nX2 is free\nX42 is free\nsum(T1, T2, s(s(T75))) !=? sum(X2, 0, X2)\nsum(T49, T50, s(T75)) !=? sum(X42, 0, X42)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL with clause\nsum(X2, 0, X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> 0,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 38 [arrowhead = none ];
38 -> 6;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 39 [arrowhead = none ];
39 -> 8;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 40 [arrowhead = none ];
40 -> 9;

41 [label="EVAL with clause\nsum(X37, s(X38), s(X39)) :- sum(X37, X38, X39).\nand substitutionT1 -> T49,\nX37 -> T49,\nX38 -> T50,\nT2 -> s(T50),\nX39 -> T48,\nT3 -> s(T48),\nT46 -> T49,\nT47 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="EVAL with clause\nsum(X6, s(X7), s(X8)) :- sum(X6, X7, X8).\nand substitutionT1 -> T12,\nX6 -> T12,\nX7 -> T13,\nT2 -> s(T13),\nX8 -> T11,\nT5 -> s(T11),\nT9 -> T12,\nT10 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="EVAL with clause\nsum(X11, 0, X11).\nand substitutionT12 -> T16,\nX11 -> T16,\nT13 -> 0,\nT11 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 46 [arrowhead = none ];
46 -> 16;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 47 [arrowhead = none ];
47 -> 17;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 48 [arrowhead = none ];
48 -> 18;

49 [label="EVAL with clause\nsum(X29, s(X30), s(X31)) :- sum(X29, X30, X31).\nand substitutionT12 -> T39,\nX29 -> T39,\nX30 -> T40,\nT13 -> s(T40),\nX31 -> T38,\nT11 -> s(T38),\nT36 -> T39,\nT37 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 50 [arrowhead = none ];
50 -> 24;

51 [label="EVAL with clause\nsum(X18, s(X19), s(X20)) :- sum(X18, X19, X20).\nand substitutionT12 -> T26,\nX18 -> T26,\nX19 -> T27,\nT13 -> s(T27),\nX20 -> T25,\nT16 -> s(T25),\nT23 -> T26,\nT24 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 51 [arrowhead = none ];
51 -> 19;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 52 [arrowhead = none ];
52 -> 20;

53 [label="INSTANCE with matching:\nT1 -> T26\nT2 -> T27\nT3 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 53 [arrowhead = none , style=dashed];
53 -> 2[style=dashed];

54 [label="INSTANCE with matching:\nT1 -> T39\nT2 -> T40\nT3 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 54 [arrowhead = none , style=dashed];
54 -> 2[style=dashed];

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="EVAL with clause\nsum(X42, 0, X42).\nand substitutionT49 -> T53,\nX42 -> T53,\nT50 -> 0,\nT48 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 56 [arrowhead = none ];
56 -> 28;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 57 [arrowhead = none ];
57 -> 29;

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 58 [arrowhead = none ];
58 -> 30;

59 [label="EVAL with clause\nsum(X60, s(X61), s(X62)) :- sum(X60, X61, X62).\nand substitutionT49 -> T76,\nX60 -> T76,\nX61 -> T77,\nT50 -> s(T77),\nX62 -> T75,\nT48 -> s(T75),\nT73 -> T76,\nT74 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 59 [arrowhead = none ];
59 -> 35;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 60 [arrowhead = none ];
60 -> 36;

61 [label="EVAL with clause\nsum(X49, s(X50), s(X51)) :- sum(X49, X50, X51).\nand substitutionT49 -> T63,\nX49 -> T63,\nX50 -> T64,\nT50 -> s(T64),\nX51 -> T62,\nT53 -> s(T62),\nT60 -> T63,\nT61 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 61 [arrowhead = none ];
61 -> 31;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 62 [arrowhead = none ];
62 -> 32;

63 [label="INSTANCE with matching:\nT1 -> T63\nT2 -> T64\nT3 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 63 [arrowhead = none , style=dashed];
63 -> 2[style=dashed];

64 [label="INSTANCE with matching:\nT1 -> T76\nT2 -> T77\nT3 -> T75", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 64 [arrowhead = none , style=dashed];
64 -> 2[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0, 0)
f2_in(s(z0)) → f2_out1(z0, s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U2(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U3(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U4(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))

Tuples:

F2_IN(s(s(z0))) → c2(U1'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c4(U3'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c5(U4'(f2_in(z0), s(s(z0))), F2_IN(z0))

S tuples:

F2_IN(s(s(z0))) → c2(U1'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c4(U3'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c5(U4'(f2_in(z0), s(s(z0))), F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5

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

Removed 4 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0, 0)
f2_in(s(z0)) → f2_out1(z0, s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U2(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U3(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U4(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))

Tuples:

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

S tuples:

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5

(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.

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

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

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

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

POL(F2_IN(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0, 0)
f2_in(s(z0)) → f2_out1(z0, s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U2(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U3(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))
U4(f2_out1(z0, z1), s(s(z2))) → f2_out1(z0, s(s(z1)))

Tuples:

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

S tuples:none
K tuples:

F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5