Runtime Complexity proof of /tmp/tmpvfqoAh/select-ffb.pl

Runtime Complexity of the query pattern
select(a,a,g)
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), 82 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 122 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 2 ms)

    ↳8 BOUNDS(O(1), O(1))

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

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 116 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Query: select(a,a,g)

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: select(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | select(T1, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX5 is free\nX6 is free\nselect(T1, T2, T3) !=? select(X5, .(X5, X6), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: select(T1, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: select(T22, T23, T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: select(T40, T41, T39)\n\nT37 is ground\nT39 is ground\nX5 is free\nX6 is free\nselect(T1, T2, .(T37, T39)) !=? select(X5, .(X5, X6), X6)", 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 -> 3;

24 [label="EVAL with clause\nselect(X5, .(X5, X6), X6).\nand substitutionT1 -> T8,\nX5 -> T8,\nX6 -> T9,\nT2 -> .(T8, T9),\nT3 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 24 [arrowhead = none ];
24 -> 5;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 25 [arrowhead = none ];
25 -> 8;

26 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 26 [arrowhead = none ];
26 -> 9;

27 [label="EVAL with clause\nselect(X31, .(X32, X33), .(X32, X34)) :- select(X31, X33, X34).\nand substitutionT1 -> T40,\nX31 -> T40,\nX32 -> T37,\nX33 -> T41,\nT2 -> .(T37, T41),\nX34 -> T39,\nT3 -> .(T37, T39),\nT36 -> T40,\nT38 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 27 [arrowhead = none ];
27 -> 21;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 28 [arrowhead = none ];
28 -> 22;

29 [label="EVAL with clause\nselect(X15, .(X16, X17), .(X16, X18)) :- select(X15, X17, X18).\nand substitutionT1 -> T22,\nX15 -> T22,\nX16 -> T19,\nX17 -> T23,\nT2 -> .(T19, T23),\nX18 -> T21,\nT9 -> .(T19, T21),\nT18 -> T22,\nT20 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 29 [arrowhead = none ];
29 -> 11;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 30 [arrowhead = none ];
30 -> 13;

31 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T23\nT3 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 31 [arrowhead = none , style=dashed];
31 -> 1[style=dashed];

32 [label="INSTANCE with matching:\nT1 -> T40\nT2 -> T41\nT3 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 32 [arrowhead = none , style=dashed];
32 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
f1_in(.(z0, z1)) → U2(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1
U2(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))
F1_IN(.(z0, z1)) → c2(U2'(f1_in(z1), .(z0, z1)), F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))
F1_IN(.(z0, z1)) → c2(U2'(f1_in(z1), .(z0, z1)), F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
f1_in(.(z0, z1)) → U2(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1
U2(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

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

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

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

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

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

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

POL(.(x₁, x₂)) = [2] + x₂
POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
f1_in(.(z0, z1)) → U2(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1
U2(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))
F1_IN(.(z0, z1)) → c2(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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

(9) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: select(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | select(T1, T2, T7), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX3 is free\nX4 is free\nselect(T1, T2, T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: select(T1, T2, T7), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: select(T16, T17, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: select(T16, T17, T15), SCOPE: 2, CLAUSE: 0 | select(T16, T17, T15), SCOPE: 2, CLAUSE: 1\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | select(T16, T17, T23), SCOPE: 2, CLAUSE: 1\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: select(T16, T17, T15), SCOPE: 2, CLAUSE: 1\n\nT15 is ground\nX17 is free\nX18 is free\nselect(T16, T17, T15) !=? select(X17, .(X17, X18), X18)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: select(T16, T17, T23), SCOPE: 2, CLAUSE: 1\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: select(T36, T37, T35)\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: select(T54, T55, T53)\n\nT51 is ground\nT53 is ground\nX17 is free\nX18 is free\nselect(T16, T17, .(T51, T53)) !=? select(X17, .(X17, X18), X18)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: select(T68, T69, T67)\n\nT65 is ground\nT67 is ground\nX3 is free\nX4 is free\nselect(T1, T2, .(T65, T67)) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: select(T68, T69, T67), SCOPE: 3, CLAUSE: 0 | select(T68, T69, T67), SCOPE: 3, CLAUSE: 1\n\nT65 is ground\nT67 is ground\nX3 is free\nX4 is free\nselect(T1, T2, .(T65, T67)) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | select(T68, T69, T75), SCOPE: 3, CLAUSE: 1\n\nT65 is ground\nT75 is ground\nX3 is free\nX4 is free\nselect(T1, T2, .(T65, T75)) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: select(T68, T69, T67), SCOPE: 3, CLAUSE: 1\n\nT65 is ground\nT67 is ground\nX3 is free\nX4 is free\nX63 is free\nX64 is free\nselect(T1, T2, .(T65, T67)) !=? select(X3, .(X3, X4), X4)\nselect(T68, T69, T67) !=? select(X63, .(X63, X64), X64)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: select(T68, T69, T75), SCOPE: 3, CLAUSE: 1\n\nT65 is ground\nT75 is ground\nX3 is free\nX4 is free\nselect(T1, T2, .(T65, T75)) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: select(T88, T89, T87)\n\nT65 is ground\nT85 is ground\nT87 is ground\nX3 is free\nX4 is free\nselect(T1, T2, .(T65, .(T85, T87))) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: select(T106, T107, T105)\n\nT65 is ground\nT103 is ground\nT105 is ground\nX3 is free\nX4 is free\nX63 is free\nX64 is free\nselect(T1, T2, .(T65, .(T103, T105))) !=? select(X3, .(X3, X4), X4)\nselect(T68, T69, .(T103, T105)) !=? select(X63, .(X63, X64), X64)", 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 -> 4;

38 [label="EVAL with clause\nselect(X3, .(X3, X4), X4).\nand substitutionT1 -> T6,\nX3 -> T6,\nX4 -> T7,\nT2 -> .(T6, T7),\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 38 [arrowhead = none ];
38 -> 6;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 39 [arrowhead = none ];
39 -> 7;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="EVAL with clause\nselect(X55, .(X56, X57), .(X56, X58)) :- select(X55, X57, X58).\nand substitutionT1 -> T68,\nX55 -> T68,\nX56 -> T65,\nX57 -> T69,\nT2 -> .(T65, T69),\nX58 -> T67,\nT3 -> .(T65, T67),\nT64 -> T68,\nT66 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="EVAL with clause\nselect(X9, .(X10, X11), .(X10, X12)) :- select(X9, X11, X12).\nand substitutionT1 -> T16,\nX9 -> T16,\nX10 -> T13,\nX11 -> T17,\nT2 -> .(T13, T17),\nX12 -> T15,\nT7 -> .(T13, T15),\nT12 -> T16,\nT14 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 43 [arrowhead = none ];
43 -> 12;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 45 [arrowhead = none ];
45 -> 15;

46 [label="EVAL with clause\nselect(X17, .(X17, X18), X18).\nand substitutionT16 -> T22,\nX17 -> T22,\nX18 -> T23,\nT17 -> .(T22, T23),\nT15 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 46 [arrowhead = none ];
46 -> 16;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 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\nselect(X43, .(X44, X45), .(X44, X46)) :- select(X43, X45, X46).\nand substitutionT16 -> T54,\nX43 -> T54,\nX44 -> T51,\nX45 -> T55,\nT17 -> .(T51, T55),\nX46 -> T53,\nT15 -> .(T51, T53),\nT50 -> T54,\nT52 -> T55", 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\nselect(X27, .(X28, X29), .(X28, X30)) :- select(X27, X29, X30).\nand substitutionT16 -> T36,\nX27 -> T36,\nX28 -> T33,\nX29 -> T37,\nT17 -> .(T33, T37),\nX30 -> T35,\nT23 -> .(T33, T35),\nT32 -> T36,\nT34 -> T37", 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 -> T36\nT2 -> T37\nT3 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 53 [arrowhead = none , style=dashed];
53 -> 2[style=dashed];

54 [label="INSTANCE with matching:\nT1 -> T54\nT2 -> T55\nT3 -> T53", 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\nselect(X63, .(X63, X64), X64).\nand substitutionT68 -> T74,\nX63 -> T74,\nX64 -> T75,\nT69 -> .(T74, T75),\nT67 -> T75", 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\nselect(X89, .(X90, X91), .(X90, X92)) :- select(X89, X91, X92).\nand substitutionT68 -> T106,\nX89 -> T106,\nX90 -> T103,\nX91 -> T107,\nT69 -> .(T103, T107),\nX92 -> T105,\nT67 -> .(T103, T105),\nT102 -> T106,\nT104 -> T107", 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\nselect(X73, .(X74, X75), .(X74, X76)) :- select(X73, X75, X76).\nand substitutionT68 -> T88,\nX73 -> T88,\nX74 -> T85,\nX75 -> T89,\nT69 -> .(T85, T89),\nX76 -> T87,\nT75 -> .(T85, T87),\nT84 -> T88,\nT86 -> T89", 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 -> T88\nT2 -> T89\nT3 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 63 [arrowhead = none , style=dashed];
63 -> 2[style=dashed];

64 [label="INSTANCE with matching:\nT1 -> T106\nT2 -> T107\nT3 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 64 [arrowhead = none , style=dashed];
64 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(U2'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(U3'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(U4'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(U2'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(U3'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(U4'(f2_in(z2), .(z0, .(z1, z2))), F2_IN(z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5

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

Removed 4 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5

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

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

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

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

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F2_IN(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

S tuples:none
K tuples:

F2_IN(.(z0, .(z1, z2))) → c2(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c3(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c4(F2_IN(z2))
F2_IN(.(z0, .(z1, z2))) → c5(F2_IN(z2))

Defined Rule Symbols:

f2_in, U1, U2, U3, U4

Defined Pair Symbols:

F2_IN

Compound Symbols:

c2, c3, c4, c5