Runtime Complexity of the query pattern
append3(a,a,g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

append1([], X, X).
append1(.(X, Y), U, .(X, Z)) :- append1(Y, U, Z).
append2([], X, X).
append2(.(X, Y), U, .(X, Z)) :- append2(Y, U, Z).
append3([], X, X).
append3(.(X, Y), U, .(X, Z)) :- append3(Y, U, Z).

Query: append3(a,a,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

append1([], X, X).
append2([], X, X).
append3([], X, X).
append1(.(X, Y), U, .(X, Z)) :- append1(Y, U, Z).
append2(.(X, Y), U, .(X, Z)) :- append2(Y, U, Z).
append3(.(X, Y), U, .(X, Z)) :- append3(Y, U, Z).

Query: append3(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];
2 [label="2: append3(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | append3(T1, T2, T6), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground\nX3 is free\nappend3(T1, T2, T3) !=? append3([], X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: append3(T1, T2, T6), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: append3(T19, T20, T18)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append3(T35, T36, T34)\n\nT31 is ground\nT34 is ground\nX3 is free\nappend3(T1, T2, .(T31, T34)) !=? append3([], X3, 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];
2 -> 23 [arrowhead = none ];
23 -> 4;

24 [label="EVAL with clause\nappend3([], X3, X3).\nand substitutionT1 -> [],\nT2 -> T6,\nX3 -> T6,\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 -> 8;

27 [label="EVAL with clause\nappend3(.(X26, X27), X28, .(X26, X29)) :- append3(X27, X28, X29).\nand substitutionX26 -> T31,\nX27 -> T35,\nT1 -> .(T31, T35),\nT2 -> T36,\nX28 -> T36,\nX29 -> T34,\nT3 -> .(T31, T34),\nT32 -> T35,\nT33 -> T36", 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\nappend3(.(X12, X13), X14, .(X12, X15)) :- append3(X13, X14, X15).\nand substitutionX12 -> T15,\nX13 -> T19,\nT1 -> .(T15, T19),\nT2 -> T20,\nX14 -> T20,\nX15 -> T18,\nT6 -> .(T15, T18),\nT16 -> T19,\nT17 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 29 [arrowhead = none ];
29 -> 12;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 30 [arrowhead = none ];
30 -> 14;

31 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T20\nT3 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 31 [arrowhead = none , style=dashed];
31 -> 2[style=dashed];

32 [label="INSTANCE with matching:\nT1 -> T35\nT2 -> T36\nT3 -> T34", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 32 [arrowhead = none , style=dashed];
32 -> 2[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1([], z0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
f2_in(.(z0, z1)) → U2(f2_in(z1), .(z0, z1))
U1(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)
U2(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)

Tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))
F2_IN(.(z0, z1)) → c2(U2'(f2_in(z1), .(z0, z1)), F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))
F2_IN(.(z0, z1)) → c2(U2'(f2_in(z1), .(z0, z1)), F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1([], z0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
f2_in(.(z0, z1)) → U2(f2_in(z1), .(z0, z1))
U1(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)
U2(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_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.

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

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

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F2_IN(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1([], z0)
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
f2_in(.(z0, z1)) → U2(f2_in(z1), .(z0, z1))
U1(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)
U2(f2_out1(z0, z1), .(z2, z3)) → f2_out1(.(z2, z0), z1)

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))
F2_IN(.(z0, z1)) → c2(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_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];
1 [label="1: append3(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | append3(T1, T2, T5), SCOPE: 1, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground\nX2 is free\nappend3(T1, T2, T3) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: append3(T1, T2, T5), SCOPE: 1, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: append3(T14, T15, T13)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: append3(T14, T15, T13), SCOPE: 2, CLAUSE: 2 | append3(T14, T15, T13), SCOPE: 2, CLAUSE: 5\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | append3(T14, T15, T18), SCOPE: 2, CLAUSE: 5\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append3(T14, T15, T13), SCOPE: 2, CLAUSE: 5\n\nT13 is ground\nX13 is free\nappend3(T14, T15, T13) !=? append3([], X13, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append3(T14, T15, T18), SCOPE: 2, CLAUSE: 5\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: append3(T31, T32, T30)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: append3(T47, T48, T46)\n\nT43 is ground\nT46 is ground\nX13 is free\nappend3(T14, T15, .(T43, T46)) !=? append3([], X13, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append3(T59, T60, T58)\n\nT55 is ground\nT58 is ground\nX2 is free\nappend3(T1, T2, .(T55, T58)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append3(T59, T60, T58), SCOPE: 3, CLAUSE: 2 | append3(T59, T60, T58), SCOPE: 3, CLAUSE: 5\n\nT55 is ground\nT58 is ground\nX2 is free\nappend3(T1, T2, .(T55, T58)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | append3(T59, T60, T63), SCOPE: 3, CLAUSE: 5\n\nT55 is ground\nT63 is ground\nX2 is free\nappend3(T1, T2, .(T55, T63)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: append3(T59, T60, T58), SCOPE: 3, CLAUSE: 5\n\nT55 is ground\nT58 is ground\nX2 is free\nX52 is free\nappend3(T1, T2, .(T55, T58)) !=? append3([], X2, X2)\nappend3(T59, T60, T58) !=? append3([], X52, X52)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append3(T59, T60, T63), SCOPE: 3, CLAUSE: 5\n\nT55 is ground\nT63 is ground\nX2 is free\nappend3(T1, T2, .(T55, T63)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: append3(T76, T77, T75)\n\nT55 is ground\nT72 is ground\nT75 is ground\nX2 is free\nappend3(T1, T2, .(T55, .(T72, T75))) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: append3(T92, T93, T91)\n\nT55 is ground\nT88 is ground\nT91 is ground\nX2 is free\nX52 is free\nappend3(T1, T2, .(T55, .(T88, T91))) !=? append3([], X2, X2)\nappend3(T59, T60, .(T88, T91)) !=? append3([], X52, X52)", 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];
1 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL with clause\nappend3([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\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 -> 9;

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

41 [label="EVAL with clause\nappend3(.(X46, X47), X48, .(X46, X49)) :- append3(X47, X48, X49).\nand substitutionX46 -> T55,\nX47 -> T59,\nT1 -> .(T55, T59),\nT2 -> T60,\nX48 -> T60,\nX49 -> T58,\nT3 -> .(T55, T58),\nT56 -> T59,\nT57 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="EVAL with clause\nappend3(.(X7, X8), X9, .(X7, X10)) :- append3(X8, X9, X10).\nand substitutionX7 -> T10,\nX8 -> T14,\nT1 -> .(T10, T14),\nT2 -> T15,\nX9 -> T15,\nX10 -> T13,\nT5 -> .(T10, T13),\nT11 -> T14,\nT12 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 44 [arrowhead = none ];
44 -> 13;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 45 [arrowhead = none ];
45 -> 15;

46 [label="EVAL with clause\nappend3([], X13, X13).\nand substitutionT14 -> [],\nT15 -> T18,\nX13 -> T18,\nT13 -> T18", 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\nappend3(.(X36, X37), X38, .(X36, X39)) :- append3(X37, X38, X39).\nand substitutionX36 -> T43,\nX37 -> T47,\nT14 -> .(T43, T47),\nT15 -> T48,\nX38 -> T48,\nX39 -> T46,\nT13 -> .(T43, T46),\nT44 -> T47,\nT45 -> T48", 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\nappend3(.(X22, X23), X24, .(X22, X25)) :- append3(X23, X24, X25).\nand substitutionX22 -> T27,\nX23 -> T31,\nT14 -> .(T27, T31),\nT15 -> T32,\nX24 -> T32,\nX25 -> T30,\nT18 -> .(T27, T30),\nT28 -> T31,\nT29 -> T32", 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 -> T31\nT2 -> T32\nT3 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 53 [arrowhead = none , style=dashed];
53 -> 1[style=dashed];

54 [label="INSTANCE with matching:\nT1 -> T47\nT2 -> T48\nT3 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 54 [arrowhead = none , style=dashed];
54 -> 1[style=dashed];

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

56 [label="EVAL with clause\nappend3([], X52, X52).\nand substitutionT59 -> [],\nT60 -> T63,\nX52 -> T63,\nT58 -> T63", 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\nappend3(.(X75, X76), X77, .(X75, X78)) :- append3(X76, X77, X78).\nand substitutionX75 -> T88,\nX76 -> T92,\nT59 -> .(T88, T92),\nT60 -> T93,\nX77 -> T93,\nX78 -> T91,\nT58 -> .(T88, T91),\nT89 -> T92,\nT90 -> T93", 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\nappend3(.(X61, X62), X63, .(X61, X64)) :- append3(X62, X63, X64).\nand substitutionX61 -> T72,\nX62 -> T76,\nT59 -> .(T72, T76),\nT60 -> T77,\nX63 -> T77,\nX64 -> T75,\nT63 -> .(T72, T75),\nT73 -> T76,\nT74 -> T77", 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 -> T76\nT2 -> T77\nT3 -> T75", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 63 [arrowhead = none , style=dashed];
63 -> 1[style=dashed];

64 [label="INSTANCE with matching:\nT1 -> T92\nT2 -> T93\nT3 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 64 [arrowhead = none , style=dashed];
64 -> 1[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1([], z0)
f1_in(.(z0, z1)) → f1_out1(.(z0, []), z1)
f1_in(.(z0, .(z1, z2))) → U1(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U2(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U3(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U4(f1_in(z2), .(z0, .(z1, z2)))
U1(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U2(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U3(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U4(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)

Tuples:

F1_IN(.(z0, .(z1, z2))) → c2(U1'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(U2'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(U3'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(U4'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))

S tuples:

F1_IN(.(z0, .(z1, z2))) → c2(U1'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(U2'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(U3'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(U4'(f1_in(z2), .(z0, .(z1, z2))), F1_IN(z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4

Defined Pair Symbols:

F1_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:

f1_in(z0) → f1_out1([], z0)
f1_in(.(z0, z1)) → f1_out1(.(z0, []), z1)
f1_in(.(z0, .(z1, z2))) → U1(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U2(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U3(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U4(f1_in(z2), .(z0, .(z1, z2)))
U1(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U2(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U3(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U4(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)

Tuples:

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

S tuples:

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4

Defined Pair Symbols:

F1_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.

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

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

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F1_IN(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1([], z0)
f1_in(.(z0, z1)) → f1_out1(.(z0, []), z1)
f1_in(.(z0, .(z1, z2))) → U1(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U2(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U3(f1_in(z2), .(z0, .(z1, z2)))
f1_in(.(z0, .(z1, z2))) → U4(f1_in(z2), .(z0, .(z1, z2)))
U1(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U2(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U3(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)
U4(f1_out1(z0, z1), .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)), z1)

Tuples:

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

S tuples:none
K tuples:

F1_IN(.(z0, .(z1, z2))) → c2(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c3(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c4(F1_IN(z2))
F1_IN(.(z0, .(z1, z2))) → c5(F1_IN(z2))

Defined Rule Symbols:

f1_in, U1, U2, U3, U4

Defined Pair Symbols:

F1_IN

Compound Symbols:

c2, c3, c4, c5