Runtime Complexity proof of /tmp/tmpT3Of6p/append

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

0 Prolog

↳1 BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 52 ms)

↳4 CdtProblem

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

    ↳6 CdtProblem

    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 114 ms)

    ↳8 CdtProblem

    ↳9 SIsEmptyProof (⇔, 0 ms)

    ↳10 BOUNDS(O(1), O(1))

↳11 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 76 ms)

↳12 CdtProblem

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

    ↳14 CdtProblem

    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 88 ms)

    ↳16 CdtProblem

(0) Obligation:

Clauses:

append(X, Y, Z) :- ','(=(X, []), ','(!, =(Y, Z))).
append(X, Y, Z) :- ','(=(X, .(H, Xs)), ','(!, ','(=(Z, .(H, Zs)), append(Xs, Y, Zs)))).
=(X, X).

Query: append(g,a,a)

(1) BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

append(X, Y, Z) :- ','(user_defined_=(X, []), ','(!, user_defined_=(Y, Z))).
append(X, Y, Z) :- ','(user_defined_=(X, .(H, Xs)), ','(!, ','(user_defined_=(Z, .(H, Zs)), append(Xs, Y, Zs)))).
user_defined_=(X, X).

Query: append(g,a,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: append(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: append(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | append(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(user_defined_=(T12, []), ','(!_1, user_defined_=(T15, T16))) | append(T12, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(user_defined_=(T12, []), ','(!_1, user_defined_=(T15, T16))), SCOPE: 2, CLAUSE: 2 | ?_2 | append(T12, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, user_defined_=(T15, T16)) | ?_2 | append([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ?_2 | append(T12, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT12 is ground\nX12 is free\nuser_defined_=(T12, []) !=? user_defined_=(X12, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: user_defined_=(T15, T16)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: user_defined_=(T15, T16), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(T12, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT12 is ground\nX12 is free\nuser_defined_=(T12, []) !=? user_defined_=(X12, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(user_defined_=(T31, .(X28, X29)), ','(!_1, ','(user_defined_=(T34, .(X28, X30)), append(X29, T35, X30))))\n\nT31 is ground\nX12 is free\nX28 is free\nX29 is free\nX30 is free\nuser_defined_=(T31, []) !=? user_defined_=(X12, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(user_defined_=(T31, .(X28, X29)), ','(!_1, ','(user_defined_=(T34, .(X28, X30)), append(X29, T35, X30)))), SCOPE: 4, CLAUSE: 2\n\nT31 is ground\nX12 is free\nX28 is free\nX29 is free\nX30 is free\nuser_defined_=(T31, []) !=? user_defined_=(X12, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(!_1, ','(user_defined_=(T34, .(T43, X30)), append(T44, T35, X30)))\n\nT43 is ground\nT44 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(user_defined_=(T34, .(T43, X30)), append(T44, T35, X30))\n\nT43 is ground\nT44 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(user_defined_=(T34, .(T43, X30)), append(T44, T35, X30)), SCOPE: 5, CLAUSE: 2\n\nT43 is ground\nT44 is ground\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: append(T44, T56, T57)\n\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="ONLY EVAL with clause\nappend(X7, X8, X9) :- ','(user_defined_=(X7, []), ','(!_1, user_defined_=(X8, X9))).\nand substitutionT1 -> T12,\nX7 -> T12,\nT2 -> T15,\nX8 -> T15,\nT3 -> T16,\nX9 -> T16,\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 42 [arrowhead = none ];
42 -> 10;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="EVAL with clause\nuser_defined_=(X12, X12).\nand substitutionT12 -> [],\nX12 -> [],\nT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 44 [arrowhead = none ];
44 -> 13;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 47 [arrowhead = none ];
47 -> 24;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 48 [arrowhead = none ];
48 -> 20;

49 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT15 -> T22,\nX15 -> T22,\nT16 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 49 [arrowhead = none ];
49 -> 21;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 50 [arrowhead = none ];
50 -> 22;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="ONLY EVAL with clause\nappend(X25, X26, X27) :- ','(user_defined_=(X25, .(X28, X29)), ','(!_1, ','(user_defined_=(X27, .(X28, X30)), append(X29, X26, X30)))).\nand substitutionT12 -> T31,\nX25 -> T31,\nT2 -> T35,\nX26 -> T35,\nT3 -> T34,\nX27 -> T34,\nT33 -> T34,\nT32 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 52 [arrowhead = none ];
52 -> 31;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 53 [arrowhead = none ];
53 -> 34;

54 [label="EVAL with clause\nuser_defined_=(X33, X33).\nand substitutionT31 -> .(T43, T44),\nX33 -> .(T43, T44),\nX28 -> T43,\nX29 -> T44,\nT42 -> .(T43, T44)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 54 [arrowhead = none ];
54 -> 35;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 55 [arrowhead = none ];
55 -> 36;

56 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 56 [arrowhead = none ];
56 -> 37;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 57 [arrowhead = none ];
57 -> 38;

58 [label="EVAL with clause\nuser_defined_=(X40, X40).\nand substitutionT34 -> .(T54, T57),\nX40 -> .(T54, T57),\nT43 -> T54,\nX30 -> T57,\nT53 -> .(T54, T57),\nT35 -> T56,\nT55 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 58 [arrowhead = none ];
58 -> 39;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 59 [arrowhead = none ];
59 -> 40;

60 [label="INSTANCE with matching:\nT1 -> T44\nT2 -> T56\nT3 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 60 [arrowhead = none , style=dashed];
60 -> 2[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(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))

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

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F2_IN(x₁)) = x₁
POL(c1(x₁)) = x₁

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(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: append(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: append(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | append(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(user_defined_=(T7, []), ','(!_1, user_defined_=(T10, T11))) | append(T7, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(user_defined_=(T7, []), ','(!_1, user_defined_=(T10, T11))), SCOPE: 2, CLAUSE: 2 | ?_2 | append(T7, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, user_defined_=(T10, T11)) | ?_2 | append([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ?_2 | append(T7, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nX9 is free\nuser_defined_=(T7, []) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: user_defined_=(T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: user_defined_=(T10, T11), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(T7, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nX9 is free\nuser_defined_=(T7, []) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(user_defined_=(T26, .(X25, X26)), ','(!_1, ','(user_defined_=(T29, .(X25, X27)), append(X26, T30, X27))))\n\nT26 is ground\nX9 is free\nX25 is free\nX26 is free\nX27 is free\nuser_defined_=(T26, []) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(user_defined_=(T26, .(X25, X26)), ','(!_1, ','(user_defined_=(T29, .(X25, X27)), append(X26, T30, X27)))), SCOPE: 4, CLAUSE: 2\n\nT26 is ground\nX9 is free\nX25 is free\nX26 is free\nX27 is free\nuser_defined_=(T26, []) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(!_1, ','(user_defined_=(T29, .(T38, X27)), append(T39, T30, X27)))\n\nT38 is ground\nT39 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(user_defined_=(T29, .(T38, X27)), append(T39, T30, X27))\n\nT38 is ground\nT39 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(user_defined_=(T29, .(T38, X27)), append(T39, T30, X27)), SCOPE: 5, CLAUSE: 2\n\nT38 is ground\nT39 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append(T39, T51, T52)\n\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];
1 -> 34 [arrowhead = none ];
34 -> 4;

35 [label="ONLY EVAL with clause\nappend(X4, X5, X6) :- ','(user_defined_=(X4, []), ','(!_1, user_defined_=(X5, X6))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T10,\nX5 -> T10,\nT3 -> T11,\nX6 -> T11,\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="EVAL with clause\nuser_defined_=(X9, X9).\nand substitutionT7 -> [],\nX9 -> [],\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 38 [arrowhead = none ];
38 -> 8;

39 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 39 [arrowhead = none ];
39 -> 9;

40 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 41 [arrowhead = none ];
41 -> 12;

42 [label="EVAL with clause\nuser_defined_=(X12, X12).\nand substitutionT10 -> T17,\nX12 -> T17,\nT11 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 42 [arrowhead = none ];
42 -> 14;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 43 [arrowhead = none ];
43 -> 15;

44 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="ONLY EVAL with clause\nappend(X22, X23, X24) :- ','(user_defined_=(X22, .(X25, X26)), ','(!_1, ','(user_defined_=(X24, .(X25, X27)), append(X26, X23, X27)))).\nand substitutionT7 -> T26,\nX22 -> T26,\nT2 -> T30,\nX23 -> T30,\nT3 -> T29,\nX24 -> T29,\nT28 -> T29,\nT27 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 45 [arrowhead = none ];
45 -> 25;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 46 [arrowhead = none ];
46 -> 26;

47 [label="EVAL with clause\nuser_defined_=(X30, X30).\nand substitutionT26 -> .(T38, T39),\nX30 -> .(T38, T39),\nX25 -> T38,\nX26 -> T39,\nT37 -> .(T38, T39)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 47 [arrowhead = none ];
47 -> 27;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 48 [arrowhead = none ];
48 -> 28;

49 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 49 [arrowhead = none ];
49 -> 29;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 50 [arrowhead = none ];
50 -> 30;

51 [label="EVAL with clause\nuser_defined_=(X37, X37).\nand substitutionT29 -> .(T49, T52),\nX37 -> .(T49, T52),\nT38 -> T49,\nX27 -> T52,\nT48 -> .(T49, T52),\nT30 -> T51,\nT50 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 51 [arrowhead = none ];
51 -> 32;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 52 [arrowhead = none ];
52 -> 33;

53 [label="INSTANCE with matching:\nT1 -> T39\nT2 -> T51\nT3 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 53 [arrowhead = none , style=dashed];
53 -> 1[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(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)) → c1(F1_IN(z1))

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

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1