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

(0) Obligation:

Clauses:

less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Query: less(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: less(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | less(T1, s(T5)), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX4 is free\nless(T1, T2) !=? less(0, s(X4))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: less(T1, s(T5)), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T12, T11)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: less(T21, T20)\n\nT20 is ground\nX4 is free\nless(T1, s(T20)) !=? less(0, s(X4))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 21 [arrowhead = none ];
21 -> 3;

22 [label="EVAL with clause\nless(0, s(X4)).\nand substitutionT1 -> 0,\nX4 -> T5,\nT2 -> s(T5)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 22 [arrowhead = none ];
22 -> 5;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 23 [arrowhead = none ];
23 -> 7;

24 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="EVAL with clause\nless(s(X17), s(X18)) :- less(X17, X18).\nand substitutionX17 -> T21,\nT1 -> s(T21),\nX18 -> T20,\nT2 -> s(T20),\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 25 [arrowhead = none ];
25 -> 19;

26 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 26 [arrowhead = none ];
26 -> 20;

27 [label="EVAL with clause\nless(s(X9), s(X10)) :- less(X9, X10).\nand substitutionX9 -> T12,\nT1 -> s(T12),\nT5 -> T11,\nX10 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 27 [arrowhead = none ];
27 -> 11;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 28 [arrowhead = none ];
28 -> 12;

29 [label="INSTANCE with matching:\nT1 -> T12\nT2 -> T11", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 29 [arrowhead = none , style=dashed];
29 -> 1[style=dashed];

30 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 30 [arrowhead = none , style=dashed];
30 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(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), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))

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

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(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), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))

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

(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(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   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(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), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))

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

(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: less(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | less(T1, s(T4)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX3 is free\nless(T1, T2) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: less(T1, s(T4)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T9, T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(T9, T8), SCOPE: 2, CLAUSE: 0 | less(T9, T8), SCOPE: 2, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | less(T9, s(T12)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T9, T8), SCOPE: 2, CLAUSE: 1\n\nT8 is ground\nX10 is free\nless(T9, T8) !=? less(0, s(X10))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T9, s(T12)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: less(T19, T18)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T28, T27)\n\nT27 is ground\nX10 is free\nless(T9, s(T27)) !=? less(0, s(X10))", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(T35, T34)\n\nT34 is ground\nX3 is free\nless(T1, s(T34)) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T35, T34), SCOPE: 3, CLAUSE: 0 | less(T35, T34), SCOPE: 3, CLAUSE: 1\n\nT34 is ground\nX3 is free\nless(T1, s(T34)) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | less(T35, s(T38)), SCOPE: 3, CLAUSE: 1\n\nT38 is ground\nX3 is free\nless(T1, s(s(T38))) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T35, T34), SCOPE: 3, CLAUSE: 1\n\nT34 is ground\nX3 is free\nX33 is free\nless(T1, s(T34)) !=? less(0, s(X3))\nless(T35, T34) !=? less(0, s(X33))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T35, s(T38)), SCOPE: 3, CLAUSE: 1\n\nT38 is ground\nX3 is free\nless(T1, s(s(T38))) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: less(T45, T44)\n\nT44 is ground\nX3 is free\nless(T1, s(s(T44))) !=? less(0, s(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T54, T53)\n\nT53 is ground\nX3 is free\nX33 is free\nless(T1, s(s(T53))) !=? less(0, s(X3))\nless(T35, s(T53)) !=? less(0, s(X33))", 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\nless(0, s(X3)).\nand substitutionT1 -> 0,\nX3 -> T4,\nT2 -> s(T4)", 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 -> 8;

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

41 [label="EVAL with clause\nless(s(X29), s(X30)) :- less(X29, X30).\nand substitutionX29 -> T35,\nT1 -> s(T35),\nX30 -> T34,\nT2 -> s(T34),\nT33 -> T35", 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\nless(s(X6), s(X7)) :- less(X6, X7).\nand substitutionX6 -> T9,\nT1 -> s(T9),\nT4 -> T8,\nX7 -> T8,\nT7 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 43 [arrowhead = none ];
43 -> 13;

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];
13 -> 45 [arrowhead = none ];
45 -> 15;

46 [label="EVAL with clause\nless(0, s(X10)).\nand substitutionT9 -> 0,\nX10 -> T12,\nT8 -> s(T12)", 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\nless(s(X23), s(X24)) :- less(X23, X24).\nand substitutionX23 -> T28,\nT9 -> s(T28),\nX24 -> T27,\nT8 -> s(T27),\nT26 -> T28", 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\nless(s(X15), s(X16)) :- less(X15, X16).\nand substitutionX15 -> T19,\nT9 -> s(T19),\nT12 -> T18,\nX16 -> T18,\nT17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 51 [arrowhead = none ];
51 -> 21;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 52 [arrowhead = none ];
52 -> 22;

53 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 53 [arrowhead = none , style=dashed];
53 -> 2[style=dashed];

54 [label="INSTANCE with matching:\nT1 -> T28\nT2 -> T27", 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\nless(0, s(X33)).\nand substitutionT35 -> 0,\nX33 -> T38,\nT34 -> s(T38)", 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\nless(s(X46), s(X47)) :- less(X46, X47).\nand substitutionX46 -> T54,\nT35 -> s(T54),\nX47 -> T53,\nT34 -> s(T53),\nT52 -> T54", 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\nless(s(X38), s(X39)) :- less(X38, X39).\nand substitutionX38 -> T45,\nT35 -> s(T45),\nT38 -> T44,\nX39 -> T44,\nT43 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 61 [arrowhead = none ];
61 -> 33;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 62 [arrowhead = none ];
62 -> 34;

63 [label="INSTANCE with matching:\nT1 -> T45\nT2 -> T44", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 63 [arrowhead = none , style=dashed];
63 -> 2[style=dashed];

64 [label="INSTANCE with matching:\nT1 -> T54\nT2 -> T53", 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(s(z0)) → f2_out1(0)
f2_in(s(s(z0))) → f2_out1(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), s(s(z1))) → f2_out1(s(s(z0)))
U2(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U3(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U4(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))

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

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

Removed 4 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

(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(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   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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