Runtime Complexity proof of /tmp/tmpjZRcCx/gopher.pl

Runtime Complexity of the query pattern
gopher(g,a)
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), 74 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 146 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 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))), 132 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).

Query: gopher(g,a)

(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: gopher(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: gopher(T1, T2), SCOPE: 1, CLAUSE: 0 | gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\ngopher(T1, T2) !=? gopher(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | gopher(cons(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX8 is free\ngopher(T1, T2) !=? gopher(nil, nil)\ngopher(T1, T2) !=? gopher(cons(nil, X8), cons(nil, X8))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: gopher(cons(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: gopher(cons(T14, cons(T15, T16)), T18)\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL with clause\ngopher(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 32 [arrowhead = none ];
32 -> 9;

33 [label="EVAL with clause\ngopher(cons(nil, X8), cons(nil, X8)).\nand substitutionX8 -> T5,\nT1 -> cons(nil, T5),\nT2 -> cons(nil, T5)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 33 [arrowhead = none ];
33 -> 16;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 34 [arrowhead = none ];
34 -> 19;

35 [label="BACKTRACK\nfor clause: gopher(cons(nil, Y), cons(nil, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 35 [arrowhead = none ];
35 -> 12;

36 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 36 [arrowhead = none ];
36 -> 13;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 37 [arrowhead = none ];
37 -> 21;

38 [label="EVAL with clause\ngopher(cons(cons(X21, X22), X23), X24) :- gopher(cons(X21, cons(X22, X23)), X24).\nand substitutionX21 -> T14,\nX22 -> T15,\nX23 -> T16,\nT1 -> cons(cons(T14, T15), T16),\nT2 -> T18,\nX24 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 38 [arrowhead = none ];
38 -> 26;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 39 [arrowhead = none ];
39 -> 28;

40 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 40 [arrowhead = none ];
40 -> 22;

41 [label="INSTANCE with matching:\nT1 -> cons(T14, cons(T15, T16))\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 41 [arrowhead = none , style=dashed];
41 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil) → f1_out1(nil)
f1_in(cons(nil, z0)) → f1_out1(cons(nil, z0))
f1_in(cons(cons(z0, z1), z2)) → U1(f1_in(cons(z0, cons(z1, z2))), cons(cons(z0, z1), z2))
U1(f1_out1(z0), cons(cons(z1, z2), z3)) → f1_out1(z0)

Tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(U1'(f1_in(cons(z0, cons(z1, z2))), cons(cons(z0, z1), z2)), F1_IN(cons(z0, cons(z1, z2))))

S tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(U1'(f1_in(cons(z0, cons(z1, z2))), cons(cons(z0, z1), z2)), F1_IN(cons(z0, cons(z1, z2))))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c2

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil) → f1_out1(nil)
f1_in(cons(nil, z0)) → f1_out1(cons(nil, z0))
f1_in(cons(cons(z0, z1), z2)) → U1(f1_in(cons(z0, cons(z1, z2))), cons(cons(z0, z1), z2))
U1(f1_out1(z0), cons(cons(z1, z2), z3)) → f1_out1(z0)

Tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

S tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

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(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

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

F1_IN(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil) → f1_out1(nil)
f1_in(cons(nil, z0)) → f1_out1(cons(nil, z0))
f1_in(cons(cons(z0, z1), z2)) → U1(f1_in(cons(z0, cons(z1, z2))), cons(cons(z0, z1), z2))
U1(f1_out1(z0), cons(cons(z1, z2), z3)) → f1_out1(z0)

Tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

S tuples:none
K tuples:

F1_IN(cons(cons(z0, z1), z2)) → c2(F1_IN(cons(z0, cons(z1, z2))))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

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: gopher(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: gopher(T1, T2), SCOPE: 1, CLAUSE: 0 | gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\ngopher(T1, T2) !=? gopher(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: gopher(nil, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | gopher(cons(nil, T4), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX7 is free\ngopher(T1, T2) !=? gopher(nil, nil)\ngopher(T1, T2) !=? gopher(cons(nil, X7), cons(nil, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: gopher(cons(nil, T4), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: gopher(cons(T9, cons(T10, T11)), T13)\n\nT9 is ground\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 0 | gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 1 | gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 1 | gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true | gopher(cons(nil, cons(T18, T19)), T13), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: gopher(cons(T9, cons(T10, T11)), T13), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nT11 is ground\nX22 is free\ngopher(cons(T9, cons(T10, T11)), T13) !=? gopher(cons(nil, X22), cons(nil, X22))", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: gopher(cons(nil, cons(T18, T19)), T13), SCOPE: 2, CLAUSE: 2\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: gopher(cons(T30, cons(T31, cons(T32, T33))), T35)\n\nT30 is ground\nT31 is ground\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 35 [arrowhead = none ];
35 -> 4;

36 [label="EVAL with clause\ngopher(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 36 [arrowhead = none ];
36 -> 5;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 38 [arrowhead = none ];
38 -> 10;

39 [label="EVAL with clause\ngopher(cons(nil, X7), cons(nil, X7)).\nand substitutionX7 -> T4,\nT1 -> cons(nil, T4),\nT2 -> cons(nil, T4)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 40 [arrowhead = none ];
40 -> 17;

41 [label="BACKTRACK\nfor clause: gopher(cons(nil, Y), cons(nil, Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 41 [arrowhead = none ];
41 -> 11;

42 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 42 [arrowhead = none ];
42 -> 14;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="EVAL with clause\ngopher(cons(cons(X16, X17), X18), X19) :- gopher(cons(X16, cons(X17, X18)), X19).\nand substitutionX16 -> T9,\nX17 -> T10,\nX18 -> T11,\nT1 -> cons(cons(T9, T10), T11),\nT2 -> T13,\nX19 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 45 [arrowhead = none ];
45 -> 24;

46 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 47 [arrowhead = none ];
47 -> 25;

48 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 48 [arrowhead = none ];
48 -> 27;

49 [label="EVAL with clause\ngopher(cons(nil, X22), cons(nil, X22)).\nand substitutionT9 -> nil,\nT10 -> T18,\nT11 -> T19,\nX22 -> cons(T18, T19),\nT13 -> cons(nil, cons(T18, T19))", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 49 [arrowhead = none ];
49 -> 29;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 50 [arrowhead = none ];
50 -> 30;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 51 [arrowhead = none ];
51 -> 31;

52 [label="EVAL with clause\ngopher(cons(cons(X35, X36), X37), X38) :- gopher(cons(X35, cons(X36, X37)), X38).\nand substitutionX35 -> T30,\nX36 -> T31,\nT9 -> cons(T30, T31),\nT10 -> T32,\nT11 -> T33,\nX37 -> cons(T32, T33),\nT13 -> T35,\nX38 -> T35,\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 52 [arrowhead = none ];
52 -> 33;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 53 [arrowhead = none ];
53 -> 34;

54 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 54 [arrowhead = none ];
54 -> 32;

55 [label="INSTANCE with matching:\nT1 -> cons(T30, cons(T31, cons(T32, T33)))\nT2 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 55 [arrowhead = none , style=dashed];
55 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil) → f2_out1(nil)
f2_in(cons(nil, z0)) → f2_out1(cons(nil, z0))
f2_in(cons(cons(nil, z0), z1)) → f2_out1(cons(nil, cons(z0, z1)))
f2_in(cons(cons(cons(z0, z1), z2), z3)) → U1(f2_in(cons(z0, cons(z1, cons(z2, z3)))), cons(cons(cons(z0, z1), z2), z3))
U1(f2_out1(z0), cons(cons(cons(z1, z2), z3), z4)) → f2_out1(z0)

Tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(U1'(f2_in(cons(z0, cons(z1, cons(z2, z3)))), cons(cons(cons(z0, z1), z2), z3)), F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

S tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(U1'(f2_in(cons(z0, cons(z1, cons(z2, z3)))), cons(cons(cons(z0, z1), z2), z3)), F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil) → f2_out1(nil)
f2_in(cons(nil, z0)) → f2_out1(cons(nil, z0))
f2_in(cons(cons(nil, z0), z1)) → f2_out1(cons(nil, cons(z0, z1)))
f2_in(cons(cons(cons(z0, z1), z2), z3)) → U1(f2_in(cons(z0, cons(z1, cons(z2, z3)))), cons(cons(cons(z0, z1), z2), z3))
U1(f2_out1(z0), cons(cons(cons(z1, z2), z3), z4)) → f2_out1(z0)

Tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

S tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3

(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(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

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

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

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

POL(F2_IN(x₁)) = [2]x₁
POL(c3(x₁)) = x₁
POL(cons(x₁, x₂)) = [2] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil) → f2_out1(nil)
f2_in(cons(nil, z0)) → f2_out1(cons(nil, z0))
f2_in(cons(cons(nil, z0), z1)) → f2_out1(cons(nil, cons(z0, z1)))
f2_in(cons(cons(cons(z0, z1), z2), z3)) → U1(f2_in(cons(z0, cons(z1, cons(z2, z3)))), cons(cons(cons(z0, z1), z2), z3))
U1(f2_out1(z0), cons(cons(cons(z1, z2), z3), z4)) → f2_out1(z0)

Tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

S tuples:none
K tuples:

F2_IN(cons(cons(cons(z0, z1), z2), z3)) → c3(F2_IN(cons(z0, cons(z1, cons(z2, z3)))))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3