Runtime Complexity proof of /tmp/tmpc5lOH6/flat-bf.pl

Runtime Complexity of the query pattern
flat(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), 47 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

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

    ↳6 CdtProblem

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

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 28 ms)

    ↳10 CdtProblem

    ↳11 CdtKnowledgeProof (⇔, 0 ms)

    ↳12 BOUNDS(O(1), O(1))

↳13 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 71 ms)

↳14 CdtProblem

    ↳15 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 1 ms)

    ↳16 CdtProblem

    ↳17 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳18 CdtProblem

(0) Obligation:

Clauses:

flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).

Query: flat(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: flat(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: flat(T1, T2), SCOPE: 1, CLAUSE: 0 | flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nflat(T1, T2) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: flat(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nflat(T1, T2) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nflat(T1, T2) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: flat(T11, T13)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: flat(.(T23, T24), T26)\n\nT23 is ground\nT24 is ground", 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\nflat([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 38 [arrowhead = none ];
38 -> 7;

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];
7 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 41 [arrowhead = none ];
41 -> 15;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 42 [arrowhead = none ];
42 -> 16;

43 [label="BACKTRACK\nfor clause: flat(.([], T), R) :- flat(T, R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="EVAL with clause\nflat(.([], X15), X16) :- flat(X15, X16).\nand substitutionX15 -> T11,\nT1 -> .([], T11),\nT2 -> T13,\nX16 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 46 [arrowhead = none ];
46 -> 22;

47 [label="EVAL with clause\nflat(.(.(X25, X26), X27), .(X25, X28)) :- flat(.(X26, X27), X28).\nand substitutionX25 -> T22,\nX26 -> T23,\nX27 -> T24,\nT1 -> .(.(T22, T23), T24),\nX28 -> T26,\nT2 -> .(T22, T26),\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 47 [arrowhead = none ];
47 -> 35;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 48 [arrowhead = none ];
48 -> 36;

49 [label="INSTANCE with matching:\nT1 -> T11\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 49 [arrowhead = none , style=dashed];
49 -> 1[style=dashed];

50 [label="INSTANCE with matching:\nT1 -> .(T23, T24)\nT2 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 50 [arrowhead = none , style=dashed];
50 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(z0) → U1(f9_in(z0), z0)
U1(f9_out1(z0), z1) → f1_out1(z0)
U1(f9_out2(z0), z1) → f1_out1(z0)
f15_in(.([], z0)) → U2(f1_in(z0), .([], z0))
U2(f1_out1(z0), .([], z1)) → f15_out1(z0)
f16_in(.(.(z0, z1), z2)) → U3(f1_in(.(z1, z2)), .(.(z0, z1), z2))
U3(f1_out1(z0), .(.(z1, z2), z3)) → f16_out1(.(z1, z0))
f9_in(z0) → U4(f15_in(z0), f16_in(z0), z0)
U4(f15_out1(z0), z1, z2) → f9_out1(z0)
U4(z0, f16_out1(z1), z2) → f9_out2(z1)

Tuples:

F1_IN(z0) → c1(U1'(f9_in(z0), z0), F9_IN(z0))
F15_IN(.([], z0)) → c4(U2'(f1_in(z0), .([], z0)), F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(U3'(f1_in(.(z1, z2)), .(.(z0, z1), z2)), F1_IN(.(z1, z2)))
F9_IN(z0) → c8(U4'(f15_in(z0), f16_in(z0), z0), F15_IN(z0), F16_IN(z0))

S tuples:

F1_IN(z0) → c1(U1'(f9_in(z0), z0), F9_IN(z0))
F15_IN(.([], z0)) → c4(U2'(f1_in(z0), .([], z0)), F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(U3'(f1_in(.(z1, z2)), .(.(z0, z1), z2)), F1_IN(.(z1, z2)))
F9_IN(z0) → c8(U4'(f15_in(z0), f16_in(z0), z0), F15_IN(z0), F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, f16_in, U3, f9_in, U4

Defined Pair Symbols:

F1_IN, F15_IN, F16_IN, F9_IN

Compound Symbols:

c1, c4, c6, c8

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

Removed 4 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(z0) → U1(f9_in(z0), z0)
U1(f9_out1(z0), z1) → f1_out1(z0)
U1(f9_out2(z0), z1) → f1_out1(z0)
f15_in(.([], z0)) → U2(f1_in(z0), .([], z0))
U2(f1_out1(z0), .([], z1)) → f15_out1(z0)
f16_in(.(.(z0, z1), z2)) → U3(f1_in(.(z1, z2)), .(.(z0, z1), z2))
U3(f1_out1(z0), .(.(z1, z2), z3)) → f16_out1(.(z1, z0))
f9_in(z0) → U4(f15_in(z0), f16_in(z0), z0)
U4(f15_out1(z0), z1, z2) → f9_out1(z0)
U4(z0, f16_out1(z1), z2) → f9_out2(z1)

Tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(z0) → c8(F15_IN(z0), F16_IN(z0))

S tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(z0) → c8(F15_IN(z0), F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, f16_in, U3, f9_in, U4

Defined Pair Symbols:

F1_IN, F15_IN, F16_IN, F9_IN

Compound Symbols:

c1, c4, c6, c8

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

Use forward instantiation to replace F9_IN(z0) → c8(F15_IN(z0), F16_IN(z0)) by

F9_IN(.([], y0)) → c8(F15_IN(.([], y0)), F16_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F15_IN(.(.(y0, y1), y2)), F16_IN(.(.(y0, y1), y2)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(z0) → U1(f9_in(z0), z0)
U1(f9_out1(z0), z1) → f1_out1(z0)
U1(f9_out2(z0), z1) → f1_out1(z0)
f15_in(.([], z0)) → U2(f1_in(z0), .([], z0))
U2(f1_out1(z0), .([], z1)) → f15_out1(z0)
f16_in(.(.(z0, z1), z2)) → U3(f1_in(.(z1, z2)), .(.(z0, z1), z2))
U3(f1_out1(z0), .(.(z1, z2), z3)) → f16_out1(.(z1, z0))
f9_in(z0) → U4(f15_in(z0), f16_in(z0), z0)
U4(f15_out1(z0), z1, z2) → f9_out1(z0)
U4(z0, f16_out1(z1), z2) → f9_out2(z1)

Tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)), F16_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F15_IN(.(.(y0, y1), y2)), F16_IN(.(.(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)), F16_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F15_IN(.(.(y0, y1), y2)), F16_IN(.(.(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, f16_in, U3, f9_in, U4

Defined Pair Symbols:

F1_IN, F15_IN, F16_IN, F9_IN

Compound Symbols:

c1, c4, c6, c8

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(z0) → U1(f9_in(z0), z0)
U1(f9_out1(z0), z1) → f1_out1(z0)
U1(f9_out2(z0), z1) → f1_out1(z0)
f15_in(.([], z0)) → U2(f1_in(z0), .([], z0))
U2(f1_out1(z0), .([], z1)) → f15_out1(z0)
f16_in(.(.(z0, z1), z2)) → U3(f1_in(.(z1, z2)), .(.(z0, z1), z2))
U3(f1_out1(z0), .(.(z1, z2), z3)) → f16_out1(.(z1, z0))
f9_in(z0) → U4(f15_in(z0), f16_in(z0), z0)
U4(f15_out1(z0), z1, z2) → f9_out1(z0)
U4(z0, f16_out1(z1), z2) → f9_out2(z1)

Tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, f16_in, U3, f9_in, U4

Defined Pair Symbols:

F1_IN, F15_IN, F16_IN, F9_IN

Compound Symbols:

c1, c4, c6, c8

(9) 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.

F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))

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

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂
POL(F15_IN(x₁)) = [2]x₁
POL(F16_IN(x₁)) = [2]x₁
POL(F1_IN(x₁)) = [2]x₁
POL(F9_IN(x₁)) = [2]x₁
POL([]) = 0
POL(c1(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c8(x₁)) = x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1([])
f1_in(z0) → U1(f9_in(z0), z0)
U1(f9_out1(z0), z1) → f1_out1(z0)
U1(f9_out2(z0), z1) → f1_out1(z0)
f15_in(.([], z0)) → U2(f1_in(z0), .([], z0))
U2(f1_out1(z0), .([], z1)) → f15_out1(z0)
f16_in(.(.(z0, z1), z2)) → U3(f1_in(.(z1, z2)), .(.(z0, z1), z2))
U3(f1_out1(z0), .(.(z1, z2), z3)) → f16_out1(.(z1, z0))
f9_in(z0) → U4(f15_in(z0), f16_in(z0), z0)
U4(f15_out1(z0), z1, z2) → f9_out1(z0)
U4(z0, f16_out1(z1), z2) → f9_out2(z1)

Tuples:

F1_IN(z0) → c1(F9_IN(z0))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F9_IN(z0))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))

K tuples:

F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))

Defined Rule Symbols:

f1_in, U1, f15_in, U2, f16_in, U3, f9_in, U4

Defined Pair Symbols:

F1_IN, F15_IN, F16_IN, F9_IN

Compound Symbols:

c1, c4, c6, c8

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(z0) → c1(F9_IN(z0))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))
F9_IN(.([], y0)) → c8(F15_IN(.([], y0)))
F9_IN(.(.(y0, y1), y2)) → c8(F16_IN(.(.(y0, y1), y2)))
F15_IN(.([], z0)) → c4(F1_IN(z0))
F16_IN(.(.(z0, z1), z2)) → c6(F1_IN(.(z1, z2)))

Now S is empty

(12) BOUNDS(O(1), O(1))

(13) 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: flat(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: flat(T1, T2), SCOPE: 1, CLAUSE: 0 | flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nflat(T1, T2) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: flat(T5, T7) | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX9 is free\nX10 is free\nflat(T1, T2) !=? flat([], [])\nflat(T1, T2) !=? flat(.([], X9), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: flat(T5, T7), SCOPE: 2, CLAUSE: 0 | flat(T5, T7), SCOPE: 2, CLAUSE: 1 | flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true | flat([], T7), SCOPE: 2, CLAUSE: 1 | flat([], T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], []), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: flat(T5, T7), SCOPE: 2, CLAUSE: 1 | flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: flat([], T7), SCOPE: 2, CLAUSE: 1 | flat([], T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], []), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: flat([], T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], []), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_2 | flat(.([], []), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: flat(.([], []), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: flat(T5, T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flat(T16, T18)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: flat(T5, T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: flat(.(T36, T37), T39)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\nflat(T5, T7) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: flat(.(T45, T46), T48)\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 0 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: flat(T57, T59)\n\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: flat(.(T69, T70), T72)\n\nT69 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 51 [arrowhead = none ];
51 -> 4;

52 [label="EVAL with clause\nflat([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 52 [arrowhead = none ];
52 -> 5;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 53 [arrowhead = none ];
53 -> 6;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 54 [arrowhead = none ];
54 -> 8;

55 [label="EVAL with clause\nflat(.([], X9), X10) :- flat(X9, X10).\nand substitutionX9 -> T5,\nT1 -> .([], T5),\nT2 -> T7,\nX10 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="BACKTRACK\nfor clause: flat(.([], T), R) :- flat(T, R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 57 [arrowhead = none ];
57 -> 13;

58 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 59 [arrowhead = none ];
59 -> 19;

60 [label="EVAL with clause\nflat(.(.(X59, X60), X61), .(X59, X62)) :- flat(.(X60, X61), X62).\nand substitutionX59 -> T44,\nX60 -> T45,\nX61 -> T46,\nT1 -> .(.(T44, T45), T46),\nX62 -> T48,\nT2 -> .(T44, T48),\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 60 [arrowhead = none ];
60 -> 41;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 61 [arrowhead = none ];
61 -> 42;

62 [label="EVAL with clause\nflat([], []).\nand substitutionT5 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 62 [arrowhead = none ];
62 -> 21;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 63 [arrowhead = none ];
63 -> 23;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 64 [arrowhead = none ];
64 -> 24;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 65 [arrowhead = none ];
65 -> 29;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 66 [arrowhead = none ];
66 -> 30;

67 [label="BACKTRACK\nfor clause: flat(.([], T), R) :- flat(T, R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 67 [arrowhead = none ];
67 -> 25;

68 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 68 [arrowhead = none ];
68 -> 26;

69 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 69 [arrowhead = none ];
69 -> 27;

70 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="EVAL with clause\nflat(.([], X29), X30) :- flat(X29, X30).\nand substitutionX29 -> T16,\nT5 -> .([], T16),\nT7 -> T18,\nX30 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 71 [arrowhead = none ];
71 -> 31;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 72 [arrowhead = none ];
72 -> 32;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 74 [arrowhead = none ];
74 -> 34;

75 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 75 [arrowhead = none , style=dashed];
75 -> 2[style=dashed];

76 [label="EVAL with clause\nflat(.(.(X47, X48), X49), .(X47, X50)) :- flat(.(X48, X49), X50).\nand substitutionX47 -> T35,\nX48 -> T36,\nX49 -> T37,\nT5 -> .(.(T35, T36), T37),\nX50 -> T39,\nT7 -> .(T35, T39),\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 76 [arrowhead = none ];
76 -> 37;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 77 [arrowhead = none ];
77 -> 38;

78 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 78 [arrowhead = none ];
78 -> 39;

79 [label="INSTANCE with matching:\nT1 -> .(T36, T37)\nT2 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 79 [arrowhead = none , style=dashed];
79 -> 2[style=dashed];

80 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 80 [arrowhead = none ];
80 -> 40;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 81 [arrowhead = none ];
81 -> 43;

82 [label="BACKTRACK\nfor clause: flat([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 82 [arrowhead = none ];
82 -> 44;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 83 [arrowhead = none ];
83 -> 45;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 84 [arrowhead = none ];
84 -> 46;

85 [label="EVAL with clause\nflat(.([], X71), X72) :- flat(X71, X72).\nand substitutionT45 -> [],\nT46 -> T57,\nX71 -> T57,\nT48 -> T59,\nX72 -> T59,\nT58 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 85 [arrowhead = none ];
85 -> 47;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 86 [arrowhead = none ];
86 -> 48;

87 [label="EVAL with clause\nflat(.(.(X81, X82), X83), .(X81, X84)) :- flat(.(X82, X83), X84).\nand substitutionX81 -> T68,\nX82 -> T69,\nT45 -> .(T68, T69),\nT46 -> T70,\nX83 -> T70,\nX84 -> T72,\nT48 -> .(T68, T72),\nT71 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 87 [arrowhead = none ];
87 -> 49;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 88 [arrowhead = none ];
88 -> 50;

89 [label="INSTANCE with matching:\nT1 -> T57\nT2 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 89 [arrowhead = none , style=dashed];
89 -> 2[style=dashed];

90 [label="INSTANCE with matching:\nT1 -> .(T69, T70)\nT2 -> T72", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 90 [arrowhead = none , style=dashed];
90 -> 2[style=dashed];

}

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.([], [])) → f2_out1([])
f2_in(.([], z0)) → U1(f23_in(z0), .([], z0))
f2_in(.(.(z0, z1), z2)) → U2(f44_in(z1, z2), .(.(z0, z1), z2))
U1(f23_out1(z0), .([], z1)) → f2_out1(z0)
U1(f23_out2(z0), .([], z1)) → f2_out1(z0)
U1(f23_out4(z0), .([], z1)) → f2_out1(z0)
U2(f44_out1(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
U2(f44_out2(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
f29_in(.([], z0)) → U3(f2_in(z0), .([], z0))
U3(f2_out1(z0), .([], z1)) → f29_out1(z0)
f33_in(.(.(z0, z1), z2)) → U4(f2_in(.(z1, z2)), .(.(z0, z1), z2))
U4(f2_out1(z0), .(.(z1, z2), z3)) → f33_out1(.(z1, z0))
f45_in([], z0) → U5(f2_in(z0), [], z0)
U5(f2_out1(z0), [], z1) → f45_out1(z0)
f46_in(.(z0, z1), z2) → U6(f2_in(.(z1, z2)), .(z0, z1), z2)
U6(f2_out1(z0), .(z1, z2), z3) → f46_out1(.(z1, z0))
f23_in(z0) → U7(f29_in(z0), f30_in(z0), z0)
U7(f29_out1(z0), z1, z2) → f23_out1(z0)
U7(z0, f30_out1(z1), z2) → f23_out2(z1)
U7(z0, f30_out3(z1), z2) → f23_out4(z1)
f30_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f30_out1(z0)
U8(z0, f34_out2(z1), z2) → f30_out3(z1)
f44_in(z0, z1) → U9(f45_in(z0, z1), f46_in(z0, z1), z0, z1)
U9(f45_out1(z0), z1, z2, z3) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3) → f44_out2(z1)

Tuples:

F2_IN(.([], z0)) → c2(U1'(f23_in(z0), .([], z0)), F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(U2'(f44_in(z1, z2), .(.(z0, z1), z2)), F44_IN(z1, z2))
F29_IN(.([], z0)) → c9(U3'(f2_in(z0), .([], z0)), F2_IN(z0))
F33_IN(.(.(z0, z1), z2)) → c11(U4'(f2_in(.(z1, z2)), .(.(z0, z1), z2)), F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(U5'(f2_in(z0), [], z0), F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(U6'(f2_in(.(z1, z2)), .(z0, z1), z2), F2_IN(.(z1, z2)))
F23_IN(z0) → c17(U7'(f29_in(z0), f30_in(z0), z0), F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(U8'(f33_in(z0), f34_in(z0), z0), F33_IN(z0))
F44_IN(z0, z1) → c24(U9'(f45_in(z0, z1), f46_in(z0, z1), z0, z1), F45_IN(z0, z1), F46_IN(z0, z1))

S tuples:

F2_IN(.([], z0)) → c2(U1'(f23_in(z0), .([], z0)), F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(U2'(f44_in(z1, z2), .(.(z0, z1), z2)), F44_IN(z1, z2))
F29_IN(.([], z0)) → c9(U3'(f2_in(z0), .([], z0)), F2_IN(z0))
F33_IN(.(.(z0, z1), z2)) → c11(U4'(f2_in(.(z1, z2)), .(.(z0, z1), z2)), F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(U5'(f2_in(z0), [], z0), F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(U6'(f2_in(.(z1, z2)), .(z0, z1), z2), F2_IN(.(z1, z2)))
F23_IN(z0) → c17(U7'(f29_in(z0), f30_in(z0), z0), F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(U8'(f33_in(z0), f34_in(z0), z0), F33_IN(z0))
F44_IN(z0, z1) → c24(U9'(f45_in(z0, z1), f46_in(z0, z1), z0, z1), F45_IN(z0, z1), F46_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f29_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f23_in, U7, f30_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F29_IN, F33_IN, F45_IN, F46_IN, F23_IN, F30_IN, F44_IN

Compound Symbols:

c2, c3, c9, c11, c13, c15, c17, c21, c24

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

Removed 9 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.([], [])) → f2_out1([])
f2_in(.([], z0)) → U1(f23_in(z0), .([], z0))
f2_in(.(.(z0, z1), z2)) → U2(f44_in(z1, z2), .(.(z0, z1), z2))
U1(f23_out1(z0), .([], z1)) → f2_out1(z0)
U1(f23_out2(z0), .([], z1)) → f2_out1(z0)
U1(f23_out4(z0), .([], z1)) → f2_out1(z0)
U2(f44_out1(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
U2(f44_out2(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
f29_in(.([], z0)) → U3(f2_in(z0), .([], z0))
U3(f2_out1(z0), .([], z1)) → f29_out1(z0)
f33_in(.(.(z0, z1), z2)) → U4(f2_in(.(z1, z2)), .(.(z0, z1), z2))
U4(f2_out1(z0), .(.(z1, z2), z3)) → f33_out1(.(z1, z0))
f45_in([], z0) → U5(f2_in(z0), [], z0)
U5(f2_out1(z0), [], z1) → f45_out1(z0)
f46_in(.(z0, z1), z2) → U6(f2_in(.(z1, z2)), .(z0, z1), z2)
U6(f2_out1(z0), .(z1, z2), z3) → f46_out1(.(z1, z0))
f23_in(z0) → U7(f29_in(z0), f30_in(z0), z0)
U7(f29_out1(z0), z1, z2) → f23_out1(z0)
U7(z0, f30_out1(z1), z2) → f23_out2(z1)
U7(z0, f30_out3(z1), z2) → f23_out4(z1)
f30_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f30_out1(z0)
U8(z0, f34_out2(z1), z2) → f30_out3(z1)
f44_in(z0, z1) → U9(f45_in(z0, z1), f46_in(z0, z1), z0, z1)
U9(f45_out1(z0), z1, z2, z3) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3) → f44_out2(z1)

Tuples:

F2_IN(.([], z0)) → c2(F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(F44_IN(z1, z2))
F29_IN(.([], z0)) → c9(F2_IN(z0))
F33_IN(.(.(z0, z1), z2)) → c11(F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(F2_IN(.(z1, z2)))
F23_IN(z0) → c17(F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1) → c24(F45_IN(z0, z1), F46_IN(z0, z1))

S tuples:

F2_IN(.([], z0)) → c2(F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(F44_IN(z1, z2))
F29_IN(.([], z0)) → c9(F2_IN(z0))
F33_IN(.(.(z0, z1), z2)) → c11(F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(F2_IN(.(z1, z2)))
F23_IN(z0) → c17(F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1) → c24(F45_IN(z0, z1), F46_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f29_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f23_in, U7, f30_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F29_IN, F33_IN, F45_IN, F46_IN, F23_IN, F30_IN, F44_IN

Compound Symbols:

c2, c3, c9, c11, c13, c15, c17, c21, c24

(17) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace F29_IN(.([], z0)) → c9(F2_IN(z0)) by

F29_IN(.([], .([], y0))) → c9(F2_IN(.([], y0)))
F29_IN(.([], .(.(y0, y1), y2))) → c9(F2_IN(.(.(y0, y1), y2)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1([])
f2_in(.([], [])) → f2_out1([])
f2_in(.([], z0)) → U1(f23_in(z0), .([], z0))
f2_in(.(.(z0, z1), z2)) → U2(f44_in(z1, z2), .(.(z0, z1), z2))
U1(f23_out1(z0), .([], z1)) → f2_out1(z0)
U1(f23_out2(z0), .([], z1)) → f2_out1(z0)
U1(f23_out4(z0), .([], z1)) → f2_out1(z0)
U2(f44_out1(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
U2(f44_out2(z0), .(.(z1, z2), z3)) → f2_out1(.(z1, z0))
f29_in(.([], z0)) → U3(f2_in(z0), .([], z0))
U3(f2_out1(z0), .([], z1)) → f29_out1(z0)
f33_in(.(.(z0, z1), z2)) → U4(f2_in(.(z1, z2)), .(.(z0, z1), z2))
U4(f2_out1(z0), .(.(z1, z2), z3)) → f33_out1(.(z1, z0))
f45_in([], z0) → U5(f2_in(z0), [], z0)
U5(f2_out1(z0), [], z1) → f45_out1(z0)
f46_in(.(z0, z1), z2) → U6(f2_in(.(z1, z2)), .(z0, z1), z2)
U6(f2_out1(z0), .(z1, z2), z3) → f46_out1(.(z1, z0))
f23_in(z0) → U7(f29_in(z0), f30_in(z0), z0)
U7(f29_out1(z0), z1, z2) → f23_out1(z0)
U7(z0, f30_out1(z1), z2) → f23_out2(z1)
U7(z0, f30_out3(z1), z2) → f23_out4(z1)
f30_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f30_out1(z0)
U8(z0, f34_out2(z1), z2) → f30_out3(z1)
f44_in(z0, z1) → U9(f45_in(z0, z1), f46_in(z0, z1), z0, z1)
U9(f45_out1(z0), z1, z2, z3) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3) → f44_out2(z1)

Tuples:

F2_IN(.([], z0)) → c2(F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(F44_IN(z1, z2))
F33_IN(.(.(z0, z1), z2)) → c11(F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(F2_IN(.(z1, z2)))
F23_IN(z0) → c17(F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1) → c24(F45_IN(z0, z1), F46_IN(z0, z1))
F29_IN(.([], .([], y0))) → c9(F2_IN(.([], y0)))
F29_IN(.([], .(.(y0, y1), y2))) → c9(F2_IN(.(.(y0, y1), y2)))

S tuples:

F2_IN(.([], z0)) → c2(F23_IN(z0))
F2_IN(.(.(z0, z1), z2)) → c3(F44_IN(z1, z2))
F33_IN(.(.(z0, z1), z2)) → c11(F2_IN(.(z1, z2)))
F45_IN([], z0) → c13(F2_IN(z0))
F46_IN(.(z0, z1), z2) → c15(F2_IN(.(z1, z2)))
F23_IN(z0) → c17(F29_IN(z0), F30_IN(z0))
F30_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1) → c24(F45_IN(z0, z1), F46_IN(z0, z1))
F29_IN(.([], .([], y0))) → c9(F2_IN(.([], y0)))
F29_IN(.([], .(.(y0, y1), y2))) → c9(F2_IN(.(.(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f29_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f23_in, U7, f30_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F33_IN, F45_IN, F46_IN, F23_IN, F30_IN, F44_IN, F29_IN

Compound Symbols:

c2, c3, c11, c13, c15, c17, c21, c24, c9