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

(0) Obligation:

Clauses:

countstack(empty, 0).
countstack(push(nil, T), X) :- countstack(T, X).
countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X).

Query: countstack(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: countstack(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: countstack(T1, T2), SCOPE: 1, CLAUSE: 0 | countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\ncountstack(T1, T2) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: countstack(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\ncountstack(T1, T2) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\ncountstack(T1, T2) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: countstack(T11, T13)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: countstack(push(T22, push(T23, T24)), T26)\n\nT22 is ground\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 -> 4;

38 [label="EVAL with clause\ncountstack(empty, 0).\nand substitutionT1 -> empty,\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 39 [arrowhead = none ];
39 -> 7;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 41 [arrowhead = none ];
41 -> 18;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 42 [arrowhead = none ];
42 -> 19;

43 [label="BACKTRACK\nfor clause: countstack(push(nil, T), X) :- countstack(T, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 43 [arrowhead = none ];
43 -> 13;

44 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="EVAL with clause\ncountstack(push(nil, X15), X16) :- countstack(X15, X16).\nand substitutionX15 -> T11,\nT1 -> push(nil, T11),\nT2 -> T13,\nX16 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 45 [arrowhead = none ];
45 -> 31;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 46 [arrowhead = none ];
46 -> 32;

47 [label="EVAL with clause\ncountstack(push(cons(X25, X26), X27), s(X28)) :- countstack(push(X25, push(X26, X27)), X28).\nand substitutionX25 -> T22,\nX26 -> T23,\nX27 -> T24,\nT1 -> push(cons(T22, T23), T24),\nX28 -> T26,\nT2 -> s(T26),\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 47 [arrowhead = none ];
47 -> 35;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 48 [arrowhead = none ];
48 -> 36;

49 [label="INSTANCE with matching:\nT1 -> T11\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 49 [arrowhead = none , style=dashed];
49 -> 1[style=dashed];

50 [label="INSTANCE with matching:\nT1 -> push(T22, push(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(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(U1'(f7_in(z0), z0), F7_IN(z0))
F18_IN(push(nil, z0)) → c4(U2'(f1_in(z0), push(nil, z0)), F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(U3'(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2)), F1_IN(push(z0, push(z1, z2))))
F7_IN(z0) → c8(U4'(f18_in(z0), f19_in(z0), z0), F18_IN(z0), F19_IN(z0))

S tuples:

F1_IN(z0) → c1(U1'(f7_in(z0), z0), F7_IN(z0))
F18_IN(push(nil, z0)) → c4(U2'(f1_in(z0), push(nil, z0)), F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(U3'(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2)), F1_IN(push(z0, push(z1, z2))))
F7_IN(z0) → c8(U4'(f18_in(z0), f19_in(z0), z0), F18_IN(z0), F19_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_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(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(z0) → c8(F18_IN(z0), F19_IN(z0))

S tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(z0) → c8(F18_IN(z0), F19_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_IN

Compound Symbols:

c1, c4, c6, c8

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

Use forward instantiation to replace F7_IN(z0) → c8(F18_IN(z0), F19_IN(z0)) by

F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)), F19_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F18_IN(push(cons(y0, y1), y2)), F19_IN(push(cons(y0, y1), y2)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)), F19_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F18_IN(push(cons(y0, y1), y2)), F19_IN(push(cons(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)), F19_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F18_IN(push(cons(y0, y1), y2)), F19_IN(push(cons(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_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(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_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.

F18_IN(push(nil, z0)) → c4(F1_IN(z0))

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

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

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

POL(F18_IN(x₁)) = x₁   
POL(F19_IN(x₁)) = x₁   
POL(F1_IN(x₁)) = x₁   
POL(F7_IN(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(nil) = [1]   
POL(push(x₁, x₂)) = x₁ + x₂   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

S tuples:

F1_IN(z0) → c1(F7_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

K tuples:

F18_IN(push(nil, z0)) → c4(F1_IN(z0))

Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_IN

Compound Symbols:

c1, c4, c6, c8

(11) 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) → c1(F7_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))

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

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

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

POL(F18_IN(x₁)) = [2]x₁   
POL(F19_IN(x₁)) = [2]x₁   
POL(F1_IN(x₁)) = [3] + [2]x₁   
POL(F7_IN(x₁)) = [2]x₁   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(cons(x₁, x₂)) = [2] + x₁ + x₂   
POL(nil) = [2]   
POL(push(x₁, x₂)) = x₁ + x₂   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(empty) → f1_out1(0)
f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1(z0), z1) → f1_out1(z0)
U1(f7_out2(z0), z1) → f1_out1(z0)
f18_in(push(nil, z0)) → U2(f1_in(z0), push(nil, z0))
U2(f1_out1(z0), push(nil, z1)) → f18_out1(z0)
f19_in(push(cons(z0, z1), z2)) → U3(f1_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U3(f1_out1(z0), push(cons(z1, z2), z3)) → f19_out1(s(z0))
f7_in(z0) → U4(f18_in(z0), f19_in(z0), z0)
U4(f18_out1(z0), z1, z2) → f7_out1(z0)
U4(z0, f19_out1(z1), z2) → f7_out2(z1)

Tuples:

F1_IN(z0) → c1(F7_IN(z0))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))
F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

S tuples:

F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))

K tuples:

F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))

Defined Rule Symbols:

f1_in, U1, f18_in, U2, f19_in, U3, f7_in, U4

Defined Pair Symbols:

F1_IN, F18_IN, F19_IN, F7_IN

Compound Symbols:

c1, c4, c6, c8

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F7_IN(push(nil, y0)) → c8(F18_IN(push(nil, y0)))
F7_IN(push(cons(y0, y1), y2)) → c8(F19_IN(push(cons(y0, y1), y2)))
F18_IN(push(nil, z0)) → c4(F1_IN(z0))
F19_IN(push(cons(z0, z1), z2)) → c6(F1_IN(push(z0, push(z1, z2))))

Now S is empty

(14) BOUNDS(O(1), O(1))

(15) 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: countstack(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: countstack(T1, T2), SCOPE: 1, CLAUSE: 0 | countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\ncountstack(T1, T2) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: countstack(empty, 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: countstack(T5, T7) | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX9 is free\nX10 is free\ncountstack(T1, T2) !=? countstack(empty, 0)\ncountstack(T1, T2) !=? countstack(push(nil, X9), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: countstack(T5, T7), SCOPE: 2, CLAUSE: 0 | countstack(T5, T7), SCOPE: 2, CLAUSE: 1 | countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true | countstack(empty, T7), SCOPE: 2, CLAUSE: 1 | countstack(empty, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, empty), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: countstack(T5, T7), SCOPE: 2, CLAUSE: 1 | countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: countstack(empty, T7), SCOPE: 2, CLAUSE: 1 | countstack(empty, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, empty), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: countstack(empty, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, empty), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ?_2 | countstack(push(nil, empty), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: countstack(push(nil, empty), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: countstack(T5, T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: countstack(T16, T18)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: countstack(T5, T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: countstack(push(T35, push(T36, T37)), T39)\n\nT35 is ground\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: countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground\ncountstack(T5, T7) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: countstack(push(T44, push(T45, T46)), T48)\n\nT44 is ground\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: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 0 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: countstack(push(T61, T62), T64)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: countstack(push(T75, push(T76, push(T77, T78))), T80)\n\nT75 is ground\nT76 is ground\nT77 is ground\nT78 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 -> 3;

52 [label="EVAL with clause\ncountstack(empty, 0).\nand substitutionT1 -> empty,\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 52 [arrowhead = none ];
52 -> 6;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 53 [arrowhead = none ];
53 -> 8;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 9;

55 [label="EVAL with clause\ncountstack(push(nil, X9), X10) :- countstack(X9, X10).\nand substitutionX9 -> T5,\nT1 -> push(nil, T5),\nT2 -> T7,\nX10 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 55 [arrowhead = none ];
55 -> 15;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 56 [arrowhead = none ];
56 -> 16;

57 [label="BACKTRACK\nfor clause: countstack(push(nil, T), X) :- countstack(T, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 57 [arrowhead = none ];
57 -> 11;

58 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 58 [arrowhead = none ];
58 -> 12;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 59 [arrowhead = none ];
59 -> 17;

60 [label="EVAL with clause\ncountstack(push(cons(X59, X60), X61), s(X62)) :- countstack(push(X59, push(X60, X61)), X62).\nand substitutionX59 -> T44,\nX60 -> T45,\nX61 -> T46,\nT1 -> push(cons(T44, T45), T46),\nX62 -> T48,\nT2 -> s(T48),\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 60 [arrowhead = none ];
60 -> 41;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 61 [arrowhead = none ];
61 -> 42;

62 [label="EVAL with clause\ncountstack(empty, 0).\nand substitutionT5 -> empty,\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 62 [arrowhead = none ];
62 -> 20;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 63 [arrowhead = none ];
63 -> 21;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 64 [arrowhead = none ];
64 -> 22;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 65 [arrowhead = none ];
65 -> 27;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 66 [arrowhead = none ];
66 -> 28;

67 [label="BACKTRACK\nfor clause: countstack(push(nil, T), X) :- countstack(T, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 67 [arrowhead = none ];
67 -> 23;

68 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 70 [arrowhead = none ];
70 -> 26;

71 [label="EVAL with clause\ncountstack(push(nil, X29), X30) :- countstack(X29, X30).\nand substitutionX29 -> T16,\nT5 -> push(nil, T16),\nT7 -> T18,\nX30 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 71 [arrowhead = none ];
71 -> 29;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 72 [arrowhead = none ];
72 -> 30;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 74 [arrowhead = none ];
74 -> 34;

75 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 75 [arrowhead = none , style=dashed];
75 -> 2[style=dashed];

76 [label="EVAL with clause\ncountstack(push(cons(X47, X48), X49), s(X50)) :- countstack(push(X47, push(X48, X49)), X50).\nand substitutionX47 -> T35,\nX48 -> T36,\nX49 -> T37,\nT5 -> push(cons(T35, T36), T37),\nX50 -> T39,\nT7 -> s(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 -> push(T35, push(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: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)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: countstack(empty, 0)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\ncountstack(push(nil, X71), X72) :- countstack(X71, X72).\nand substitutionT44 -> nil,\nT45 -> T61,\nT46 -> T62,\nX71 -> push(T61, T62),\nT48 -> T64,\nX72 -> T64,\nT63 -> T64", 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\ncountstack(push(cons(X81, X82), X83), s(X84)) :- countstack(push(X81, push(X82, X83)), X84).\nand substitutionX81 -> T75,\nX82 -> T76,\nT44 -> cons(T75, T76),\nT45 -> T77,\nT46 -> T78,\nX83 -> push(T77, T78),\nX84 -> T80,\nT48 -> s(T80),\nT79 -> T80", 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 -> push(T61, T62)\nT2 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 89 [arrowhead = none , style=dashed];
89 -> 2[style=dashed];

90 [label="INSTANCE with matching:\nT1 -> push(T75, push(T76, push(T77, T78)))\nT2 -> T80", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 90 [arrowhead = none , style=dashed];
90 -> 2[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(empty) → f2_out1(0)
f2_in(push(nil, empty)) → f2_out1(0)
f2_in(push(nil, z0)) → U1(f21_in(z0), push(nil, z0))
f2_in(push(cons(z0, z1), z2)) → U2(f44_in(z0, z1, z2), push(cons(z0, z1), z2))
U1(f21_out1(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out2(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out4(z0), push(nil, z1)) → f2_out1(z0)
U2(f44_out1(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
U2(f44_out2(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
f27_in(push(nil, z0)) → U3(f2_in(z0), push(nil, z0))
U3(f2_out1(z0), push(nil, z1)) → f27_out1(z0)
f33_in(push(cons(z0, z1), z2)) → U4(f2_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U4(f2_out1(z0), push(cons(z1, z2), z3)) → f33_out1(s(z0))
f45_in(nil, z0, z1) → U5(f2_in(push(z0, z1)), nil, z0, z1)
U5(f2_out1(z0), nil, z1, z2) → f45_out1(z0)
f46_in(cons(z0, z1), z2, z3) → U6(f2_in(push(z0, push(z1, push(z2, z3)))), cons(z0, z1), z2, z3)
U6(f2_out1(z0), cons(z1, z2), z3, z4) → f46_out1(s(z0))
f21_in(z0) → U7(f27_in(z0), f28_in(z0), z0)
U7(f27_out1(z0), z1, z2) → f21_out1(z0)
U7(z0, f28_out1(z1), z2) → f21_out2(z1)
U7(z0, f28_out3(z1), z2) → f21_out4(z1)
f28_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f28_out1(z0)
U8(z0, f34_out2(z1), z2) → f28_out3(z1)
f44_in(z0, z1, z2) → U9(f45_in(z0, z1, z2), f46_in(z0, z1, z2), z0, z1, z2)
U9(f45_out1(z0), z1, z2, z3, z4) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3, z4) → f44_out2(z1)

Tuples:

F2_IN(push(nil, z0)) → c2(U1'(f21_in(z0), push(nil, z0)), F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(U2'(f44_in(z0, z1, z2), push(cons(z0, z1), z2)), F44_IN(z0, z1, z2))
F27_IN(push(nil, z0)) → c9(U3'(f2_in(z0), push(nil, z0)), F2_IN(z0))
F33_IN(push(cons(z0, z1), z2)) → c11(U4'(f2_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2)), F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(U5'(f2_in(push(z0, z1)), nil, z0, z1), F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(U6'(f2_in(push(z0, push(z1, push(z2, z3)))), cons(z0, z1), z2, z3), F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(U7'(f27_in(z0), f28_in(z0), z0), F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(U8'(f33_in(z0), f34_in(z0), z0), F33_IN(z0))
F44_IN(z0, z1, z2) → c24(U9'(f45_in(z0, z1, z2), f46_in(z0, z1, z2), z0, z1, z2), F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))

S tuples:

F2_IN(push(nil, z0)) → c2(U1'(f21_in(z0), push(nil, z0)), F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(U2'(f44_in(z0, z1, z2), push(cons(z0, z1), z2)), F44_IN(z0, z1, z2))
F27_IN(push(nil, z0)) → c9(U3'(f2_in(z0), push(nil, z0)), F2_IN(z0))
F33_IN(push(cons(z0, z1), z2)) → c11(U4'(f2_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2)), F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(U5'(f2_in(push(z0, z1)), nil, z0, z1), F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(U6'(f2_in(push(z0, push(z1, push(z2, z3)))), cons(z0, z1), z2, z3), F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(U7'(f27_in(z0), f28_in(z0), z0), F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(U8'(f33_in(z0), f34_in(z0), z0), F33_IN(z0))
F44_IN(z0, z1, z2) → c24(U9'(f45_in(z0, z1, z2), f46_in(z0, z1, z2), z0, z1, z2), F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f27_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f21_in, U7, f28_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F27_IN, F33_IN, F45_IN, F46_IN, F21_IN, F28_IN, F44_IN

Compound Symbols:

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

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

Removed 9 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(empty) → f2_out1(0)
f2_in(push(nil, empty)) → f2_out1(0)
f2_in(push(nil, z0)) → U1(f21_in(z0), push(nil, z0))
f2_in(push(cons(z0, z1), z2)) → U2(f44_in(z0, z1, z2), push(cons(z0, z1), z2))
U1(f21_out1(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out2(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out4(z0), push(nil, z1)) → f2_out1(z0)
U2(f44_out1(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
U2(f44_out2(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
f27_in(push(nil, z0)) → U3(f2_in(z0), push(nil, z0))
U3(f2_out1(z0), push(nil, z1)) → f27_out1(z0)
f33_in(push(cons(z0, z1), z2)) → U4(f2_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U4(f2_out1(z0), push(cons(z1, z2), z3)) → f33_out1(s(z0))
f45_in(nil, z0, z1) → U5(f2_in(push(z0, z1)), nil, z0, z1)
U5(f2_out1(z0), nil, z1, z2) → f45_out1(z0)
f46_in(cons(z0, z1), z2, z3) → U6(f2_in(push(z0, push(z1, push(z2, z3)))), cons(z0, z1), z2, z3)
U6(f2_out1(z0), cons(z1, z2), z3, z4) → f46_out1(s(z0))
f21_in(z0) → U7(f27_in(z0), f28_in(z0), z0)
U7(f27_out1(z0), z1, z2) → f21_out1(z0)
U7(z0, f28_out1(z1), z2) → f21_out2(z1)
U7(z0, f28_out3(z1), z2) → f21_out4(z1)
f28_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f28_out1(z0)
U8(z0, f34_out2(z1), z2) → f28_out3(z1)
f44_in(z0, z1, z2) → U9(f45_in(z0, z1, z2), f46_in(z0, z1, z2), z0, z1, z2)
U9(f45_out1(z0), z1, z2, z3, z4) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3, z4) → f44_out2(z1)

Tuples:

F2_IN(push(nil, z0)) → c2(F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(F44_IN(z0, z1, z2))
F27_IN(push(nil, z0)) → c9(F2_IN(z0))
F33_IN(push(cons(z0, z1), z2)) → c11(F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1, z2) → c24(F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))

S tuples:

F2_IN(push(nil, z0)) → c2(F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(F44_IN(z0, z1, z2))
F27_IN(push(nil, z0)) → c9(F2_IN(z0))
F33_IN(push(cons(z0, z1), z2)) → c11(F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1, z2) → c24(F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f27_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f21_in, U7, f28_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F27_IN, F33_IN, F45_IN, F46_IN, F21_IN, F28_IN, F44_IN

Compound Symbols:

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

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

Use forward instantiation to replace F27_IN(push(nil, z0)) → c9(F2_IN(z0)) by

F27_IN(push(nil, push(nil, y0))) → c9(F2_IN(push(nil, y0)))
F27_IN(push(nil, push(cons(y0, y1), y2))) → c9(F2_IN(push(cons(y0, y1), y2)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(empty) → f2_out1(0)
f2_in(push(nil, empty)) → f2_out1(0)
f2_in(push(nil, z0)) → U1(f21_in(z0), push(nil, z0))
f2_in(push(cons(z0, z1), z2)) → U2(f44_in(z0, z1, z2), push(cons(z0, z1), z2))
U1(f21_out1(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out2(z0), push(nil, z1)) → f2_out1(z0)
U1(f21_out4(z0), push(nil, z1)) → f2_out1(z0)
U2(f44_out1(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
U2(f44_out2(z0), push(cons(z1, z2), z3)) → f2_out1(s(z0))
f27_in(push(nil, z0)) → U3(f2_in(z0), push(nil, z0))
U3(f2_out1(z0), push(nil, z1)) → f27_out1(z0)
f33_in(push(cons(z0, z1), z2)) → U4(f2_in(push(z0, push(z1, z2))), push(cons(z0, z1), z2))
U4(f2_out1(z0), push(cons(z1, z2), z3)) → f33_out1(s(z0))
f45_in(nil, z0, z1) → U5(f2_in(push(z0, z1)), nil, z0, z1)
U5(f2_out1(z0), nil, z1, z2) → f45_out1(z0)
f46_in(cons(z0, z1), z2, z3) → U6(f2_in(push(z0, push(z1, push(z2, z3)))), cons(z0, z1), z2, z3)
U6(f2_out1(z0), cons(z1, z2), z3, z4) → f46_out1(s(z0))
f21_in(z0) → U7(f27_in(z0), f28_in(z0), z0)
U7(f27_out1(z0), z1, z2) → f21_out1(z0)
U7(z0, f28_out1(z1), z2) → f21_out2(z1)
U7(z0, f28_out3(z1), z2) → f21_out4(z1)
f28_in(z0) → U8(f33_in(z0), f34_in(z0), z0)
U8(f33_out1(z0), z1, z2) → f28_out1(z0)
U8(z0, f34_out2(z1), z2) → f28_out3(z1)
f44_in(z0, z1, z2) → U9(f45_in(z0, z1, z2), f46_in(z0, z1, z2), z0, z1, z2)
U9(f45_out1(z0), z1, z2, z3, z4) → f44_out1(z0)
U9(z0, f46_out1(z1), z2, z3, z4) → f44_out2(z1)

Tuples:

F2_IN(push(nil, z0)) → c2(F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(F44_IN(z0, z1, z2))
F33_IN(push(cons(z0, z1), z2)) → c11(F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1, z2) → c24(F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))
F27_IN(push(nil, push(nil, y0))) → c9(F2_IN(push(nil, y0)))
F27_IN(push(nil, push(cons(y0, y1), y2))) → c9(F2_IN(push(cons(y0, y1), y2)))

S tuples:

F2_IN(push(nil, z0)) → c2(F21_IN(z0))
F2_IN(push(cons(z0, z1), z2)) → c3(F44_IN(z0, z1, z2))
F33_IN(push(cons(z0, z1), z2)) → c11(F2_IN(push(z0, push(z1, z2))))
F45_IN(nil, z0, z1) → c13(F2_IN(push(z0, z1)))
F46_IN(cons(z0, z1), z2, z3) → c15(F2_IN(push(z0, push(z1, push(z2, z3)))))
F21_IN(z0) → c17(F27_IN(z0), F28_IN(z0))
F28_IN(z0) → c21(F33_IN(z0))
F44_IN(z0, z1, z2) → c24(F45_IN(z0, z1, z2), F46_IN(z0, z1, z2))
F27_IN(push(nil, push(nil, y0))) → c9(F2_IN(push(nil, y0)))
F27_IN(push(nil, push(cons(y0, y1), y2))) → c9(F2_IN(push(cons(y0, y1), y2)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f27_in, U3, f33_in, U4, f45_in, U5, f46_in, U6, f21_in, U7, f28_in, U8, f44_in, U9

Defined Pair Symbols:

F2_IN, F33_IN, F45_IN, F46_IN, F21_IN, F28_IN, F44_IN, F27_IN

Compound Symbols:

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