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

(0) Obligation:

Clauses:

bin_tree(void) :- !.
bin_tree(T) :- ','(left(T, L), ','(right(T, R), ','(bin_tree(L), bin_tree(R)))).
left(void, void).
left(tree(X1, L, X2), L).
right(void, void).
right(tree(X3, X4, R), R).

Query: bin_tree(g)

(1) 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: bin_tree(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: bin_tree(T1), SCOPE: 1, CLAUSE: 0 | bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nbin_tree(T1) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(left(T4, X10), ','(right(T4, X11), ','(bin_tree(X10), bin_tree(X11))))\n\nT4 is ground\nX10 is free\nX11 is free\nbin_tree(T4) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(left(T4, X10), ','(right(T4, X11), ','(bin_tree(X10), bin_tree(X11)))), SCOPE: 2, CLAUSE: 2 | ','(left(T4, X10), ','(right(T4, X11), ','(bin_tree(X10), bin_tree(X11)))), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nX10 is free\nX11 is free\nbin_tree(T4) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(left(T4, X10), ','(right(T4, X11), ','(bin_tree(X10), bin_tree(X11)))), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nX10 is free\nX11 is free\nbin_tree(T4) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(right(tree(T11, T12, T13), X11), ','(bin_tree(T12), bin_tree(X11)))\n\nT11 is ground\nT12 is ground\nT13 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(right(tree(T11, T12, T13), X11), ','(bin_tree(T12), bin_tree(X11))), SCOPE: 3, CLAUSE: 4 | ','(right(tree(T11, T12, T13), X11), ','(bin_tree(T12), bin_tree(X11))), SCOPE: 3, CLAUSE: 5\n\nT11 is ground\nT12 is ground\nT13 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(right(tree(T11, T12, T13), X11), ','(bin_tree(T12), bin_tree(X11))), SCOPE: 3, CLAUSE: 5\n\nT11 is ground\nT12 is ground\nT13 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(bin_tree(T21), bin_tree(T22))\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: bin_tree(T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: bin_tree(T22)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 33 [arrowhead = none ];
33 -> 4;

34 [label="EVAL with clause\nbin_tree(void) :- !_1.\nand substitutionT1 -> void", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 34 [arrowhead = none ];
34 -> 6;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 35 [arrowhead = none ];
35 -> 8;

36 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 36 [arrowhead = none ];
36 -> 11;

37 [label="ONLY EVAL with clause\nbin_tree(X9) :- ','(left(X9, X10), ','(right(X9, X11), ','(bin_tree(X10), bin_tree(X11)))).\nand substitutionT1 -> T4,\nX9 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 37 [arrowhead = none ];
37 -> 16;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 39 [arrowhead = none ];
39 -> 17;

40 [label="BACKTRACK\nfor clause: left(void, void)\nwith clash: (bin_tree(T4), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="EVAL with clause\nleft(tree(X18, X19, X20), X19).\nand substitutionX18 -> T11,\nX19 -> T12,\nX20 -> T13,\nT4 -> tree(T11, T12, T13),\nX10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 41 [arrowhead = none ];
41 -> 23;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 42 [arrowhead = none ];
42 -> 24;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 43 [arrowhead = none ];
43 -> 25;

44 [label="BACKTRACK\nfor clause: right(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 44 [arrowhead = none ];
44 -> 26;

45 [label="ONLY EVAL with clause\nright(tree(X27, X28, X29), X29).\nand substitutionT11 -> T20,\nX27 -> T20,\nT12 -> T21,\nX28 -> T21,\nT13 -> T22,\nX29 -> T22,\nX11 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 45 [arrowhead = none ];
45 -> 28;

46 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
28 -> 46 [arrowhead = none ];
46 -> 30;

47 [label="SPLIT 2\nnew knowledge:\nT21 is ground", fontsize=14, style = filled, fillcolor = violet];
28 -> 47 [arrowhead = none ];
47 -> 32;

48 [label="INSTANCE with matching:\nT1 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 48 [arrowhead = none , style=dashed];
48 -> 2[style=dashed];

49 [label="INSTANCE with matching:\nT1 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 49 [arrowhead = none , style=dashed];
49 -> 2[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

Tuples:

F2_IN(tree(z0, z1, z2)) → c1(U1'(f28_in(z1, z2), tree(z0, z1, z2)), F28_IN(z1, z2))
F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
U2'(f2_out1, z0, z1) → c4(U3'(f2_in(z1), z0, z1), F2_IN(z1))

S tuples:

F2_IN(tree(z0, z1, z2)) → c1(U1'(f28_in(z1, z2), tree(z0, z1, z2)), F28_IN(z1, z2))
F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
U2'(f2_out1, z0, z1) → c4(U3'(f2_in(z1), z0, z1), F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f28_in, U2, U3

Defined Pair Symbols:

F2_IN, F28_IN, U2'

Compound Symbols:

c1, c3, c4

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

Tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

S tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f28_in, U2, U3

Defined Pair Symbols:

F28_IN, F2_IN, U2'

Compound Symbols:

c3, c1, c4

(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.

U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

We considered the (Usable) Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

And the Tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

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

POL(F28_IN(x₁, x₂)) = [2] + x₁ + x₂   
POL(F2_IN(x₁)) = x₁   
POL(U1(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U2'(x₁, x₂, x₃)) = [2] + x₃   
POL(U3(x₁, x₂, x₃)) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f28_in(x₁, x₂)) = 0   
POL(f28_out1) = 0   
POL(f2_in(x₁)) = 0   
POL(f2_out1) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₂ + x₃   
POL(void) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

Tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

S tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))

K tuples:

U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1, f28_in, U2, U3

Defined Pair Symbols:

F28_IN, F2_IN, U2'

Compound Symbols:

c3, c1, c4

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))

We considered the (Usable) Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

And the Tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

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

POL(F28_IN(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(F2_IN(x₁)) = [2]x₁   
POL(U1(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = [3] + [3]x₁ + x₃   
POL(U2'(x₁, x₂, x₃)) = [2]x₃   
POL(U3(x₁, x₂, x₃)) = x₃   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f28_in(x₁, x₂)) = [3] + x₁ + [2]x₂   
POL(f28_out1) = 0   
POL(f2_in(x₁)) = 0   
POL(f2_out1) = [2]   
POL(tree(x₁, x₂, x₃)) = [1] + x₂ + x₃   
POL(void) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(void) → f2_out1
f2_in(tree(z0, z1, z2)) → U1(f28_in(z1, z2), tree(z0, z1, z2))
U1(f28_out1, tree(z0, z1, z2)) → f2_out1
f28_in(z0, z1) → U2(f2_in(z0), z0, z1)
U2(f2_out1, z0, z1) → U3(f2_in(z1), z0, z1)
U3(f2_out1, z0, z1) → f28_out1

Tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

S tuples:

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))

K tuples:

U2'(f2_out1, z0, z1) → c4(F2_IN(z1))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))

Defined Rule Symbols:

f2_in, U1, f28_in, U2, U3

Defined Pair Symbols:

F28_IN, F2_IN, U2'

Compound Symbols:

c3, c1, c4

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F28_IN(z0, z1) → c3(U2'(f2_in(z0), z0, z1), F2_IN(z0))
F2_IN(tree(z0, z1, z2)) → c1(F28_IN(z1, z2))
U2'(f2_out1, z0, z1) → c4(F2_IN(z1))

Now S is empty

(10) BOUNDS(O(1), O(1))

(11) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: bin_tree(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: bin_tree(T1), SCOPE: 1, CLAUSE: 0 | bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nbin_tree(T1) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(left(T3, X7), ','(right(T3, X8), ','(bin_tree(X7), bin_tree(X8))))\n\nT3 is ground\nX7 is free\nX8 is free\nbin_tree(T3) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(left(T3, X7), ','(right(T3, X8), ','(bin_tree(X7), bin_tree(X8)))), SCOPE: 2, CLAUSE: 2 | ','(left(T3, X7), ','(right(T3, X8), ','(bin_tree(X7), bin_tree(X8)))), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX7 is free\nX8 is free\nbin_tree(T3) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(left(T3, X7), ','(right(T3, X8), ','(bin_tree(X7), bin_tree(X8)))), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX7 is free\nX8 is free\nbin_tree(T3) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(right(tree(T10, T11, T12), X8), ','(bin_tree(T11), bin_tree(X8)))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(right(tree(T10, T11, T12), X8), ','(bin_tree(T11), bin_tree(X8))), SCOPE: 3, CLAUSE: 4 | ','(right(tree(T10, T11, T12), X8), ','(bin_tree(T11), bin_tree(X8))), SCOPE: 3, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nT12 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(right(tree(T10, T11, T12), X8), ','(bin_tree(T11), bin_tree(X8))), SCOPE: 3, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nT12 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(bin_tree(T20), bin_tree(T21))\n\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: bin_tree(T20)\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: bin_tree(T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 32 [arrowhead = none ];
32 -> 3;

33 [label="EVAL with clause\nbin_tree(void) :- !_1.\nand substitutionT1 -> void", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 33 [arrowhead = none ];
33 -> 5;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 34 [arrowhead = none ];
34 -> 7;

35 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="ONLY EVAL with clause\nbin_tree(X6) :- ','(left(X6, X7), ','(right(X6, X8), ','(bin_tree(X7), bin_tree(X8)))).\nand substitutionT1 -> T3,\nX6 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 36 [arrowhead = none ];
36 -> 13;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 37 [arrowhead = none ];
37 -> 10;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="BACKTRACK\nfor clause: left(void, void)\nwith clash: (bin_tree(T3), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="EVAL with clause\nleft(tree(X15, X16, X17), X16).\nand substitutionX15 -> T10,\nX16 -> T11,\nX17 -> T12,\nT3 -> tree(T10, T11, T12),\nX7 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 40 [arrowhead = none ];
40 -> 19;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 41 [arrowhead = none ];
41 -> 20;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 42 [arrowhead = none ];
42 -> 21;

43 [label="BACKTRACK\nfor clause: right(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 43 [arrowhead = none ];
43 -> 22;

44 [label="ONLY EVAL with clause\nright(tree(X24, X25, X26), X26).\nand substitutionT10 -> T19,\nX24 -> T19,\nT11 -> T20,\nX25 -> T20,\nT12 -> T21,\nX26 -> T21,\nX8 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 44 [arrowhead = none ];
44 -> 27;

45 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="SPLIT 2\nnew knowledge:\nT20 is ground", fontsize=14, style = filled, fillcolor = violet];
27 -> 46 [arrowhead = none ];
46 -> 31;

47 [label="INSTANCE with matching:\nT1 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 47 [arrowhead = none , style=dashed];
47 -> 1[style=dashed];

48 [label="INSTANCE with matching:\nT1 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 48 [arrowhead = none , style=dashed];
48 -> 1[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

Tuples:

F1_IN(tree(z0, z1, z2)) → c1(U1'(f27_in(z1, z2), tree(z0, z1, z2)), F27_IN(z1, z2))
F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
U2'(f1_out1, z0, z1) → c4(U3'(f1_in(z1), z0, z1), F1_IN(z1))

S tuples:

F1_IN(tree(z0, z1, z2)) → c1(U1'(f27_in(z1, z2), tree(z0, z1, z2)), F27_IN(z1, z2))
F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
U2'(f1_out1, z0, z1) → c4(U3'(f1_in(z1), z0, z1), F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f27_in, U2, U3

Defined Pair Symbols:

F1_IN, F27_IN, U2'

Compound Symbols:

c1, c3, c4

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

Removed 2 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

Tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

S tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f27_in, U2, U3

Defined Pair Symbols:

F27_IN, F1_IN, U2'

Compound Symbols:

c3, c1, c4

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

We considered the (Usable) Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

And the Tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

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

POL(F1_IN(x₁)) = x₁   
POL(F27_IN(x₁, x₂)) = [2] + x₁ + x₂   
POL(U1(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U2'(x₁, x₂, x₃)) = [2] + x₃   
POL(U3(x₁, x₂, x₃)) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f1_in(x₁)) = 0   
POL(f1_out1) = 0   
POL(f27_in(x₁, x₂)) = 0   
POL(f27_out1) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₂ + x₃   
POL(void) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

Tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

S tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))

K tuples:

U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1, f27_in, U2, U3

Defined Pair Symbols:

F27_IN, F1_IN, U2'

Compound Symbols:

c3, c1, c4

(17) 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(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))

We considered the (Usable) Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

And the Tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

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

POL(F1_IN(x₁)) = [2]x₁   
POL(F27_IN(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(U1(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = [3] + [3]x₁ + x₃   
POL(U2'(x₁, x₂, x₃)) = [2]x₃   
POL(U3(x₁, x₂, x₃)) = x₃   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁)) = x₁   
POL(f1_in(x₁)) = 0   
POL(f1_out1) = [2]   
POL(f27_in(x₁, x₂)) = [3] + x₁ + [2]x₂   
POL(f27_out1) = 0   
POL(tree(x₁, x₂, x₃)) = [1] + x₂ + x₃   
POL(void) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(void) → f1_out1
f1_in(tree(z0, z1, z2)) → U1(f27_in(z1, z2), tree(z0, z1, z2))
U1(f27_out1, tree(z0, z1, z2)) → f1_out1
f27_in(z0, z1) → U2(f1_in(z0), z0, z1)
U2(f1_out1, z0, z1) → U3(f1_in(z1), z0, z1)
U3(f1_out1, z0, z1) → f27_out1

Tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))
U2'(f1_out1, z0, z1) → c4(F1_IN(z1))

S tuples:

F27_IN(z0, z1) → c3(U2'(f1_in(z0), z0, z1), F1_IN(z0))

K tuples:

U2'(f1_out1, z0, z1) → c4(F1_IN(z1))
F1_IN(tree(z0, z1, z2)) → c1(F27_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f27_in, U2, U3

Defined Pair Symbols:

F27_IN, F1_IN, U2'

Compound Symbols:

c3, c1, c4