Runtime Complexity proof of /tmp/tmpKmDLI3/paper1.pl

Runtime Complexity of the query pattern
p(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), 79 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 196 ms)

    ↳6 CdtProblem

    ↳7 CdtKnowledgeProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 62 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 148 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

p(X, X).
p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y))).

Query: p(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: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\np(T1, T2) !=? p(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(f(T10), f(X11)), p(X11, g(T12)))\n\nT10 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(f(T10), f(X11))\n\nT10 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T15, g(T12))\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(f(T10), f(X11)), SCOPE: 2, CLAUSE: 0 | p(f(T10), f(X11)), SCOPE: 2, CLAUSE: 1\n\nT10 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | p(f(T19), f(X11)), SCOPE: 2, CLAUSE: 1\n\nT19 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(f(T19), f(X11)), SCOPE: 2, CLAUSE: 1\n\nT19 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(f(T29), f(X30)), p(X30, g(T31)))\n\nT29 is ground\nX3 is free\nX30 is free\np(f(T29), T2) !=? p(X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL with clause\np(X3, X3).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="EVAL with clause\np(f(X28), g(X29)) :- ','(p(f(X28), f(X30)), p(X30, g(X29))).\nand substitutionX28 -> T29,\nT1 -> f(T29),\nX29 -> T31,\nT2 -> g(T31),\nT30 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 35 [arrowhead = none ];
35 -> 29;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 36 [arrowhead = none ];
36 -> 30;

37 [label="EVAL with clause\np(f(X9), g(X10)) :- ','(p(f(X9), f(X11)), p(X11, g(X10))).\nand substitutionX9 -> T10,\nT5 -> f(T10),\nX10 -> T12,\nT2 -> g(T12),\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 39 [arrowhead = none ];
39 -> 17;

40 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nT15 is ground\nreplacements:X11 -> T15", fontsize=14, style = filled, fillcolor = violet];
13 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 41 [arrowhead = none ];
41 -> 21;

42 [label="INSTANCE with matching:\nT1 -> T15\nT2 -> g(T12)", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="ONLY EVAL with clause\np(X18, X18).\nand substitutionT10 -> T19,\nX18 -> f(T19),\nX11 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 43 [arrowhead = none ];
43 -> 22;

44 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 45 [arrowhead = none ];
45 -> 24;

46 [label="INSTANCE with matching:\nT10 -> T29\nX11 -> X30\nT12 -> T31", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 46 [arrowhead = none , style=dashed];
46 -> 13[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0)
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
U2(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
f17_in(z0) → f17_out1(z0)
f13_in(z0) → U3(f17_in(z0), z0)
U3(f17_out1(z0), z1) → U4(f1_in(z0), z1, z0)
U4(f1_out1(g(z0)), z1, z2) → f13_out1(z2, z0)

Tuples:

F1_IN(f(z0)) → c1(U1'(f13_in(z0), f(z0)), F13_IN(z0))
F1_IN(f(z0)) → c2(U2'(f13_in(z0), f(z0)), F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0), F17_IN(z0))
U3'(f17_out1(z0), z1) → c7(U4'(f1_in(z0), z1, z0), F1_IN(z0))

S tuples:

F1_IN(f(z0)) → c1(U1'(f13_in(z0), f(z0)), F13_IN(z0))
F1_IN(f(z0)) → c2(U2'(f13_in(z0), f(z0)), F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0), F17_IN(z0))
U3'(f17_out1(z0), z1) → c7(U4'(f1_in(z0), z1, z0), F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f17_in, f13_in, U3, U4

Defined Pair Symbols:

F1_IN, F13_IN, U3'

Compound Symbols:

c1, c2, c6, c7

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

Removed 4 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0)
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
U2(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
f17_in(z0) → f17_out1(z0)
f13_in(z0) → U3(f17_in(z0), z0)
U3(f17_out1(z0), z1) → U4(f1_in(z0), z1, z0)
U4(f1_out1(g(z0)), z1, z2) → f13_out1(z2, z0)

Tuples:

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))

S tuples:

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f17_in, f13_in, U3, U4

Defined Pair Symbols:

F1_IN, F13_IN, U3'

Compound Symbols:

c1, c2, c6, c7

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

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

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))

We considered the (Usable) Rules:

f17_in(z0) → f17_out1(z0)

And the Tuples:

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))

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

POL(F13_IN(x₁)) = [2]x₁
POL(F1_IN(x₁)) = [2]x₁
POL(U3'(x₁, x₂)) = [2]x₁
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(f(x₁)) = [1] + x₁
POL(f17_in(x₁)) = x₁
POL(f17_out1(x₁)) = x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(z0)
f1_in(f(z0)) → U1(f13_in(z0), f(z0))
f1_in(f(z0)) → U2(f13_in(z0), f(z0))
U1(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
U2(f13_out1(z0, z1), f(z2)) → f1_out1(g(z1))
f17_in(z0) → f17_out1(z0)
f13_in(z0) → U3(f17_in(z0), z0)
U3(f17_out1(z0), z1) → U4(f1_in(z0), z1, z0)
U4(f1_out1(g(z0)), z1, z2) → f13_out1(z2, z0)

Tuples:

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))
F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))

S tuples:

F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))

K tuples:

F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, f17_in, f13_in, U3, U4

Defined Pair Symbols:

F1_IN, F13_IN, U3'

Compound Symbols:

c1, c2, c6, c7

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F13_IN(z0) → c6(U3'(f17_in(z0), z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))
U3'(f17_out1(z0), z1) → c7(F1_IN(z0))
F1_IN(f(z0)) → c1(F13_IN(z0))
F1_IN(f(z0)) → c2(F13_IN(z0))

Now S is empty

(8) BOUNDS(O(1), O(1))

(9) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\np(T1, T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(f(T7), f(X7)), p(X7, g(T9)))\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(f(T7), f(X7)), p(X7, g(T9))), SCOPE: 2, CLAUSE: 0 | ','(p(f(T7), f(X7)), p(X7, g(T9))), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(T16, g(T9)) | ','(p(f(T16), f(X7)), p(X7, g(T9))), SCOPE: 2, CLAUSE: 1\n\nT16 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(T16, g(T9))\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(p(f(T16), f(X7)), p(X7, g(T9))), SCOPE: 2, CLAUSE: 1\n\nT16 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(f(T28), f(X23)), p(X23, g(T30)))\n\nT28 is ground\nX2 is free\nX23 is free\np(f(T28), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(p(f(T28), f(X23)), p(X23, g(T30))), SCOPE: 3, CLAUSE: 0 | ','(p(f(T28), f(X23)), p(X23, g(T30))), SCOPE: 3, CLAUSE: 1\n\nT28 is ground\nX2 is free\nX23 is free\np(f(T28), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T37, g(T30)) | ','(p(f(T37), f(X23)), p(X23, g(T30))), SCOPE: 3, CLAUSE: 1\n\nT37 is ground\nX2 is free\nX23 is free\np(f(T37), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(T37, g(T30))\n\nT37 is ground\nX2 is free\np(f(T37), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(f(T37), f(X23)), p(X23, g(T30))), SCOPE: 3, CLAUSE: 1\n\nT37 is ground\nX2 is free\nX23 is free\np(f(T37), T2) !=? p(X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 35 [arrowhead = none ];
35 -> 4;

36 [label="EVAL with clause\np(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 37 [arrowhead = none ];
37 -> 8;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 38 [arrowhead = none ];
38 -> 10;

39 [label="EVAL with clause\np(f(X21), g(X22)) :- ','(p(f(X21), f(X23)), p(X23, g(X22))).\nand substitutionX21 -> T28,\nT1 -> f(T28),\nX22 -> T30,\nT2 -> g(T30),\nT29 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 39 [arrowhead = none ];
39 -> 26;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 40 [arrowhead = none ];
40 -> 27;

41 [label="EVAL with clause\np(f(X5), g(X6)) :- ','(p(f(X5), f(X7)), p(X7, g(X6))).\nand substitutionX5 -> T7,\nT4 -> f(T7),\nX6 -> T9,\nT2 -> g(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 41 [arrowhead = none ];
41 -> 11;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 42 [arrowhead = none ];
42 -> 12;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 43 [arrowhead = none ];
43 -> 15;

44 [label="ONLY EVAL with clause\np(X12, X12).\nand substitutionT7 -> T16,\nX12 -> f(T16),\nX7 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 45 [arrowhead = none ];
45 -> 19;

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> g(T9)", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 47 [arrowhead = none , style=dashed];
47 -> 2[style=dashed];

48 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 48 [arrowhead = none ];
48 -> 25;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 49 [arrowhead = none ];
49 -> 28;

50 [label="ONLY EVAL with clause\np(X28, X28).\nand substitutionT28 -> T37,\nX28 -> f(T37),\nX23 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 50 [arrowhead = none ];
50 -> 31;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 51 [arrowhead = none ];
51 -> 32;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 52 [arrowhead = none ];
52 -> 33;

53 [label="INSTANCE with matching:\nT1 -> T37\nT2 -> g(T30)", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 53 [arrowhead = none , style=dashed];
53 -> 2[style=dashed];

54 [label="BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 54 [arrowhead = none ];
54 -> 34;

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(f(z0)) → U1(f16_in(z0), f(z0))
f2_in(f(z0)) → U2(f31_in(z0), f(z0))
U1(f16_out1(z0), f(z1)) → f2_out1(g(z0))
U1(f16_out2(z0, z1), f(z2)) → f2_out1(g(z1))
U2(f31_out1(z0), f(z1)) → f2_out1(g(z0))
U2(f31_out2(z0, z1), f(z2)) → f2_out1(g(z1))
f16_in(z0) → U3(f2_in(z0), f20_in(z0), z0)
U3(f2_out1(g(z0)), z1, z2) → f16_out1(z0)
U3(z0, f20_out1(z1, z2), z3) → f16_out2(z1, z2)
f31_in(z0) → U4(f2_in(z0), f33_in(z0), z0)
U4(f2_out1(g(z0)), z1, z2) → f31_out1(z0)
U4(z0, f33_out1(z1, z2), z3) → f31_out2(z1, z2)

Tuples:

F2_IN(f(z0)) → c1(U1'(f16_in(z0), f(z0)), F16_IN(z0))
F2_IN(f(z0)) → c2(U2'(f31_in(z0), f(z0)), F31_IN(z0))
F16_IN(z0) → c7(U3'(f2_in(z0), f20_in(z0), z0), F2_IN(z0))
F31_IN(z0) → c10(U4'(f2_in(z0), f33_in(z0), z0), F2_IN(z0))

S tuples:

F2_IN(f(z0)) → c1(U1'(f16_in(z0), f(z0)), F16_IN(z0))
F2_IN(f(z0)) → c2(U2'(f31_in(z0), f(z0)), F31_IN(z0))
F16_IN(z0) → c7(U3'(f2_in(z0), f20_in(z0), z0), F2_IN(z0))
F31_IN(z0) → c10(U4'(f2_in(z0), f33_in(z0), z0), F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f16_in, U3, f31_in, U4

Defined Pair Symbols:

F2_IN, F16_IN, F31_IN

Compound Symbols:

c1, c2, c7, c10

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

Removed 4 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(f(z0)) → U1(f16_in(z0), f(z0))
f2_in(f(z0)) → U2(f31_in(z0), f(z0))
U1(f16_out1(z0), f(z1)) → f2_out1(g(z0))
U1(f16_out2(z0, z1), f(z2)) → f2_out1(g(z1))
U2(f31_out1(z0), f(z1)) → f2_out1(g(z0))
U2(f31_out2(z0, z1), f(z2)) → f2_out1(g(z1))
f16_in(z0) → U3(f2_in(z0), f20_in(z0), z0)
U3(f2_out1(g(z0)), z1, z2) → f16_out1(z0)
U3(z0, f20_out1(z1, z2), z3) → f16_out2(z1, z2)
f31_in(z0) → U4(f2_in(z0), f33_in(z0), z0)
U4(f2_out1(g(z0)), z1, z2) → f31_out1(z0)
U4(z0, f33_out1(z1, z2), z3) → f31_out2(z1, z2)

Tuples:

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))
F16_IN(z0) → c7(F2_IN(z0))
F31_IN(z0) → c10(F2_IN(z0))

S tuples:

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))
F16_IN(z0) → c7(F2_IN(z0))
F31_IN(z0) → c10(F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f16_in, U3, f31_in, U4

Defined Pair Symbols:

F2_IN, F16_IN, F31_IN

Compound Symbols:

c1, c2, c7, c10

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

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

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))

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

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))
F16_IN(z0) → c7(F2_IN(z0))
F31_IN(z0) → c10(F2_IN(z0))

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

POL(F16_IN(x₁)) = x₁
POL(F2_IN(x₁)) = x₁
POL(F31_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(f(x₁)) = [1] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(z0)
f2_in(f(z0)) → U1(f16_in(z0), f(z0))
f2_in(f(z0)) → U2(f31_in(z0), f(z0))
U1(f16_out1(z0), f(z1)) → f2_out1(g(z0))
U1(f16_out2(z0, z1), f(z2)) → f2_out1(g(z1))
U2(f31_out1(z0), f(z1)) → f2_out1(g(z0))
U2(f31_out2(z0, z1), f(z2)) → f2_out1(g(z1))
f16_in(z0) → U3(f2_in(z0), f20_in(z0), z0)
U3(f2_out1(g(z0)), z1, z2) → f16_out1(z0)
U3(z0, f20_out1(z1, z2), z3) → f16_out2(z1, z2)
f31_in(z0) → U4(f2_in(z0), f33_in(z0), z0)
U4(f2_out1(g(z0)), z1, z2) → f31_out1(z0)
U4(z0, f33_out1(z1, z2), z3) → f31_out2(z1, z2)

Tuples:

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))
F16_IN(z0) → c7(F2_IN(z0))
F31_IN(z0) → c10(F2_IN(z0))

S tuples:

F16_IN(z0) → c7(F2_IN(z0))
F31_IN(z0) → c10(F2_IN(z0))

K tuples:

F2_IN(f(z0)) → c1(F16_IN(z0))
F2_IN(f(z0)) → c2(F31_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, f16_in, U3, f31_in, U4

Defined Pair Symbols:

F2_IN, F16_IN, F31_IN

Compound Symbols:

c1, c2, c7, c10