Runtime Complexity proof of /tmp/tmpjYSdDG/pplus2.pl

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

0 Prolog

↳1 LPReorderTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 59 ms)

↳4 CdtProblem

    ↳5 CdtGraphSplitRhsProof (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))), 259 ms)

    ↳10 CdtProblem

    ↳11 CdtKnowledgeProof (⇔, 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 90 ms)

    ↳14 CdtProblem

    ↳15 SIsEmptyProof (⇔, 0 ms)

    ↳16 BOUNDS(O(1), O(1))

↳17 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 75 ms)

↳18 CdtProblem

    ↳19 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳20 CdtProblem

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

    ↳22 CdtProblem

    ↳23 CdtKnowledgeProof (⇔, 0 ms)

    ↳24 CdtProblem

    ↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 219 ms)

    ↳26 CdtProblem

    ↳27 CdtKnowledgeProof (⇔, 0 ms)

    ↳28 CdtProblem

    ↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 76 ms)

    ↳30 CdtProblem

(0) Obligation:

Clauses:

p(s(0), 0).
p(s(s(X)), s(s(Y))) :- p(s(X), s(Y)).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z)).

Query: plus(g,a,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

p(s(0), 0).
plus(0, Y, Y).
p(s(s(X)), s(s(Y))) :- p(s(X), s(Y)).
plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z)).

Query: plus(g,a,a)

(3) 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: plus(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: plus(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | plus(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | plus(0, T2, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: plus(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX3 is free\nplus(T1, T2, T3) !=? plus(0, X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: plus(0, T2, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(s(T13), X17), plus(X17, T16, T17))\n\nT13 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(s(T13), X17)\n\nT13 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: plus(T18, T16, T17)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(s(T13), X17), SCOPE: 2, CLAUSE: 0 | p(s(T13), X17), SCOPE: 2, CLAUSE: 2\n\nT13 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true | p(s(0), X17), SCOPE: 2, CLAUSE: 2\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(s(T13), X17), SCOPE: 2, CLAUSE: 2\n\nT13 is ground\nX17 is free\np(s(T13), X17) !=? p(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(s(0), X17), SCOPE: 2, CLAUSE: 2\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(s(T21), s(X28))\n\nT21 is ground\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(s(T21), s(X28)), SCOPE: 3, CLAUSE: 0 | p(s(T21), s(X28)), SCOPE: 3, CLAUSE: 2\n\nT21 is ground\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: p(s(T21), s(X28)), SCOPE: 3, CLAUSE: 2\n\nT21 is ground\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(s(T24), s(X37))\n\nT24 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 48 [arrowhead = none ];
48 -> 3;

49 [label="EVAL with clause\nplus(0, X3, X3).\nand substitutionT1 -> 0,\nT2 -> T6,\nX3 -> T6,\nT3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 49 [arrowhead = none ];
49 -> 6;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 50 [arrowhead = none ];
50 -> 10;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 51 [arrowhead = none ];
51 -> 11;

52 [label="EVAL with clause\nplus(s(X14), X15, s(X16)) :- ','(p(s(X14), X17), plus(X17, X15, X16)).\nand substitutionX14 -> T13,\nT1 -> s(T13),\nT2 -> T16,\nX15 -> T16,\nX16 -> T17,\nT3 -> s(T17),\nT14 -> T16,\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 53 [arrowhead = none ];
53 -> 19;

54 [label="BACKTRACK\nfor clause: plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 54 [arrowhead = none ];
54 -> 12;

55 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 55 [arrowhead = none ];
55 -> 23;

56 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT18 is ground\nreplacements:X17 -> T18", fontsize=14, style = filled, fillcolor = violet];
17 -> 56 [arrowhead = none ];
56 -> 24;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 57 [arrowhead = none ];
57 -> 31;

58 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> T16\nT3 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 58 [arrowhead = none , style=dashed];
58 -> 2[style=dashed];

59 [label="EVAL with clause\np(s(0), 0).\nand substitutionT13 -> 0,\nX17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 59 [arrowhead = none ];
59 -> 32;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 60 [arrowhead = none ];
60 -> 33;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 61 [arrowhead = none ];
61 -> 34;

62 [label="EVAL with clause\np(s(s(X26)), s(s(X27))) :- p(s(X26), s(X27)).\nand substitutionX26 -> T21,\nT13 -> s(T21),\nX27 -> X28,\nX17 -> s(s(X28))", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 62 [arrowhead = none ];
62 -> 38;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 63 [arrowhead = none ];
63 -> 39;

64 [label="BACKTRACK\nfor clause: p(s(s(X)), s(s(Y))) :- p(s(X), s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 64 [arrowhead = none ];
64 -> 35;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 65 [arrowhead = none ];
65 -> 42;

66 [label="BACKTRACK\nfor clause: p(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 66 [arrowhead = none ];
66 -> 43;

67 [label="EVAL with clause\np(s(s(X35)), s(s(X36))) :- p(s(X35), s(X36)).\nand substitutionX35 -> T24,\nT21 -> s(T24),\nX36 -> X37,\nX28 -> s(X37)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 67 [arrowhead = none ];
67 -> 46;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 68 [arrowhead = none ];
68 -> 47;

69 [label="INSTANCE with matching:\nT21 -> T24\nX28 -> X37", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 69 [arrowhead = none , style=dashed];
69 -> 38[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F2_IN(s(z0)) → c1(U1'(f17_in(z0), s(z0)), F17_IN(z0))
F38_IN(s(z0)) → c3(U2'(f38_in(z0), s(z0)), F38_IN(z0))
F23_IN(s(z0)) → c6(U3'(f38_in(z0), s(z0)), F38_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
U4'(f23_out1(z0), z1) → c9(U5'(f2_in(z0), z1, z0), F2_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(U1'(f17_in(z0), s(z0)), F17_IN(z0))
F38_IN(s(z0)) → c3(U2'(f38_in(z0), s(z0)), F38_IN(z0))
F23_IN(s(z0)) → c6(U3'(f38_in(z0), s(z0)), F38_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
U4'(f23_out1(z0), z1) → c9(U5'(f2_in(z0), z1, z0), F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F2_IN, F38_IN, F23_IN, F17_IN, U4'

Compound Symbols:

c1, c3, c6, c8, c9

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F2_IN(s(z0)) → c1(U1'(f17_in(z0), s(z0)), F17_IN(z0))
F38_IN(s(z0)) → c3(U2'(f38_in(z0), s(z0)), F38_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
U4'(f23_out1(z0), z1) → c9(U5'(f2_in(z0), z1, z0), F2_IN(z0))
F23_IN(s(z0)) → c(U3'(f38_in(z0), s(z0)))
F23_IN(s(z0)) → c(F38_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(U1'(f17_in(z0), s(z0)), F17_IN(z0))
F38_IN(s(z0)) → c3(U2'(f38_in(z0), s(z0)), F38_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
U4'(f23_out1(z0), z1) → c9(U5'(f2_in(z0), z1, z0), F2_IN(z0))
F23_IN(s(z0)) → c(U3'(f38_in(z0), s(z0)))
F23_IN(s(z0)) → c(F38_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F2_IN, F38_IN, F17_IN, U4', F23_IN

Compound Symbols:

c1, c3, c8, c9, c

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

S tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F17_IN, F23_IN, F2_IN, F38_IN, U4'

Compound Symbols:

c8, c, c1, c3, c9, c

(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.

F2_IN(s(z0)) → c1(F17_IN(z0))

We considered the (Usable) Rules:

f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))

And the Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

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

POL(0) = 0
POL(F17_IN(x₁)) = 0
POL(F23_IN(x₁)) = 0
POL(F2_IN(x₁)) = x₁
POL(F38_IN(x₁)) = 0
POL(U2(x₁, x₂)) = [2]x₁
POL(U3(x₁, x₂)) = x₁
POL(U4'(x₁, x₂)) = [2]x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(f23_in(x₁)) = 0
POL(f23_out1(x₁)) = x₁
POL(f38_in(x₁)) = 0
POL(f38_out1(x₁)) = [1] + x₁
POL(s(x₁)) = [1]

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

S tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

K tuples:

F2_IN(s(z0)) → c1(F17_IN(z0))

Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F17_IN, F23_IN, F2_IN, F38_IN, U4'

Compound Symbols:

c8, c, c1, c3, c9, c

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c
F23_IN(s(z0)) → c(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c
F2_IN(s(z0)) → c1(F17_IN(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

S tuples:

F38_IN(s(z0)) → c3(F38_IN(z0))

K tuples:

F2_IN(s(z0)) → c1(F17_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F17_IN, F23_IN, F2_IN, F38_IN, U4'

Compound Symbols:

c8, c, c1, c3, c9, c

(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.

F38_IN(s(z0)) → c3(F38_IN(z0))

We considered the (Usable) Rules:

f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))

And the Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

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

POL(0) = [1]
POL(F17_IN(x₁)) = [2] + x₁
POL(F23_IN(x₁)) = x₁
POL(F2_IN(x₁)) = [2]x₁
POL(F38_IN(x₁)) = [2] + x₁
POL(U2(x₁, x₂)) = [2]x₁
POL(U3(x₁, x₂)) = [1] + x₁
POL(U4'(x₁, x₂)) = [2]x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(f23_in(x₁)) = [1]
POL(f23_out1(x₁)) = x₁
POL(f38_in(x₁)) = 0
POL(f38_out1(x₁)) = [3] + x₁
POL(s(x₁)) = [2] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f17_in(z0), s(z0))
U1(f17_out1(z0), s(z1)) → f2_out1
f38_in(s(z0)) → U2(f38_in(z0), s(z0))
U2(f38_out1(z0), s(z1)) → f38_out1(s(z0))
f23_in(0) → f23_out1(0)
f23_in(s(z0)) → U3(f38_in(z0), s(z0))
U3(f38_out1(z0), s(z1)) → f23_out1(s(s(z0)))
f17_in(z0) → U4(f23_in(z0), z0)
U4(f23_out1(z0), z1) → U5(f2_in(z0), z1, z0)
U5(f2_out1, z0, z1) → f17_out1(z1)

Tuples:

F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
F2_IN(s(z0)) → c1(F17_IN(z0))
F38_IN(s(z0)) → c3(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c

S tuples:none
K tuples:

F2_IN(s(z0)) → c1(F17_IN(z0))
F17_IN(z0) → c8(U4'(f23_in(z0), z0), F23_IN(z0))
F23_IN(s(z0)) → c(F38_IN(z0))
U4'(f23_out1(z0), z1) → c9(F2_IN(z0))
F23_IN(s(z0)) → c
F38_IN(s(z0)) → c3(F38_IN(z0))

Defined Rule Symbols:

f2_in, U1, f38_in, U2, f23_in, U3, f17_in, U4, U5

Defined Pair Symbols:

F17_IN, F23_IN, F2_IN, F38_IN, U4'

Compound Symbols:

c8, c, c1, c3, c9, c

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))

(17) 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: plus(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: plus(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | plus(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | plus(0, T2, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: plus(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX2 is free\nplus(T1, T2, T3) !=? plus(0, X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: plus(0, T2, T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(s(T9), X12), plus(X12, T12, T13))\n\nT9 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(s(T9), X12), plus(X12, T12, T13)), SCOPE: 2, CLAUSE: 0 | ','(p(s(T9), X12), plus(X12, T12, T13)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: plus(0, T12, T13) | ','(p(s(0), X12), plus(X12, T12, T13)), SCOPE: 2, CLAUSE: 2\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(s(T9), X12), plus(X12, T12, T13)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nX12 is free\np(s(T9), X12) !=? p(s(0), 0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: plus(0, T12, T13)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(s(0), X12), plus(X12, T12, T13)), SCOPE: 2, CLAUSE: 2\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: plus(0, T12, T13), SCOPE: 3, CLAUSE: 1 | plus(0, T12, T13), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | plus(0, T12, T13), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: plus(0, T12, T13), SCOPE: 3, CLAUSE: 3\n\nX19 is free\nplus(0, T12, T13) !=? plus(0, X19, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: plus(0, T12, T13), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(s(T23), s(X36)), plus(s(s(X36)), T12, T13))\n\nT23 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: p(s(T23), s(X36))\n\nT23 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: plus(s(s(T24)), T12, T13)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(s(T23), s(X36)), SCOPE: 4, CLAUSE: 0 | p(s(T23), s(X36)), SCOPE: 4, CLAUSE: 2\n\nT23 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(s(T23), s(X36)), SCOPE: 4, CLAUSE: 2\n\nT23 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: p(s(T27), s(X45))\n\nT27 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 50 [arrowhead = none ];
50 -> 4;

51 [label="EVAL with clause\nplus(0, X2, X2).\nand substitutionT1 -> 0,\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 51 [arrowhead = none ];
51 -> 5;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 52 [arrowhead = none ];
52 -> 7;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 53 [arrowhead = none ];
53 -> 8;

54 [label="EVAL with clause\nplus(s(X9), X10, s(X11)) :- ','(p(s(X9), X12), plus(X12, X10, X11)).\nand substitutionX9 -> T9,\nT1 -> s(T9),\nT2 -> T12,\nX10 -> T12,\nX11 -> T13,\nT3 -> s(T13),\nT10 -> T12,\nT11 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 55 [arrowhead = none ];
55 -> 14;

56 [label="BACKTRACK\nfor clause: plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 56 [arrowhead = none ];
56 -> 9;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 57 [arrowhead = none ];
57 -> 15;

58 [label="EVAL with clause\np(s(0), 0).\nand substitutionT9 -> 0,\nX12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 58 [arrowhead = none ];
58 -> 16;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 59 [arrowhead = none ];
59 -> 18;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 60 [arrowhead = none ];
60 -> 20;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 61 [arrowhead = none ];
61 -> 21;

62 [label="EVAL with clause\np(s(s(X34)), s(s(X35))) :- p(s(X34), s(X35)).\nand substitutionX34 -> T23,\nT9 -> s(T23),\nX35 -> X36,\nX12 -> s(s(X36))", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 62 [arrowhead = none ];
62 -> 36;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 63 [arrowhead = none ];
63 -> 37;

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

65 [label="BACKTRACK\nfor clause: p(s(s(X)), s(s(Y))) :- p(s(X), s(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="EVAL with clause\nplus(0, X19, X19).\nand substitutionT12 -> T20,\nX19 -> T20,\nT13 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 66 [arrowhead = none ];
66 -> 25;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 67 [arrowhead = none ];
67 -> 26;

68 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 68 [arrowhead = none ];
68 -> 27;

69 [label="BACKTRACK\nfor clause: plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 69 [arrowhead = none ];
69 -> 29;

70 [label="BACKTRACK\nfor clause: plus(s(X), Y, s(Z)) :- ','(p(s(X), U), plus(U, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 71 [arrowhead = none ];
71 -> 40;

72 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT24 is ground\nreplacements:X36 -> T24", fontsize=14, style = filled, fillcolor = violet];
36 -> 72 [arrowhead = none ];
72 -> 41;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 73 [arrowhead = none ];
73 -> 44;

74 [label="INSTANCE with matching:\nT1 -> s(s(T24))\nT2 -> T12\nT3 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 74 [arrowhead = none , style=dashed];
74 -> 1[style=dashed];

75 [label="BACKTRACK\nfor clause: p(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 75 [arrowhead = none ];
75 -> 45;

76 [label="EVAL with clause\np(s(s(X43)), s(s(X44))) :- p(s(X43), s(X44)).\nand substitutionX43 -> T27,\nT23 -> s(T27),\nX44 -> X45,\nX36 -> s(X45)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 76 [arrowhead = none ];
76 -> 48;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 77 [arrowhead = none ];
77 -> 49;

78 [label="INSTANCE with matching:\nT23 -> T27\nX36 -> X45", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 78 [arrowhead = none , style=dashed];
78 -> 40[style=dashed];

}

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F1_IN(s(0)) → c1(U1'(f16_in, s(0)), F16_IN)
F1_IN(s(s(z0))) → c2(U2'(f36_in(z0), s(s(z0))), F36_IN(z0))
F40_IN(s(z0)) → c6(U3'(f40_in(z0), s(z0)), F40_IN(z0))
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(U5'(f1_in(s(s(z0))), z1, z0), F1_IN(s(s(z0))))
F16_IN → c12(U6'(f20_in, f21_in), F20_IN)

S tuples:

F1_IN(s(0)) → c1(U1'(f16_in, s(0)), F16_IN)
F1_IN(s(s(z0))) → c2(U2'(f36_in(z0), s(s(z0))), F36_IN(z0))
F40_IN(s(z0)) → c6(U3'(f40_in(z0), s(z0)), F40_IN(z0))
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(U5'(f1_in(s(s(z0))), z1, z0), F1_IN(s(s(z0))))
F16_IN → c12(U6'(f20_in, f21_in), F20_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F1_IN, F40_IN, F36_IN, U4', F16_IN

Compound Symbols:

c1, c2, c6, c9, c10, c12

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

Split RHS of tuples not part of any SCC

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F1_IN(s(s(z0))) → c2(U2'(f36_in(z0), s(s(z0))), F36_IN(z0))
F40_IN(s(z0)) → c6(U3'(f40_in(z0), s(z0)), F40_IN(z0))
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(U5'(f1_in(s(s(z0))), z1, z0), F1_IN(s(s(z0))))
F1_IN(s(0)) → c(U1'(f16_in, s(0)))
F1_IN(s(0)) → c(F16_IN)
F16_IN → c(U6'(f20_in, f21_in))
F16_IN → c(F20_IN)

S tuples:

F1_IN(s(s(z0))) → c2(U2'(f36_in(z0), s(s(z0))), F36_IN(z0))
F40_IN(s(z0)) → c6(U3'(f40_in(z0), s(z0)), F40_IN(z0))
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(U5'(f1_in(s(s(z0))), z1, z0), F1_IN(s(s(z0))))
F1_IN(s(0)) → c(U1'(f16_in, s(0)))
F1_IN(s(0)) → c(F16_IN)
F16_IN → c(U6'(f20_in, f21_in))
F16_IN → c(F20_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F1_IN, F40_IN, F36_IN, U4', F16_IN

Compound Symbols:

c2, c6, c9, c10, c

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

Removed 6 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

S tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F36_IN, F1_IN, F40_IN, U4', F16_IN

Compound Symbols:

c9, c, c2, c6, c10, c

(23) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(s(0)) → c(F16_IN)
F1_IN(s(0)) → c
F16_IN → c
F16_IN → c
F16_IN → c
F16_IN → c

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

S tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))

K tuples:

F1_IN(s(0)) → c(F16_IN)
F1_IN(s(0)) → c
F16_IN → c

Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F36_IN, F1_IN, F40_IN, U4', F16_IN

Compound Symbols:

c9, c, c2, c6, c10, c

(25) 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.

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))

We considered the (Usable) Rules:

f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))

And the Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

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

POL(0) = 0
POL(F16_IN) = [1]
POL(F1_IN(x₁)) = [1] + x₁
POL(F36_IN(x₁)) = [2]
POL(F40_IN(x₁)) = [1]
POL(U3(x₁, x₂)) = x₁
POL(U4'(x₁, x₂)) = [2]x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f40_in(x₁)) = 0
POL(f40_out1(x₁)) = [1]
POL(s(x₁)) = [1]

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

S tuples:

F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))

K tuples:

F1_IN(s(0)) → c(F16_IN)
F1_IN(s(0)) → c
F16_IN → c
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F36_IN, F1_IN, F40_IN, U4', F16_IN

Compound Symbols:

c9, c, c2, c6, c10, c

(27) CdtKnowledgeProof (EQUIVALENT transformation)

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

U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

S tuples:

F40_IN(s(z0)) → c6(F40_IN(z0))

K tuples:

F1_IN(s(0)) → c(F16_IN)
F1_IN(s(0)) → c
F16_IN → c
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(s(z0))) → c2(F36_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F36_IN, F1_IN, F40_IN, U4', F16_IN

Compound Symbols:

c9, c, c2, c6, c10, c

(29) 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.

F40_IN(s(z0)) → c6(F40_IN(z0))

We considered the (Usable) Rules:

f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))

And the Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

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

POL(0) = [1]
POL(F16_IN) = 0
POL(F1_IN(x₁)) = x₁
POL(F36_IN(x₁)) = [3] + x₁
POL(F40_IN(x₁)) = x₁
POL(U3(x₁, x₂)) = [3]x₁
POL(U4'(x₁, x₂)) = [3] + x₁
POL(c) = 0
POL(c(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f40_in(x₁)) = 0
POL(f40_out1(x₁)) = [2] + x₁
POL(s(x₁)) = [2] + x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(0)) → U1(f16_in, s(0))
f1_in(s(s(z0))) → U2(f36_in(z0), s(s(z0)))
U1(f16_out1, s(0)) → f1_out1
U1(f16_out2(z0), s(0)) → f1_out1
U2(f36_out1(z0), s(s(z1))) → f1_out1
f40_in(s(z0)) → U3(f40_in(z0), s(z0))
U3(f40_out1(z0), s(z1)) → f40_out1(s(z0))
f20_in → f20_out1
f36_in(z0) → U4(f40_in(z0), z0)
U4(f40_out1(z0), z1) → U5(f1_in(s(s(z0))), z1, z0)
U5(f1_out1, z0, z1) → f36_out1(z1)
f16_in → U6(f20_in, f21_in)
U6(f20_out1, z0) → f16_out1
U6(z0, f21_out1(z1)) → f16_out2(z1)

Tuples:

F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
F1_IN(s(0)) → c(F16_IN)
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(0)) → c
F16_IN → c

S tuples:none
K tuples:

F1_IN(s(0)) → c(F16_IN)
F1_IN(s(0)) → c
F16_IN → c
F36_IN(z0) → c9(U4'(f40_in(z0), z0), F40_IN(z0))
U4'(f40_out1(z0), z1) → c10(F1_IN(s(s(z0))))
F1_IN(s(s(z0))) → c2(F36_IN(z0))
F40_IN(s(z0)) → c6(F40_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, f40_in, U3, f20_in, f36_in, U4, U5, f16_in, U6

Defined Pair Symbols:

F36_IN, F1_IN, F40_IN, U4', F16_IN

Compound Symbols:

c9, c, c2, c6, c10, c