YES(O(1), O(n^2)) 5.03/1.79 YES(O(1), O(n^2)) 5.46/1.83 5.46/1.83 5.46/1.83 5.46/1.83 5.46/1.83 5.46/1.83 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 5.46/1.83 5.46/1.83 5.46/1.83
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The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxRelTRS
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↳1 CpxRelTrsToCDT (UPPER BOUND (ID))
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↳2 CdtProblem
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↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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↳4 CdtProblem
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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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5.46/1.83

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

isort(Cons(x, xs), r) → isort(xs, insert(x, r)) 5.46/1.83
insert(x', Cons(x, xs)) → insert[Ite][False][Ite](<(x', x), x', Cons(x, xs)) 5.46/1.83
isort(Nil, r) → r 5.46/1.83
insert(x, Nil) → Cons(x, Nil) 5.46/1.83
inssort(xs) → isort(xs, Nil)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y) 5.46/1.83
<(0, S(y)) → True 5.46/1.83
<(x, 0) → False 5.46/1.83
insert[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) 5.46/1.83
insert[Ite][False][Ite](True, x, r) → Cons(x, r)

Rewrite Strategy: INNERMOST
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5.46/1.83

(1) CpxRelTrsToCDT (UPPER BOUND (ID) transformation)

Relative innermost TRS to CDT Problem.
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5.46/1.83

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1) 5.46/1.83
isort(Cons(z0, z1), z2) → isort(z1, insert(z0, z2)) 5.46/1.83
isort(Nil, z0) → z0 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
inssort(z0) → isort(z0, Nil)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1)) 5.46/1.83
INSSORT(z0) → c9(ISORT(z0, Nil))
S tuples:

ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1)) 5.46/1.83
INSSORT(z0) → c9(ISORT(z0, Nil))
K tuples:none
Defined Rule Symbols:

isort, insert, inssort, <, insert[Ite][False][Ite]

Defined Pair Symbols:

<', INSERT[ITE][FALSE][ITE], ISORT, INSERT, INSSORT

Compound Symbols:

c, c3, c5, c7, c9

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5.46/1.83

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

INSSORT(z0) → c9(ISORT(z0, Nil))
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5.46/1.83

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1) 5.46/1.83
isort(Cons(z0, z1), z2) → isort(z1, insert(z0, z2)) 5.46/1.83
isort(Nil, z0) → z0 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
inssort(z0) → isort(z0, Nil)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
S tuples:

ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
K tuples:none
Defined Rule Symbols:

isort, insert, inssort, <, insert[Ite][False][Ite]

Defined Pair Symbols:

<', INSERT[ITE][FALSE][ITE], ISORT, INSERT

Compound Symbols:

c, c3, c5, c7

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5.46/1.83

(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.

ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2))
We considered the (Usable) Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1)
And the Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.46/1.83

POL(0) = [3]    5.46/1.83
POL(<(x1, x2)) = 0    5.46/1.83
POL(<'(x1, x2)) = 0    5.46/1.83
POL(Cons(x1, x2)) = [2] + x2    5.46/1.83
POL(False) = 0    5.46/1.83
POL(INSERT(x1, x2)) = [1]    5.46/1.83
POL(INSERT[ITE][FALSE][ITE](x1, x2, x3)) = [1]    5.46/1.83
POL(ISORT(x1, x2)) = [2]x1    5.46/1.83
POL(Nil) = 0    5.46/1.83
POL(S(x1)) = [3] + x1    5.46/1.83
POL(True) = [5]    5.46/1.83
POL(c(x1)) = x1    5.46/1.83
POL(c3(x1)) = x1    5.46/1.83
POL(c5(x1, x2)) = x1 + x2    5.46/1.83
POL(c7(x1, x2)) = x1 + x2    5.46/1.83
POL(insert(x1, x2)) = 0    5.46/1.83
POL(insert[Ite][False][Ite](x1, x2, x3)) = [3] + [3]x2 + [3]x3   
5.46/1.83
5.46/1.83

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1) 5.46/1.83
isort(Cons(z0, z1), z2) → isort(z1, insert(z0, z2)) 5.46/1.83
isort(Nil, z0) → z0 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
inssort(z0) → isort(z0, Nil)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
S tuples:

INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
K tuples:

ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2))
Defined Rule Symbols:

isort, insert, inssort, <, insert[Ite][False][Ite]

Defined Pair Symbols:

<', INSERT[ITE][FALSE][ITE], ISORT, INSERT

Compound Symbols:

c, c3, c5, c7

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5.46/1.83

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
We considered the (Usable) Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1)
And the Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.46/1.83

POL(0) = 0    5.46/1.83
POL(<(x1, x2)) = 0    5.46/1.83
POL(<'(x1, x2)) = 0    5.46/1.83
POL(Cons(x1, x2)) = [1] + x2    5.46/1.83
POL(False) = 0    5.46/1.83
POL(INSERT(x1, x2)) = [1] + x2    5.46/1.83
POL(INSERT[ITE][FALSE][ITE](x1, x2, x3)) = x3    5.46/1.83
POL(ISORT(x1, x2)) = [2]x1·x2 + x12    5.46/1.83
POL(Nil) = 0    5.46/1.83
POL(S(x1)) = 0    5.46/1.83
POL(True) = 0    5.46/1.83
POL(c(x1)) = x1    5.46/1.83
POL(c3(x1)) = x1    5.46/1.83
POL(c5(x1, x2)) = x1 + x2    5.46/1.83
POL(c7(x1, x2)) = x1 + x2    5.46/1.83
POL(insert(x1, x2)) = [1] + x2    5.46/1.83
POL(insert[Ite][False][Ite](x1, x2, x3)) = [1] + x3   
5.46/1.83
5.46/1.83

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

<(S(z0), S(z1)) → <(z0, z1) 5.46/1.83
<(0, S(z0)) → True 5.46/1.83
<(z0, 0) → False 5.46/1.83
insert[Ite][False][Ite](False, z0, Cons(z1, z2)) → Cons(z1, insert(z0, z2)) 5.46/1.83
insert[Ite][False][Ite](True, z0, z1) → Cons(z0, z1) 5.46/1.83
isort(Cons(z0, z1), z2) → isort(z1, insert(z0, z2)) 5.46/1.83
isort(Nil, z0) → z0 5.46/1.83
insert(z0, Cons(z1, z2)) → insert[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2)) 5.46/1.83
insert(z0, Nil) → Cons(z0, Nil) 5.46/1.83
inssort(z0) → isort(z0, Nil)
Tuples:

<'(S(z0), S(z1)) → c(<'(z0, z1)) 5.46/1.83
INSERT[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) → c3(INSERT(z0, z2)) 5.46/1.83
ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
S tuples:none
K tuples:

ISORT(Cons(z0, z1), z2) → c5(ISORT(z1, insert(z0, z2)), INSERT(z0, z2)) 5.46/1.83
INSERT(z0, Cons(z1, z2)) → c7(INSERT[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
Defined Rule Symbols:

isort, insert, inssort, <, insert[Ite][False][Ite]

Defined Pair Symbols:

<', INSERT[ITE][FALSE][ITE], ISORT, INSERT

Compound Symbols:

c, c3, c5, c7

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5.46/1.83

(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(10) BOUNDS(O(1), O(1))

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5.46/1.89 EOF