3.41/1.35 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))

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↳2 CdtProblem

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↳3 CdtUnreachableProof (⇔)

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↳4 CdtProblem

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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳8 CdtProblem

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↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳10 CdtProblem

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↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳12 CdtProblem

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

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↳14 CdtProblem

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↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID))

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↳16 BOUNDS(O(1), O(1))

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(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N))) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(X)) → s(add(sqr(X), dbl(X))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(X)) → s(s(dbl(X))) 3.41/1.35
add(0, X) → X 3.41/1.35
add(s(X), Y) → s(add(X, Y)) 3.41/1.35
first(0, X) → nil 3.41/1.35
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(X))) → s(half(X)) 3.41/1.35
half(dbl(X)) → X 3.41/1.35
terms(X) → n__terms(X) 3.41/1.35
s(X) → n__s(X) 3.41/1.35
first(X1, X2) → n__first(X1, X2) 3.41/1.35
activate(n__terms(X)) → terms(activate(X)) 3.41/1.35
activate(n__s(X)) → s(activate(X)) 3.41/1.35
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) 3.41/1.35
activate(X) → X

Rewrite Strategy: INNERMOST

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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

TERMS(z0) → c(SQR(z0)) 3.41/1.35
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0)) 3.41/1.35
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0)) 3.41/1.35
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1)) 3.41/1.35
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2)) 3.41/1.35
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0)) 3.41/1.35
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

TERMS(z0) → c(SQR(z0)) 3.41/1.35
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0)) 3.41/1.35
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0)) 3.41/1.35
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1)) 3.41/1.35
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2)) 3.41/1.35
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0)) 3.41/1.35
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST, HALF, ACTIVATE

Compound Symbols:

c, c3, c5, c7, c9, c13, c16, c17, c18

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(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0)) 3.41/1.35
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0)) 3.41/1.35
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1)) 3.41/1.35
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2)) 3.41/1.35
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0))

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

TERMS(z0) → c(SQR(z0)) 3.41/1.35
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

TERMS(z0) → c(SQR(z0)) 3.41/1.35
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

TERMS, ACTIVATE

Compound Symbols:

c, c16, c17, c18

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

Removed 3 trailing tuple parts

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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c16, c, c17, c18

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

Removed 1 trailing nodes:

TERMS(z0) → c

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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c16, c, c17, c18

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(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.

ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0

And the Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 3.41/1.35

POL(0) = 0 3.41/1.35
POL(ACTIVATE(x₁)) = x₁ 3.41/1.35
POL(TERMS(x₁)) = 0 3.41/1.35
POL(activate(x₁)) = 0 3.41/1.35
POL(c) = 0 3.41/1.35
POL(c16(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(c17(x₁)) = x₁ 3.41/1.35
POL(c18(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(cons(x₁, x₂)) = [3] + x₁ 3.41/1.35
POL(first(x₁, x₂)) = [3] 3.41/1.35
POL(n__first(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(n__s(x₁)) = [1] + x₁ 3.41/1.35
POL(n__terms(x₁)) = x₁ 3.41/1.35
POL(nil) = [3] 3.41/1.35
POL(recip(x₁)) = [3] + x₁ 3.41/1.35
POL(s(x₁)) = [3] 3.41/1.35
POL(sqr(x₁)) = [1] 3.41/1.35
POL(terms(x₁)) = [3]

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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))

Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c16, c, c17, c18

3.41/1.35

(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.

ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0

And the Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 3.41/1.35

POL(0) = [2] 3.41/1.35
POL(ACTIVATE(x₁)) = [5] + [4]x₁ 3.41/1.35
POL(TERMS(x₁)) = 0 3.41/1.35
POL(activate(x₁)) = 0 3.41/1.35
POL(c) = 0 3.41/1.35
POL(c16(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(c17(x₁)) = x₁ 3.41/1.35
POL(c18(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(cons(x₁, x₂)) = [3] + x₁ 3.41/1.35
POL(first(x₁, x₂)) = [3] 3.41/1.35
POL(n__first(x₁, x₂)) = [4] + x₁ + x₂ 3.41/1.35
POL(n__s(x₁)) = x₁ 3.41/1.35
POL(n__terms(x₁)) = x₁ 3.41/1.35
POL(nil) = [3] 3.41/1.35
POL(recip(x₁)) = [3] + x₁ 3.41/1.35
POL(s(x₁)) = [3] 3.41/1.35
POL(sqr(x₁)) = [4] 3.41/1.35
POL(terms(x₁)) = [3]

3.41/1.35

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c

K tuples:

ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c16, c, c17, c18

3.41/1.35

(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.

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c

We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0

And the Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 3.41/1.35

POL(0) = [2] 3.41/1.35
POL(ACTIVATE(x₁)) = [2]x₁ 3.41/1.35
POL(TERMS(x₁)) = [3] 3.41/1.35
POL(activate(x₁)) = [2]x₁ 3.41/1.35
POL(c) = 0 3.41/1.35
POL(c16(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(c17(x₁)) = x₁ 3.41/1.35
POL(c18(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(cons(x₁, x₂)) = [3] + x₁ 3.41/1.35
POL(first(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(n__first(x₁, x₂)) = x₁ + x₂ 3.41/1.35
POL(n__s(x₁)) = x₁ 3.41/1.35
POL(n__terms(x₁)) = [2] + x₁ 3.41/1.35
POL(nil) = [2] 3.41/1.35
POL(recip(x₁)) = [1] 3.41/1.35
POL(s(x₁)) = x₁ 3.41/1.35
POL(sqr(x₁)) = [4] 3.41/1.35
POL(terms(x₁)) = [4] + x₁

3.41/1.35

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 3.41/1.35
terms(z0) → n__terms(z0) 3.41/1.35
sqr(0) → 0 3.41/1.35
sqr(s(z0)) → s(add(sqr(z0), dbl(z0))) 3.41/1.35
dbl(0) → 0 3.41/1.35
dbl(s(z0)) → s(s(dbl(z0))) 3.41/1.35
add(0, z0) → z0 3.41/1.35
add(s(z0), z1) → s(add(z0, z1)) 3.41/1.35
first(0, z0) → nil 3.41/1.35
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 3.41/1.35
first(z0, z1) → n__first(z0, z1) 3.41/1.35
half(0) → 0 3.41/1.35
half(s(0)) → 0 3.41/1.35
half(s(s(z0))) → s(half(z0)) 3.41/1.35
half(dbl(z0)) → z0 3.41/1.35
s(z0) → n__s(z0) 3.41/1.35
activate(n__terms(z0)) → terms(activate(z0)) 3.41/1.35
activate(n__s(z0)) → s(activate(z0)) 3.41/1.35
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 3.41/1.35
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c 3.41/1.35
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))

S tuples:none
K tuples:

ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0)) 3.41/1.35
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1)) 3.41/1.35
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0)) 3.41/1.35
TERMS(z0) → c

Defined Rule Symbols:

terms, sqr, dbl, add, first, half, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c16, c, c17, c18

3.41/1.35

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

The set S is empty

3.41/1.35

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

3.41/1.35