The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳6 CdtProblem
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtUsableRulesProof (⇔, 0 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 51 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of terms: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 0th argument of activate: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 0th argument of first: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 1th argument of first: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 0th argument of sqr: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 0th argument of s: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 0th argument of add: s, sqr, terms, add, dbl, first, activate
The following defined symbols can occur below the 1th argument of add: dbl

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
dbl(s(X)) → s(s(dbl(X)))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(0) → 0
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(0, z0) → nil
first(z0, z1) → n__first(z0, z1)
terms(z0) → n__terms(z0)
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__terms(z0)) → terms(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
s(z0) → n__s(z0)
Tuples:

DBL(0) → c
FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
FIRST(0, z0) → c2
FIRST(z0, z1) → c3
TERMS(z0) → c4
TERMS(z0) → c5(SQR(z0))
ACTIVATE(z0) → c6
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
SQR(0) → c10
SQR(s(z0)) → c11(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ADD(0, z0) → c12
ADD(s(z0), z1) → c13(S(add(z0, z1)), ADD(z0, z1))
S(z0) → c14
S tuples:

DBL(0) → c
FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
FIRST(0, z0) → c2
FIRST(z0, z1) → c3
TERMS(z0) → c4
TERMS(z0) → c5(SQR(z0))
ACTIVATE(z0) → c6
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
SQR(0) → c10
SQR(s(z0)) → c11(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ADD(0, z0) → c12
ADD(s(z0), z1) → c13(S(add(z0, z1)), ADD(z0, z1))
S(z0) → c14
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

DBL, FIRST, TERMS, ACTIVATE, SQR, ADD, S

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
Removed 11 trailing nodes:

FIRST(0, z0) → c2
SQR(0) → c10
FIRST(z0, z1) → c3
DBL(0) → c
ADD(s(z0), z1) → c13(S(add(z0, z1)), ADD(z0, z1))
ADD(0, z0) → c12
TERMS(z0) → c4
TERMS(z0) → c5(SQR(z0))
ACTIVATE(z0) → c6
SQR(s(z0)) → c11(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
S(z0) → c14

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(0) → 0
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(0, z0) → nil
first(z0, z1) → n__first(z0, z1)
terms(z0) → n__terms(z0)
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__terms(z0)) → terms(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
s(z0) → n__s(z0)
Tuples:

ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7, c8, c9

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

Removed 3 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(0) → 0
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(0, z0) → nil
first(z0, z1) → n__first(z0, z1)
terms(z0) → n__terms(z0)
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__terms(z0)) → terms(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
s(z0) → n__s(z0)
Tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7, c8, c9

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7, c8, c9

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [1] + x1 + x12   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2)) = x1 + x2   
POL(n__first(x1, x2)) = [1] + x1 + x2   
POL(n__s(x1)) = [1] + x1   
POL(n__terms(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__terms(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c9(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c7, c8, c9

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

The set S is empty

(14) BOUNDS(1, 1)