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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 15 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 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^1)), 59 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^1).


The TRS R consists of the following rules:

from(X) → cons(X, n__from(n__s(X)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 1th argument of 2ndspos: cons, from
The following defined symbols can occur below the 0th argument of from: s, cons, from, activate
The following defined symbols can occur below the 0th argument of s: s, cons, from, activate
The following defined symbols can occur below the 0th argument of cons: s, cons, from, activate
The following defined symbols can occur below the 1th argument of cons: s, cons, from, activate

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
plus(s(X), Y) → s(plus(X, Y))
times(s(X), Y) → plus(Y, times(X, Y))

(2) Obligation:

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


The TRS R consists of the following rules:

2ndspos(0, Z) → rnil
plus(0, Y) → Y
activate(X) → X
times(0, Y) → 0
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
square(X) → times(X, X)
activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
pi(X) → 2ndspos(X, from(0))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
2ndsneg(0, Z) → rnil

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

2ndspos(0, z0) → rnil
plus(0, z0) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
times(0, z0) → 0
cons(z0, z1) → n__cons(z0, z1)
square(z0) → times(z0, z0)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
pi(z0) → 2ndspos(z0, from(0))
s(z0) → n__s(z0)
2ndsneg(0, z0) → rnil
Tuples:

2NDSPOS(0, z0) → c
PLUS(0, z0) → c1
ACTIVATE(z0) → c2
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
TIMES(0, z0) → c6
CONS(z0, z1) → c7
SQUARE(z0) → c8(TIMES(z0, z0))
FROM(z0) → c9(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c10
PI(z0) → c11(2NDSPOS(z0, from(0)), FROM(0))
S(z0) → c12
2NDSNEG(0, z0) → c13
S tuples:

2NDSPOS(0, z0) → c
PLUS(0, z0) → c1
ACTIVATE(z0) → c2
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
TIMES(0, z0) → c6
CONS(z0, z1) → c7
SQUARE(z0) → c8(TIMES(z0, z0))
FROM(z0) → c9(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c10
PI(z0) → c11(2NDSPOS(z0, from(0)), FROM(0))
S(z0) → c12
2NDSNEG(0, z0) → c13
K tuples:none
Defined Rule Symbols:

2ndspos, plus, activate, times, cons, square, from, pi, s, 2ndsneg

Defined Pair Symbols:

2NDSPOS, PLUS, ACTIVATE, TIMES, CONS, SQUARE, FROM, PI, S, 2NDSNEG

Compound Symbols:

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

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

Removed 11 trailing nodes:

SQUARE(z0) → c8(TIMES(z0, z0))
TIMES(0, z0) → c6
2NDSPOS(0, z0) → c
2NDSNEG(0, z0) → c13
PI(z0) → c11(2NDSPOS(z0, from(0)), FROM(0))
FROM(z0) → c9(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c10
S(z0) → c12
ACTIVATE(z0) → c2
PLUS(0, z0) → c1
CONS(z0, z1) → c7

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

2ndspos(0, z0) → rnil
plus(0, z0) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
times(0, z0) → 0
cons(z0, z1) → n__cons(z0, z1)
square(z0) → times(z0, z0)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
pi(z0) → 2ndspos(z0, from(0))
s(z0) → n__s(z0)
2ndsneg(0, z0) → rnil
Tuples:

ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2ndspos, plus, activate, times, cons, square, from, pi, s, 2ndsneg

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4, c5

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

Removed 3 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

2ndspos(0, z0) → rnil
plus(0, z0) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
times(0, z0) → 0
cons(z0, z1) → n__cons(z0, z1)
square(z0) → times(z0, z0)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
pi(z0) → 2ndspos(z0, from(0))
s(z0) → n__s(z0)
2ndsneg(0, z0) → rnil
Tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2ndspos, plus, activate, times, cons, square, from, pi, s, 2ndsneg

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4, c5

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

2ndspos(0, z0) → rnil
plus(0, z0) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
times(0, z0) → 0
cons(z0, z1) → n__cons(z0, z1)
square(z0) → times(z0, z0)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
pi(z0) → 2ndspos(z0, from(0))
s(z0) → n__s(z0)
2ndsneg(0, z0) → rnil

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4, c5

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [2]x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(n__cons(x1, x2)) = [2] + x1   
POL(n__from(x1)) = [2] + x1   
POL(n__s(x1)) = [2] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4, c5

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

The set S is empty

(14) BOUNDS(1, 1)