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

(0) Obligation:

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


The TRS R consists of the following rules:

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Rewrite Strategy: FULL

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)

(2) Obligation:

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


The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X)))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(4) Obligation:

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


The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X)))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(z0, z1)
cons(z0, z1) → n__cons(z0, z1)

Tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0))))
FROM(z0) → c1
ACTIVATE(n__from(z0)) → c2(FROM(z0))
ACTIVATE(z0) → c3
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
CONS(z0, z1) → c5

S tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0))))
FROM(z0) → c1
ACTIVATE(n__from(z0)) → c2(FROM(z0))
ACTIVATE(z0) → c3
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
CONS(z0, z1) → c5

K tuples:none
Defined Rule Symbols:

from, activate, cons

Defined Pair Symbols:

FROM, ACTIVATE, CONS

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 6 trailing nodes:

ACTIVATE(z0) → c3
ACTIVATE(n__from(z0)) → c2(FROM(z0))
CONS(z0, z1) → c5
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
FROM(z0) → c1
FROM(z0) → c(CONS(z0, n__from(s(z0))))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(z0, z1)
cons(z0, z1) → n__cons(z0, z1)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

from, activate, cons

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)