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:

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X), cons(Y)) → cons(app(Y, cons(X)))
prefix(L) → cons(nil)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

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

(2) 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:

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X), cons(Y)) → cons(app(Y, cons(X)))
prefix(L) → cons(nil)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
app(cons(z0), z1) → cons(z0)
from(z0) → cons(z0)
zWadr(nil, z0) → nil
zWadr(z0, nil) → nil
zWadr(cons(z0), cons(z1)) → cons(app(z1, cons(z0)))
prefix(z0) → cons(nil)

Tuples:

APP(nil, z0) → c
APP(cons(z0), z1) → c1
FROM(z0) → c2
ZWADR(nil, z0) → c3
ZWADR(z0, nil) → c4
ZWADR(cons(z0), cons(z1)) → c5(APP(z1, cons(z0)))
PREFIX(z0) → c6

S tuples:

APP(nil, z0) → c
APP(cons(z0), z1) → c1
FROM(z0) → c2
ZWADR(nil, z0) → c3
ZWADR(z0, nil) → c4
ZWADR(cons(z0), cons(z1)) → c5(APP(z1, cons(z0)))
PREFIX(z0) → c6

K tuples:none
Defined Rule Symbols:

app, from, zWadr, prefix

Defined Pair Symbols:

APP, FROM, ZWADR, PREFIX

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 7 trailing nodes:

FROM(z0) → c2
PREFIX(z0) → c6
ZWADR(nil, z0) → c3
APP(nil, z0) → c
ZWADR(cons(z0), cons(z1)) → c5(APP(z1, cons(z0)))
APP(cons(z0), z1) → c1
ZWADR(z0, nil) → c4

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
app(cons(z0), z1) → cons(z0)
from(z0) → cons(z0)
zWadr(nil, z0) → nil
zWadr(z0, nil) → nil
zWadr(cons(z0), cons(z1)) → cons(app(z1, cons(z0)))
prefix(z0) → cons(nil)

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

app, from, zWadr, prefix

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)