(0) 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
(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
nil0() → 0
cons0(0) → 0
app0(0, 0) → 1
from0(0) → 2
zWadr0(0, 0) → 3
prefix0(0) → 4
cons1(0) → 1
cons1(0) → 2
nil1() → 3
cons1(0) → 6
app1(0, 6) → 5
cons1(5) → 3
nil1() → 7
cons1(7) → 4
0 → 1
6 → 5
(2) BOUNDS(1, n^1)
(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:
APP(nil, z0) → c
PREFIX(z0) → c6
ZWADR(cons(z0), cons(z1)) → c5(APP(z1, cons(z0)))
ZWADR(z0, nil) → c4
FROM(z0) → c2
ZWADR(nil, z0) → c3
APP(cons(z0), z1) → c1
(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)