(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) 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
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing nodes:
APP(nil, z0) → c
ZWADR(nil, z0) → c3
FROM(z0) → c2
APP(cons(z0), z1) → c1
ZWADR(cons(z0), cons(z1)) → c5(APP(z1, cons(z0)))
PREFIX(z0) → c6
ZWADR(z0, nil) → c4
(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:none
S tuples:none
K tuples:none
Defined Rule Symbols:
app, from, zWadr, prefix
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)