(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(cons(X, XS), YS) → APP(XS, YS)
FROM(X) → FROM(s(X))
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, nil))
ZWADR(cons(X, XS), cons(Y, YS)) → ZWADR(XS, YS)
PREFIX(L) → ZWADR(L, prefix(L))
PREFIX(L) → PREFIX(L)
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 2 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FROM(X) → FROM(s(X))
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x0 / s(x0)] on the rule
FROM(s(x0))[ ]n[ ] → FROM(s(x0))[ ]n[x0 / s(x0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
FROM(X)[ ]n[ ] → FROM(s(X))[ ]n[ ]
by OriginalRule from TRS P
(7) NO
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(cons(X, XS), YS) → APP(XS, YS)
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- APP(cons(X, XS), YS) → APP(XS, YS)
The graph contains the following edges 1 > 1, 2 >= 2
(10) YES
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ZWADR(cons(X, XS), cons(Y, YS)) → ZWADR(XS, YS)
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- ZWADR(cons(X, XS), cons(Y, YS)) → ZWADR(XS, YS)
The graph contains the following edges 1 > 1, 2 > 2
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PREFIX(L) → PREFIX(L)
The TRS R consists of the following rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [ ] on the rule
PREFIX(x0)[ ]n[ ] → PREFIX(x0)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Instantiation
PREFIX(L)[ ]n[ ] → PREFIX(L)[ ]n[ ]
by OriginalRule from TRS P
(16) NO