Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
ADDLISTS(xs, ys, zs) → ISZERO(head(xs))
ADDLISTS(xs, ys, zs) → ISEMPTY(xs)
ADDLISTS(xs, ys, zs) → ISEMPTY(ys)
ADDLISTS(xs, ys, zs) → INC(head(ys))
ADDLISTS(xs, ys, zs) → TAIL(ys)
INC(s(x)) → INC(x)
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
ADDLISTS(xs, ys, zs) → HEAD(ys)
ADDLISTS(xs, ys, zs) → TAIL(xs)
ADDLISTS(xs, ys, zs) → APPEND(zs, head(ys))
ADDLISTS(xs, ys, zs) → P(head(xs))
ADDLISTS(xs, ys, zs) → HEAD(xs)
P(s(s(x))) → P(s(x))
ADDLIST(xs, ys) → ADDLISTS(xs, ys, nil)
APPEND(cons(y, ys), x) → APPEND(ys, x)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
ADDLISTS(xs, ys, zs) → ISZERO(head(xs))
ADDLISTS(xs, ys, zs) → ISEMPTY(xs)
ADDLISTS(xs, ys, zs) → ISEMPTY(ys)
ADDLISTS(xs, ys, zs) → INC(head(ys))
ADDLISTS(xs, ys, zs) → TAIL(ys)
INC(s(x)) → INC(x)
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
ADDLISTS(xs, ys, zs) → HEAD(ys)
ADDLISTS(xs, ys, zs) → TAIL(xs)
ADDLISTS(xs, ys, zs) → APPEND(zs, head(ys))
ADDLISTS(xs, ys, zs) → P(head(xs))
ADDLISTS(xs, ys, zs) → HEAD(xs)
P(s(s(x))) → P(s(x))
ADDLIST(xs, ys) → ADDLISTS(xs, ys, nil)
APPEND(cons(y, ys), x) → APPEND(ys, x)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 11 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
INC(s(x)) → INC(x)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
INC(s(x)) → INC(x)
R is empty.
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
INC(s(x)) → INC(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- INC(s(x)) → INC(x)
The graph contains the following edges 1 > 1
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
P(s(s(x))) → P(s(x))
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
P(s(s(x))) → P(s(x))
R is empty.
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
P(s(s(x))) → P(s(x))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- P(s(s(x))) → P(s(x))
The graph contains the following edges 1 > 1
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
APPEND(cons(y, ys), x) → APPEND(ys, x)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
APPEND(cons(y, ys), x) → APPEND(ys, x)
R is empty.
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
APPEND(cons(y, ys), x) → APPEND(ys, x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- APPEND(cons(y, ys), x) → APPEND(ys, x)
The graph contains the following edges 1 > 1, 2 >= 2
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
isZero(0) → true
isZero(s(x)) → false
head(cons(x, xs)) → x
tail(cons(x, xs)) → xs
tail(nil) → nil
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
addLists(xs, ys, zs) → if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
if(true, true, b, xs, ys, xs2, ys2, zs, zs2) → zs
if(true, false, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, true, b, xs, ys, xs2, ys2, zs, zs2) → differentLengthError
if(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists(xs2, ys2, zs)
if(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists(xs, ys, zs2)
addList(xs, ys) → addLists(xs, ys, nil)
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
addLists(x0, x1, x2)
if(true, true, x0, x1, x2, x3, x4, x5, x6)
if(true, false, x0, x1, x2, x3, x4, x5, x6)
if(false, true, x0, x1, x2, x3, x4, x5, x6)
if(false, false, false, x0, x1, x2, x3, x4, x5)
if(false, false, true, x0, x1, x2, x3, x4, x5)
addList(x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ADDLISTS(xs, ys, zs) → IF(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) at position [0] we obtained the following new rules:
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(head(cons(x0, x1))), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
ADDLISTS(nil, y1, y2) → IF(true, isEmpty(y1), isZero(head(nil)), tail(nil), tail(y1), cons(p(head(nil)), tail(nil)), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(head(cons(x0, x1))), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
ADDLISTS(nil, y1, y2) → IF(true, isEmpty(y1), isZero(head(nil)), tail(nil), tail(y1), cons(p(head(nil)), tail(nil)), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(head(cons(x0, x1))), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(head(cons(x0, x1))), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1))) at position [2,0] we obtained the following new rules:
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), tail(cons(x0, x1)), tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1))) at position [3] we obtained the following new rules:
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(head(cons(x0, x1))), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1))) at position [5,0,0] we obtained the following new rules:
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), tail(cons(x0, x1))), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1))) at position [5,1] we obtained the following new rules:
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), x1), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), x1), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ADDLISTS(cons(x0, x1), y1, y2) → IF(false, isEmpty(y1), isZero(x0), x1, tail(y1), cons(p(x0), x1), cons(inc(head(y1)), tail(y1)), y2, append(y2, head(y1))) at position [1] we obtained the following new rules:
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
ADDLISTS(cons(y0, y1), nil, y3) → IF(false, true, isZero(y0), y1, tail(nil), cons(p(y0), y1), cons(inc(head(nil)), tail(nil)), y3, append(y3, head(nil)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
ADDLISTS(cons(y0, y1), nil, y3) → IF(false, true, isZero(y0), y1, tail(nil), cons(p(y0), y1), cons(inc(head(nil)), tail(nil)), y3, append(y3, head(nil)))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isEmpty(cons(x, xs)) → false
isEmpty(nil) → true
head(cons(x, xs)) → x
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
tail(nil) → nil
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
head(cons(x, xs)) → x
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isEmpty(cons(x0, x1))
isEmpty(nil)
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
isEmpty(cons(x0, x1))
isEmpty(nil)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
head(cons(x, xs)) → x
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, tail(cons(x0, x1)), cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1)))) at position [4] we obtained the following new rules:
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
head(cons(x, xs)) → x
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(head(cons(x0, x1))), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1)))) at position [6,0,0] we obtained the following new rules:
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
head(cons(x, xs)) → x
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), tail(cons(x0, x1))), y3, append(y3, head(cons(x0, x1)))) at position [6,1] we obtained the following new rules:
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, head(cons(x0, x1))))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
tail(cons(x, xs)) → xs
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
head(cons(x, xs)) → x
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
head(cons(x, xs)) → x
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
tail(cons(x0, x1))
tail(nil)
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
tail(cons(x0, x1))
tail(nil)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, head(cons(x0, x1))))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
head(cons(x, xs)) → x
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, head(cons(x0, x1)))) at position [8,1] we obtained the following new rules:
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, x0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, x0))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
head(cons(x, xs)) → x
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, x0))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
head(cons(x0, x1))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
head(cons(x0, x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, x0))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ADDLISTS(cons(y0, y1), cons(x0, x1), y3) → IF(false, false, isZero(y0), y1, x1, cons(p(y0), y1), cons(inc(x0), x1), y3, append(y3, x0)) at position [2] we obtained the following new rules:
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(p(0), y1), cons(inc(y2), y3), y4, append(y4, y2))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(p(0), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
isZero(0) → true
isZero(s(x)) → false
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
p(0) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(p(0), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(0) → 0
The set Q consists of the following terms:
isZero(0)
isZero(s(x0))
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
isZero(0)
isZero(s(x0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(p(0), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(0) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(p(0), y1), cons(inc(y2), y3), y4, append(y4, y2)) at position [5,0] we obtained the following new rules:
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(0) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2)
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF(false, false, true, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs, ys, zs2) we obtained the following new rules:
IF(false, false, true, z0, z2, cons(0, z0), cons(y_0, z2), z3, y_1) → ADDLISTS(z0, z2, y_1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, true, z0, z2, cons(0, z0), cons(y_0, z2), z3, y_1) → ADDLISTS(z0, z2, y_1)
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
IF(false, false, true, z0, z2, cons(0, z0), cons(y_0, z2), z3, y_1) → ADDLISTS(z0, z2, y_1)
The remaining pairs can at least be oriented weakly.
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
Used ordering: Polynomial interpretation [25]:
POL(0) = 1
POL(ADDLISTS(x1, x2, x3)) = x1
POL(IF(x1, x2, x3, x4, x5, x6, x7, x8, x9)) = x6
POL(append(x1, x2)) = 0
POL(cons(x1, x2)) = x1 + x2
POL(false) = 0
POL(inc(x1)) = 0
POL(nil) = 0
POL(p(x1)) = x1
POL(s(x1)) = x1
POL(true) = 0
The following usable rules [17] were oriented:
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ADDLISTS(cons(0, y1), cons(y2, y3), y4) → IF(false, false, true, y1, y3, cons(0, y1), cons(inc(y2), y3), y4, append(y4, y2))
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ADDLISTS(cons(s(x0), y1), cons(y2, y3), y4) → IF(false, false, false, y1, y3, cons(p(s(x0)), y1), cons(inc(y2), y3), y4, append(y4, y2))
The remaining pairs can at least be oriented weakly.
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( append(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( IF(x1, ..., x9) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 | + | | · | x6 | + | | · | x7 | + | | · | x8 | + | | · | x9 |
| M( ADDLISTS(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IF(false, false, false, xs, ys, xs2, ys2, zs, zs2) → ADDLISTS(xs2, ys2, zs)
The TRS R consists of the following rules:
inc(s(x)) → s(inc(x))
inc(0) → s(0)
append(nil, x) → cons(x, nil)
append(cons(y, ys), x) → cons(y, append(ys, x))
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
The set Q consists of the following terms:
append(nil, x0)
append(cons(x0, x1), x2)
p(s(s(x0)))
p(s(0))
p(0)
inc(s(x0))
inc(0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.