(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isList(ok(X)) → ok(isList(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
isPal(ok(X)) → ok(isPal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
top(ok(X)) → top(active(X))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isPal(ok(X)) → ok(isPal(X))
isNePal(ok(X)) → ok(isNePal(X))
and(mark(X1), X2) → mark(and(X1, X2))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
__(mark(X1), X2) → mark(__(X1, X2))
proper(a) → ok(a)
proper(i) → ok(i)
and(ok(X1), ok(X2)) → ok(and(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(o) → ok(o)
proper(u) → ok(u)
top(mark(X)) → top(proper(X))
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))
Rewrite Strategy: INNERMOST
(3) 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 2.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9]
transitions:
ok0(0) → 0
active0(0) → 0
nil0() → 0
tt0() → 0
mark0(0) → 0
a0() → 0
i0() → 0
o0() → 0
u0() → 0
e0() → 0
top0(0) → 1
proper0(0) → 2
isNeList0(0) → 3
isQid0(0) → 4
isPal0(0) → 5
isNePal0(0) → 6
and0(0, 0) → 7
__0(0, 0) → 8
isList0(0) → 9
active1(0) → 10
top1(10) → 1
nil1() → 11
ok1(11) → 2
tt1() → 12
ok1(12) → 2
isNeList1(0) → 13
ok1(13) → 3
isQid1(0) → 14
ok1(14) → 4
isPal1(0) → 15
ok1(15) → 5
isNePal1(0) → 16
ok1(16) → 6
and1(0, 0) → 17
mark1(17) → 7
__1(0, 0) → 18
ok1(18) → 8
__1(0, 0) → 19
mark1(19) → 8
a1() → 20
ok1(20) → 2
i1() → 21
ok1(21) → 2
and1(0, 0) → 22
ok1(22) → 7
o1() → 23
ok1(23) → 2
u1() → 24
ok1(24) → 2
proper1(0) → 25
top1(25) → 1
e1() → 26
ok1(26) → 2
isList1(0) → 27
ok1(27) → 9
ok1(11) → 25
ok1(12) → 25
ok1(13) → 13
ok1(14) → 14
ok1(15) → 15
ok1(16) → 16
mark1(17) → 17
mark1(17) → 22
ok1(18) → 18
ok1(18) → 19
mark1(19) → 18
mark1(19) → 19
ok1(20) → 25
ok1(21) → 25
ok1(22) → 17
ok1(22) → 22
ok1(23) → 25
ok1(24) → 25
ok1(26) → 25
ok1(27) → 27
active2(11) → 28
top2(28) → 1
active2(12) → 28
active2(20) → 28
active2(21) → 28
active2(23) → 28
active2(24) → 28
active2(26) → 28
(4) BOUNDS(1, n^1)