(0) Obligation:
The Runtime Complexity (full) 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(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL
(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(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(U11(X)) → U11(proper(X))
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
(2) Obligation:
The Runtime Complexity (full) 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)
isNePal(ok(X)) → ok(isNePal(X))
U12(mark(X)) → mark(U12(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
__(mark(X1), X2) → mark(__(X1, X2))
U12(ok(X)) → ok(U12(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
top(mark(X)) → top(proper(X))
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS is a non-duplicating overlay system, we have rc = irc.
(4) 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)
isNePal(ok(X)) → ok(isNePal(X))
U12(mark(X)) → mark(U12(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
__(mark(X1), X2) → mark(__(X1, X2))
U12(ok(X)) → ok(U12(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
top(mark(X)) → top(proper(X))
Rewrite Strategy: INNERMOST
(5) 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]
transitions:
ok0(0) → 0
active0(0) → 0
nil0() → 0
tt0() → 0
mark0(0) → 0
top0(0) → 1
proper0(0) → 2
isNePal0(0) → 3
U120(0) → 4
__0(0, 0) → 5
U110(0) → 6
active1(0) → 7
top1(7) → 1
nil1() → 8
ok1(8) → 2
tt1() → 9
ok1(9) → 2
isNePal1(0) → 10
ok1(10) → 3
U121(0) → 11
mark1(11) → 4
__1(0, 0) → 12
ok1(12) → 5
__1(0, 0) → 13
mark1(13) → 5
U121(0) → 14
ok1(14) → 4
U111(0) → 15
mark1(15) → 6
U111(0) → 16
ok1(16) → 6
isNePal1(0) → 17
mark1(17) → 3
proper1(0) → 18
top1(18) → 1
ok1(8) → 18
ok1(9) → 18
ok1(10) → 10
ok1(10) → 17
mark1(11) → 11
mark1(11) → 14
ok1(12) → 12
ok1(12) → 13
mark1(13) → 12
mark1(13) → 13
ok1(14) → 11
ok1(14) → 14
mark1(15) → 15
mark1(15) → 16
ok1(16) → 15
ok1(16) → 16
mark1(17) → 10
mark1(17) → 17
active2(8) → 19
top2(19) → 1
active2(9) → 19
(6) BOUNDS(1, n^1)