(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(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(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(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(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:
p(mark(X)) → mark(p(X))
top(ok(X)) → top(active(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
p(ok(X)) → ok(p(X))
f(mark(X)) → mark(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(0) → ok(0)
f(ok(X)) → ok(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
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:
p(mark(X)) → mark(p(X))
top(ok(X)) → top(active(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
p(ok(X)) → ok(p(X))
f(mark(X)) → mark(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(0) → ok(0)
f(ok(X)) → ok(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
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:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
00() → 0
p0(0) → 1
top0(0) → 2
s0(0) → 3
f0(0) → 4
cons0(0, 0) → 5
proper0(0) → 6
p1(0) → 7
mark1(7) → 1
active1(0) → 8
top1(8) → 2
s1(0) → 9
ok1(9) → 3
s1(0) → 10
mark1(10) → 3
p1(0) → 11
ok1(11) → 1
f1(0) → 12
mark1(12) → 4
cons1(0, 0) → 13
ok1(13) → 5
01() → 14
ok1(14) → 6
f1(0) → 15
ok1(15) → 4
cons1(0, 0) → 16
mark1(16) → 5
proper1(0) → 17
top1(17) → 2
mark1(7) → 7
mark1(7) → 11
ok1(9) → 9
ok1(9) → 10
mark1(10) → 9
mark1(10) → 10
ok1(11) → 7
ok1(11) → 11
mark1(12) → 12
mark1(12) → 15
ok1(13) → 13
ok1(13) → 16
ok1(14) → 17
ok1(15) → 12
ok1(15) → 15
mark1(16) → 13
mark1(16) → 16
active2(14) → 18
top2(18) → 2
(6) BOUNDS(1, n^1)