(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(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(fib(X)) → fib(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(fib1(X1, X2)) → fib1(active(X1), X2)
active(fib1(X1, X2)) → fib1(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fib(mark(X)) → mark(fib(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fib(X)) → fib(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) → fib1(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
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(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(fib(X)) → fib(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(fib1(X1, X2)) → fib1(active(X1), X2)
active(fib1(X1, X2)) → fib1(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
proper(fib(X)) → fib(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) → fib1(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
(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:
fib1(mark(X1), X2) → mark(fib1(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(ok(X)) → top(active(X))
fib(ok(X)) → ok(fib(X))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
proper(0) → ok(0)
add(mark(X1), X2) → mark(add(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib(mark(X)) → mark(fib(X))
sel(X1, mark(X2)) → mark(sel(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(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]
transitions:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
00() → 0
fib10(0, 0) → 1
add0(0, 0) → 2
top0(0) → 3
fib0(0) → 4
cons0(0, 0) → 5
sel0(0, 0) → 6
s0(0) → 7
proper0(0) → 8
fib11(0, 0) → 9
mark1(9) → 1
add1(0, 0) → 10
ok1(10) → 2
active1(0) → 11
top1(11) → 3
fib1(0) → 12
ok1(12) → 4
cons1(0, 0) → 13
ok1(13) → 5
sel1(0, 0) → 14
mark1(14) → 6
add1(0, 0) → 15
mark1(15) → 2
fib11(0, 0) → 16
ok1(16) → 1
s1(0) → 17
ok1(17) → 7
s1(0) → 18
mark1(18) → 7
01() → 19
ok1(19) → 8
sel1(0, 0) → 20
ok1(20) → 6
fib1(0) → 21
mark1(21) → 4
cons1(0, 0) → 22
mark1(22) → 5
proper1(0) → 23
top1(23) → 3
mark1(9) → 9
mark1(9) → 16
ok1(10) → 10
ok1(10) → 15
ok1(12) → 12
ok1(12) → 21
ok1(13) → 13
ok1(13) → 22
mark1(14) → 14
mark1(14) → 20
mark1(15) → 10
mark1(15) → 15
ok1(16) → 9
ok1(16) → 16
ok1(17) → 17
ok1(17) → 18
mark1(18) → 17
mark1(18) → 18
ok1(19) → 23
ok1(20) → 14
ok1(20) → 20
mark1(21) → 12
mark1(21) → 21
mark1(22) → 13
mark1(22) → 22
active2(19) → 24
top2(24) → 3
(4) BOUNDS(1, n^1)