(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(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
x(ok(X1), ok(X2)) → ok(x(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(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(x(X1, X2)) → x(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:
top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
and(mark(X1), X2) → mark(and(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(0) → ok(0)
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(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]
transitions:
ok0(0) → 0
active0(0) → 0
tt0() → 0
mark0(0) → 0
00() → 0
top0(0) → 1
proper0(0) → 2
and0(0, 0) → 3
x0(0, 0) → 4
s0(0) → 5
plus0(0, 0) → 6
active1(0) → 7
top1(7) → 1
tt1() → 8
ok1(8) → 2
and1(0, 0) → 9
mark1(9) → 3
x1(0, 0) → 10
mark1(10) → 4
x1(0, 0) → 11
ok1(11) → 4
and1(0, 0) → 12
ok1(12) → 3
s1(0) → 13
ok1(13) → 5
s1(0) → 14
mark1(14) → 5
plus1(0, 0) → 15
mark1(15) → 6
01() → 16
ok1(16) → 2
plus1(0, 0) → 17
ok1(17) → 6
proper1(0) → 18
top1(18) → 1
ok1(8) → 18
mark1(9) → 9
mark1(9) → 12
mark1(10) → 10
mark1(10) → 11
ok1(11) → 10
ok1(11) → 11
ok1(12) → 9
ok1(12) → 12
ok1(13) → 13
ok1(13) → 14
mark1(14) → 13
mark1(14) → 14
mark1(15) → 15
mark1(15) → 17
ok1(16) → 18
ok1(17) → 15
ok1(17) → 17
active2(8) → 19
top2(19) → 1
active2(16) → 19
(4) BOUNDS(1, n^1)