(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(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(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(f(X, X)) → mark(f(a, b))
active(f(X1, X2)) → f(active(X1), X2)
proper(f(X1, X2)) → f(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:
f(mark(X1), X2) → mark(f(X1, X2))
top(ok(X)) → top(active(X))
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
active(b) → mark(a)
top(mark(X)) → top(proper(X))
proper(a) → ok(a)
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 4.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
mark0(0) → 0
ok0(0) → 0
b0() → 0
a0() → 0
f0(0, 0) → 1
top0(0) → 2
proper0(0) → 3
active0(0) → 4
f1(0, 0) → 5
mark1(5) → 1
active1(0) → 6
top1(6) → 2
b1() → 7
ok1(7) → 3
f1(0, 0) → 8
ok1(8) → 1
a1() → 9
mark1(9) → 4
proper1(0) → 10
top1(10) → 2
a1() → 11
ok1(11) → 3
mark1(5) → 5
mark1(5) → 8
ok1(7) → 10
ok1(8) → 5
ok1(8) → 8
mark1(9) → 6
ok1(11) → 10
active2(7) → 12
top2(12) → 2
active2(11) → 12
proper2(9) → 13
top2(13) → 2
a2() → 14
mark2(14) → 12
a2() → 15
ok2(15) → 13
active3(15) → 16
top3(16) → 2
proper3(14) → 17
top3(17) → 2
a3() → 18
ok3(18) → 17
active4(18) → 19
top4(19) → 2
(4) BOUNDS(1, n^1)