(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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
(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:
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))
g(mark(X1), X2) → mark(g(X1, X2))
active(a) → mark(b)
top(mark(X)) → top(proper(X))
h(mark(X)) → mark(h(X))
proper(a) → ok(a)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
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:
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))
g(mark(X1), X2) → mark(g(X1, X2))
active(a) → mark(b)
top(mark(X)) → top(proper(X))
h(mark(X)) → mark(h(X))
proper(a) → ok(a)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
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 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, 5, 6]
transitions:
mark0(0) → 0
ok0(0) → 0
b0() → 0
a0() → 0
f0(0, 0) → 1
top0(0) → 2
proper0(0) → 3
g0(0, 0) → 4
active0(0) → 5
h0(0) → 6
f1(0, 0) → 7
mark1(7) → 1
active1(0) → 8
top1(8) → 2
b1() → 9
ok1(9) → 3
f1(0, 0) → 10
ok1(10) → 1
g1(0, 0) → 11
mark1(11) → 4
b1() → 12
mark1(12) → 5
proper1(0) → 13
top1(13) → 2
h1(0) → 14
mark1(14) → 6
a1() → 15
ok1(15) → 3
h1(0) → 16
ok1(16) → 6
g1(0, 0) → 17
ok1(17) → 4
mark1(7) → 7
mark1(7) → 10
ok1(9) → 13
ok1(10) → 7
ok1(10) → 10
mark1(11) → 11
mark1(11) → 17
mark1(12) → 8
mark1(14) → 14
mark1(14) → 16
ok1(15) → 13
ok1(16) → 14
ok1(16) → 16
ok1(17) → 11
ok1(17) → 17
active2(9) → 18
top2(18) → 2
active2(15) → 18
proper2(12) → 19
top2(19) → 2
b2() → 20
ok2(20) → 19
b2() → 21
mark2(21) → 18
active3(20) → 22
top3(22) → 2
proper3(21) → 23
top3(23) → 2
b3() → 24
ok3(24) → 23
active4(24) → 25
top4(25) → 2
(6) BOUNDS(1, n^1)