(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, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
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, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(g(X)) → g(active(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
(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:
g(ok(X)) → ok(g(X))
top(ok(X)) → top(active(X))
active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(mark(X)) → mark(g(X))
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 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]
transitions:
ok0(0) → 0
b0() → 0
mark0(0) → 0
c0() → 0
g0(0) → 1
top0(0) → 2
active0(0) → 3
proper0(0) → 4
f0(0, 0, 0) → 5
g1(0) → 6
ok1(6) → 1
active1(0) → 7
top1(7) → 2
c1() → 8
mark1(8) → 3
b1() → 9
ok1(9) → 4
c1() → 10
ok1(10) → 4
f1(0, 0, 0) → 11
ok1(11) → 5
g1(0) → 12
mark1(12) → 1
proper1(0) → 13
top1(13) → 2
ok1(6) → 6
ok1(6) → 12
mark1(8) → 7
ok1(9) → 13
ok1(10) → 13
ok1(11) → 11
mark1(12) → 6
mark1(12) → 12
active2(9) → 14
top2(14) → 2
active2(10) → 14
proper2(8) → 15
top2(15) → 2
c2() → 16
mark2(16) → 14
c2() → 17
ok2(17) → 15
active3(17) → 18
top3(18) → 2
proper3(16) → 19
top3(19) → 2
c3() → 20
ok3(20) → 19
active4(20) → 21
top4(21) → 2
(4) BOUNDS(1, n^1)