(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) 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]
transitions:
mark0(0) → 0
c0() → 0
true0() → 0
false0() → 0
ok0(0) → 0
active0(0) → 1
f0(0) → 2
if0(0, 0, 0) → 3
proper0(0) → 4
top0(0) → 5
f1(0) → 6
mark1(6) → 2
if1(0, 0, 0) → 7
mark1(7) → 3
c1() → 8
ok1(8) → 4
true1() → 9
ok1(9) → 4
false1() → 10
ok1(10) → 4
f1(0) → 11
ok1(11) → 2
if1(0, 0, 0) → 12
ok1(12) → 3
proper1(0) → 13
top1(13) → 5
active1(0) → 14
top1(14) → 5
mark1(6) → 6
mark1(6) → 11
mark1(7) → 7
mark1(7) → 12
ok1(8) → 13
ok1(9) → 13
ok1(10) → 13
ok1(11) → 6
ok1(11) → 11
ok1(12) → 7
ok1(12) → 12
active2(8) → 15
top2(15) → 5
active2(9) → 15
active2(10) → 15
(2) BOUNDS(1, n^1)