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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, M, N)) → mark(U12(tt, M, N))
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active(U12(tt, M, N)) → mark(s(plus(N, M)))
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active(plus(N, 0)) → mark(N)
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active(plus(N, s(M))) → mark(U11(tt, M, N))
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active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
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active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
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active(s(X)) → s(active(X))
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active(plus(X1, X2)) → plus(active(X1), X2)
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active(plus(X1, X2)) → plus(X1, active(X2))
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U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
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U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
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s(mark(X)) → mark(s(X))
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plus(mark(X1), X2) → mark(plus(X1, X2))
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plus(X1, mark(X2)) → mark(plus(X1, X2))
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proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
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proper(tt) → ok(tt)
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proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
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proper(s(X)) → s(proper(X))
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proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
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proper(0) → ok(0)
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U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
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U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
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s(ok(X)) → ok(s(X))
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plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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top(mark(X)) → top(proper(X))
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top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
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(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, 6, 7]
transitions:
tt0() → 0
mark0(0) → 0
00() → 0
ok0(0) → 0
active0(0) → 1
U110(0, 0, 0) → 2
U120(0, 0, 0) → 3
s0(0) → 4
plus0(0, 0) → 5
proper0(0) → 6
top0(0) → 7
U111(0, 0, 0) → 8
mark1(8) → 2
U121(0, 0, 0) → 9
mark1(9) → 3
s1(0) → 10
mark1(10) → 4
plus1(0, 0) → 11
mark1(11) → 5
tt1() → 12
ok1(12) → 6
01() → 13
ok1(13) → 6
U111(0, 0, 0) → 14
ok1(14) → 2
U121(0, 0, 0) → 15
ok1(15) → 3
s1(0) → 16
ok1(16) → 4
plus1(0, 0) → 17
ok1(17) → 5
proper1(0) → 18
top1(18) → 7
active1(0) → 19
top1(19) → 7
mark1(8) → 8
mark1(8) → 14
mark1(9) → 9
mark1(9) → 15
mark1(10) → 10
mark1(10) → 16
mark1(11) → 11
mark1(11) → 17
ok1(12) → 18
ok1(13) → 18
ok1(14) → 8
ok1(14) → 14
ok1(15) → 9
ok1(15) → 15
ok1(16) → 10
ok1(16) → 16
ok1(17) → 11
ok1(17) → 17
active2(12) → 20
top2(20) → 7
active2(13) → 20
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(2) BOUNDS(O(1), O(n^1))
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