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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
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active(__(X, nil)) → mark(X)
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active(__(nil, X)) → mark(X)
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active(U11(tt)) → mark(U12(tt))
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active(U12(tt)) → mark(tt)
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active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
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active(__(X1, X2)) → __(active(X1), X2)
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active(__(X1, X2)) → __(X1, active(X2))
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active(U11(X)) → U11(active(X))
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active(U12(X)) → U12(active(X))
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active(isNePal(X)) → isNePal(active(X))
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__(mark(X1), X2) → mark(__(X1, X2))
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__(X1, mark(X2)) → mark(__(X1, X2))
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U11(mark(X)) → mark(U11(X))
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U12(mark(X)) → mark(U12(X))
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isNePal(mark(X)) → mark(isNePal(X))
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proper(__(X1, X2)) → __(proper(X1), proper(X2))
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proper(nil) → ok(nil)
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proper(U11(X)) → U11(proper(X))
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proper(tt) → ok(tt)
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proper(U12(X)) → U12(proper(X))
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proper(isNePal(X)) → isNePal(proper(X))
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__(ok(X1), ok(X2)) → ok(__(X1, X2))
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U11(ok(X)) → ok(U11(X))
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U12(ok(X)) → ok(U12(X))
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isNePal(ok(X)) → ok(isNePal(X))
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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(-raise)-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.
The compatible tree automaton used to show the Match(-raise)-Boundedness (for constructor-based start-terms) is represented by:
final states : [7, 2, 3, 4, 5, 6, 1, 20, 21, 22, 23, 26, 27, 30, 38, 39, 40, 41, 42, 43]
transitions:
mark0(0) → 0
ok0(0) → 0
top0(0) → 7
__1(0, 0) → 8
mark1(8) → 2
__1(0, 0) → 2
U111(0) → 9
mark1(9) → 3
U111(0) → 3
U121(0) → 10
mark1(10) → 4
U121(0) → 4
isNePal1(0) → 11
mark1(11) → 5
isNePal1(0) → 5
ok1(12) → 6
ok1(13) → 6
__1(0, 0) → 14
ok1(14) → 2
U111(0) → 15
ok1(15) → 3
U121(0) → 16
ok1(16) → 4
isNePal1(0) → 17
ok1(17) → 5
proper1(0) → 18
top1(18) → 7
proper1(0) → 6
active1(0) → 19
top1(19) → 7
active1(0) → 1
__1(0, 0) → 20
U111(0) → 21
U121(0) → 22
isNePal1(0) → 23
proper1(0) → 26
active1(0) → 27
mark0(24) → 0
mark0(25) → 0
ok0(24) → 0
ok0(25) → 0
top0(24) → 7
top0(25) → 7
mark1(20) → 2
mark1(21) → 3
mark1(22) → 4
mark1(23) → 5
__1(24, 0) → 20
__1(0, 24) → 20
__1(25, 0) → 20
__1(0, 25) → 20
__1(24, 24) → 20
__1(24, 25) → 20
__1(25, 24) → 20
__1(25, 25) → 20
U111(24) → 21
U111(25) → 21
U121(24) → 22
U121(25) → 22
isNePal1(24) → 23
isNePal1(25) → 23
ok1(20) → 2
ok1(21) → 3
ok1(22) → 4
ok1(23) → 5
ok1(24) → 6
ok1(25) → 6
top1(26) → 7
top1(27) → 7
proper1(24) → 26
proper1(25) → 26
active1(24) → 27
active1(25) → 27
ok1(24) → 0
ok1(25) → 0
mark1(8) → 20
__1(24, 0) → 8
__1(25, 0) → 8
__1(0, 24) → 8
__1(24, 24) → 8
__1(25, 24) → 8
__1(0, 25) → 8
__1(24, 25) → 8
__1(25, 25) → 8
mark1(9) → 21
U111(24) → 9
U111(25) → 9
mark1(10) → 22
U121(24) → 10
U121(25) → 10
mark1(11) → 23
isNePal1(24) → 11
isNePal1(25) → 11
nil2() → 28
ok2(28) → 26
nil2() → 12
nil2() → 0
nil2() → 24
tt2() → 29
ok2(29) → 26
tt2() → 13
tt2() → 0
tt2() → 25
ok1(14) → 20
__1(0, 24) → 14
__1(0, 25) → 14
__1(24, 0) → 14
__1(24, 24) → 14
__1(24, 25) → 14
__1(25, 0) → 14
__1(25, 24) → 14
__1(25, 25) → 14
ok1(15) → 21
U111(24) → 15
U111(25) → 15
ok1(16) → 22
U121(24) → 16
U121(25) → 16
ok1(17) → 23
isNePal1(24) → 17
isNePal1(25) → 17
ok1(24) → 30
ok1(25) → 30
nil2() → 31
tt2() → 32
mark0(30) → 0
mark0(31) → 0
mark0(32) → 0
ok0(30) → 0
ok0(31) → 0
ok0(32) → 0
top0(30) → 7
top0(31) → 7
top0(32) → 7
mark1(20) → 20
mark1(21) → 21
mark1(22) → 22
mark1(23) → 23
__1(30, 0) → 20
__1(30, 24) → 20
__1(30, 25) → 20
__1(0, 30) → 20
__1(24, 30) → 20
__1(25, 30) → 20
__1(31, 0) → 20
__1(31, 24) → 20
__1(31, 25) → 20
__1(0, 31) → 20
__1(24, 31) → 20
__1(25, 31) → 20
__1(32, 0) → 20
__1(32, 24) → 20
__1(32, 25) → 20
__1(0, 32) → 20
__1(24, 32) → 20
__1(25, 32) → 20
__1(30, 30) → 20
__1(30, 31) → 20
__1(30, 32) → 20
__1(31, 30) → 20
__1(32, 30) → 20
U111(30) → 21
U121(30) → 22
isNePal1(30) → 23
ok1(20) → 20
ok1(21) → 21
ok1(22) → 22
ok1(23) → 23
ok1(31) → 30
ok1(32) → 30
proper1(30) → 26
proper1(31) → 26
proper1(32) → 26
active1(30) → 27
ok2(31) → 26
ok2(32) → 26
ok1(31) → 0
ok2(31) → 0
ok1(32) → 0
ok2(32) → 0
ok2(31) → 30
ok2(32) → 30
__2(31, 31) → 33
ok2(33) → 20
__2(31, 31) → 20
__2(31, 32) → 33
__2(31, 32) → 20
__2(32, 31) → 33
__2(32, 31) → 20
__2(32, 32) → 33
__2(32, 32) → 20
U112(31) → 34
ok2(34) → 21
U112(31) → 21
U112(32) → 34
U112(32) → 21
U122(31) → 35
ok2(35) → 22
U122(31) → 22
U122(32) → 35
U122(32) → 22
isNePal2(31) → 36
ok2(36) → 23
isNePal2(31) → 23
isNePal2(32) → 36
isNePal2(32) → 23
active2(31) → 37
top2(37) → 7
active2(31) → 27
active2(32) → 37
active2(32) → 27
ok2(31) → 38
ok2(32) → 38
__2(31, 31) → 39
__2(31, 32) → 39
__2(32, 31) → 39
__2(32, 32) → 39
U112(31) → 40
U112(32) → 40
U122(31) → 41
U122(32) → 41
isNePal2(31) → 42
isNePal2(32) → 42
active2(31) → 43
active2(32) → 43
mark0(38) → 0
ok0(38) → 0
top0(38) → 7
mark1(39) → 20
mark1(40) → 21
mark1(41) → 22
mark1(42) → 23
__1(31, 31) → 20
__1(31, 32) → 20
__1(32, 31) → 20
__1(32, 32) → 20
__1(38, 0) → 20
__1(38, 24) → 20
__1(38, 25) → 20
__1(38, 30) → 20
__1(38, 31) → 20
__1(38, 32) → 20
__1(0, 38) → 20
__1(24, 38) → 20
__1(25, 38) → 20
__1(30, 38) → 20
__1(31, 38) → 20
__1(32, 38) → 20
__1(38, 38) → 20
U111(31) → 21
U111(32) → 21
U111(38) → 21
U121(31) → 22
U121(32) → 22
U121(38) → 22
isNePal1(31) → 23
isNePal1(32) → 23
isNePal1(38) → 23
ok1(39) → 20
ok1(40) → 21
ok1(41) → 22
ok1(42) → 23
top1(38) → 7
top1(43) → 7
proper1(38) → 26
active1(31) → 27
active1(32) → 27
active1(38) → 27
ok2(39) → 20
ok2(40) → 21
ok2(41) → 22
ok2(42) → 23
top2(43) → 7
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(2) BOUNDS(O(1), O(n^1))
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