The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
proper(dbl(X)) → dbl(proper(X))
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))

(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:

top(ok(X)) → top(active(X))
from(ok(X)) → ok(from(X))
dbl(mark(X)) → mark(dbl(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
proper(01) → ok(01)
dbl1(ok(X)) → ok(dbl1(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
indx(mark(X1), X2) → mark(indx(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s(ok(X)) → ok(s(X))
dbls(mark(X)) → mark(dbls(X))
proper(0) → ok(0)
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
dbl(ok(X)) → ok(dbl(X))
dbls(ok(X)) → ok(dbls(X))
quote(mark(X)) → mark(quote(X))
top(mark(X)) → top(proper(X))
s1(mark(X)) → mark(s1(X))

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:

top(ok(X)) → top(active(X))
from(ok(X)) → ok(from(X))
dbl(mark(X)) → mark(dbl(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
proper(01) → ok(01)
dbl1(ok(X)) → ok(dbl1(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
indx(mark(X1), X2) → mark(indx(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s(ok(X)) → ok(s(X))
dbls(mark(X)) → mark(dbls(X))
proper(0) → ok(0)
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
dbl(ok(X)) → ok(dbl(X))
dbls(ok(X)) → ok(dbls(X))
quote(mark(X)) → mark(quote(X))
top(mark(X)) → top(proper(X))
s1(mark(X)) → mark(s1(X))

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 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, 8, 9, 10, 11, 12, 13]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
010() → 0
nil0() → 0
00() → 0
top0(0) → 1
from0(0) → 2
dbl0(0) → 3
sel10(0, 0) → 4
cons0(0, 0) → 5
dbl10(0) → 6
proper0(0) → 7
sel0(0, 0) → 8
indx0(0, 0) → 9
quote0(0) → 10
s10(0) → 11
s0(0) → 12
dbls0(0) → 13
active1(0) → 14
top1(14) → 1
from1(0) → 15
ok1(15) → 2
dbl1(0) → 16
mark1(16) → 3
sel11(0, 0) → 17
mark1(17) → 4
cons1(0, 0) → 18
ok1(18) → 5
dbl11(0) → 19
mark1(19) → 6
011() → 20
ok1(20) → 7
dbl11(0) → 21
ok1(21) → 6
sel1(0, 0) → 22
ok1(22) → 8
sel1(0, 0) → 23
mark1(23) → 8
nil1() → 24
ok1(24) → 7
indx1(0, 0) → 25
mark1(25) → 9
quote1(0) → 26
ok1(26) → 10
s11(0) → 27
ok1(27) → 11
sel11(0, 0) → 28
ok1(28) → 4
s1(0) → 29
ok1(29) → 12
dbls1(0) → 30
mark1(30) → 13
01() → 31
ok1(31) → 7
indx1(0, 0) → 32
ok1(32) → 9
dbl1(0) → 33
ok1(33) → 3
dbls1(0) → 34
ok1(34) → 13
quote1(0) → 35
mark1(35) → 10
proper1(0) → 36
top1(36) → 1
s11(0) → 37
mark1(37) → 11
ok1(15) → 15
mark1(16) → 16
mark1(16) → 33
mark1(17) → 17
mark1(17) → 28
ok1(18) → 18
mark1(19) → 19
mark1(19) → 21
ok1(20) → 36
ok1(21) → 19
ok1(21) → 21
ok1(22) → 22
ok1(22) → 23
mark1(23) → 22
mark1(23) → 23
ok1(24) → 36
mark1(25) → 25
mark1(25) → 32
ok1(26) → 26
ok1(26) → 35
ok1(27) → 27
ok1(27) → 37
ok1(28) → 17
ok1(28) → 28
ok1(29) → 29
mark1(30) → 30
mark1(30) → 34
ok1(31) → 36
ok1(32) → 25
ok1(32) → 32
ok1(33) → 16
ok1(33) → 33
ok1(34) → 30
ok1(34) → 34
mark1(35) → 26
mark1(35) → 35
mark1(37) → 27
mark1(37) → 37
active2(20) → 38
top2(38) → 1
active2(24) → 38
active2(31) → 38

(6) BOUNDS(1, n^1)