(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(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(cons2(X1, X2)) → cons2(X1, active(X2))
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) → cons2(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(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(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(cons2(X1, X2)) → cons2(X1, active(X2))
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) → cons2(proper(X1), proper(X2))
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(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:
negrecip(mark(X)) → mark(negrecip(X))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
top(ok(X)) → top(active(X))
posrecip(mark(X)) → mark(posrecip(X))
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
square(mark(X)) → mark(square(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
posrecip(ok(X)) → ok(posrecip(X))
square(ok(X)) → ok(square(X))
proper(rnil) → ok(rnil)
times(X1, mark(X2)) → mark(times(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
pi(mark(X)) → mark(pi(X))
proper(nil) → ok(nil)
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
cons(mark(X1), X2) → mark(cons(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
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 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, 14, 15]
transitions:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
rnil0() → 0
nil0() → 0
00() → 0
negrecip0(0) → 1
times0(0, 0) → 2
top0(0) → 3
posrecip0(0) → 4
from0(0) → 5
cons20(0, 0) → 6
rcons0(0, 0) → 7
cons0(0, 0) → 8
2ndsneg0(0, 0) → 9
square0(0) → 10
2ndspos0(0, 0) → 11
proper0(0) → 12
pi0(0) → 13
plus0(0, 0) → 14
s0(0) → 15
negrecip1(0) → 16
mark1(16) → 1
times1(0, 0) → 17
ok1(17) → 2
active1(0) → 18
top1(18) → 3
posrecip1(0) → 19
mark1(19) → 4
from1(0) → 20
ok1(20) → 5
from1(0) → 21
mark1(21) → 5
cons21(0, 0) → 22
ok1(22) → 6
rcons1(0, 0) → 23
mark1(23) → 7
negrecip1(0) → 24
ok1(24) → 1
cons1(0, 0) → 25
ok1(25) → 8
rcons1(0, 0) → 26
ok1(26) → 7
2ndsneg1(0, 0) → 27
mark1(27) → 9
square1(0) → 28
mark1(28) → 10
2ndspos1(0, 0) → 29
ok1(29) → 11
posrecip1(0) → 30
ok1(30) → 4
square1(0) → 31
ok1(31) → 10
rnil1() → 32
ok1(32) → 12
times1(0, 0) → 33
mark1(33) → 2
pi1(0) → 34
ok1(34) → 13
plus1(0, 0) → 35
ok1(35) → 14
plus1(0, 0) → 36
mark1(36) → 14
pi1(0) → 37
mark1(37) → 13
nil1() → 38
ok1(38) → 12
2ndsneg1(0, 0) → 39
ok1(39) → 9
cons21(0, 0) → 40
mark1(40) → 6
2ndspos1(0, 0) → 41
mark1(41) → 11
s1(0) → 42
ok1(42) → 15
s1(0) → 43
mark1(43) → 15
01() → 44
ok1(44) → 12
cons1(0, 0) → 45
mark1(45) → 8
proper1(0) → 46
top1(46) → 3
mark1(16) → 16
mark1(16) → 24
ok1(17) → 17
ok1(17) → 33
mark1(19) → 19
mark1(19) → 30
ok1(20) → 20
ok1(20) → 21
mark1(21) → 20
mark1(21) → 21
ok1(22) → 22
ok1(22) → 40
mark1(23) → 23
mark1(23) → 26
ok1(24) → 16
ok1(24) → 24
ok1(25) → 25
ok1(25) → 45
ok1(26) → 23
ok1(26) → 26
mark1(27) → 27
mark1(27) → 39
mark1(28) → 28
mark1(28) → 31
ok1(29) → 29
ok1(29) → 41
ok1(30) → 19
ok1(30) → 30
ok1(31) → 28
ok1(31) → 31
ok1(32) → 46
mark1(33) → 17
mark1(33) → 33
ok1(34) → 34
ok1(34) → 37
ok1(35) → 35
ok1(35) → 36
mark1(36) → 35
mark1(36) → 36
mark1(37) → 34
mark1(37) → 37
ok1(38) → 46
ok1(39) → 27
ok1(39) → 39
mark1(40) → 22
mark1(40) → 40
mark1(41) → 29
mark1(41) → 41
ok1(42) → 42
ok1(42) → 43
mark1(43) → 42
mark1(43) → 43
ok1(44) → 46
mark1(45) → 25
mark1(45) → 45
active2(32) → 47
top2(47) → 3
active2(38) → 47
active2(44) → 47
(4) BOUNDS(1, n^1)