(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(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(add(X1, X2)) → add(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
add(mark(X1), X2) → mark(add(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(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(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(add(X1, X2)) → add(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(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:
proper(true) → ok(true)
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(ok(X)) → top(active(X))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
s(ok(X)) → ok(s(X))
proper(false) → ok(false)
proper(0) → ok(0)
first(X1, mark(X2)) → mark(first(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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]
transitions:
true0() → 0
ok0(0) → 0
active0(0) → 0
nil0() → 0
mark0(0) → 0
false0() → 0
00() → 0
proper0(0) → 1
add0(0, 0) → 2
top0(0) → 3
from0(0) → 4
cons0(0, 0) → 5
and0(0, 0) → 6
first0(0, 0) → 7
s0(0) → 8
if0(0, 0, 0) → 9
true1() → 10
ok1(10) → 1
add1(0, 0) → 11
ok1(11) → 2
active1(0) → 12
top1(12) → 3
nil1() → 13
ok1(13) → 1
from1(0) → 14
ok1(14) → 4
cons1(0, 0) → 15
ok1(15) → 5
and1(0, 0) → 16
mark1(16) → 6
first1(0, 0) → 17
ok1(17) → 7
first1(0, 0) → 18
mark1(18) → 7
and1(0, 0) → 19
ok1(19) → 6
s1(0) → 20
ok1(20) → 8
false1() → 21
ok1(21) → 1
01() → 22
ok1(22) → 1
add1(0, 0) → 23
mark1(23) → 2
if1(0, 0, 0) → 24
mark1(24) → 9
if1(0, 0, 0) → 25
ok1(25) → 9
proper1(0) → 26
top1(26) → 3
ok1(10) → 26
ok1(11) → 11
ok1(11) → 23
ok1(13) → 26
ok1(14) → 14
ok1(15) → 15
mark1(16) → 16
mark1(16) → 19
ok1(17) → 17
ok1(17) → 18
mark1(18) → 17
mark1(18) → 18
ok1(19) → 16
ok1(19) → 19
ok1(20) → 20
ok1(21) → 26
ok1(22) → 26
mark1(23) → 11
mark1(23) → 23
mark1(24) → 24
mark1(24) → 25
ok1(25) → 24
ok1(25) → 25
active2(10) → 27
top2(27) → 3
active2(13) → 27
active2(21) → 27
active2(22) → 27
(4) BOUNDS(1, n^1)