The Runtime Complexity (full) of the given

0 CpxTRS

↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CpxTRS

↳3 RcToIrcProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CpxTRS

↳5 CpxTrsMatchBoundsTAProof (⇔, 38 ms)

↳6 BOUNDS(1, n^1)

The TRS R consists of the following rules:

active(2nd(cons(X, cons(Y, Z)))) → mark(Y)

active(from(X)) → mark(cons(X, from(s(X))))

active(2nd(X)) → 2nd(active(X))

active(cons(X1, X2)) → cons(active(X1), X2)

active(from(X)) → from(active(X))

active(s(X)) → s(active(X))

2nd(mark(X)) → mark(2nd(X))

cons(mark(X1), X2) → mark(cons(X1, X2))

from(mark(X)) → mark(from(X))

s(mark(X)) → mark(s(X))

proper(2nd(X)) → 2nd(proper(X))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(from(X)) → from(proper(X))

proper(s(X)) → s(proper(X))

2nd(ok(X)) → ok(2nd(X))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

from(ok(X)) → ok(from(X))

s(ok(X)) → ok(s(X))

top(mark(X)) → top(proper(X))

top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

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(2nd(cons(X, cons(Y, Z)))) → mark(Y)

active(from(X)) → mark(cons(X, from(s(X))))

active(2nd(X)) → 2nd(active(X))

active(cons(X1, X2)) → cons(active(X1), X2)

active(from(X)) → from(active(X))

active(s(X)) → s(active(X))

proper(2nd(X)) → 2nd(proper(X))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(from(X)) → from(proper(X))

proper(s(X)) → s(proper(X))

The TRS R consists of the following rules:

top(ok(X)) → top(active(X))

from(ok(X)) → ok(from(X))

from(mark(X)) → mark(from(X))

s(ok(X)) → ok(s(X))

s(mark(X)) → mark(s(X))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

2nd(mark(X)) → mark(2nd(X))

cons(mark(X1), X2) → mark(cons(X1, X2))

2nd(ok(X)) → ok(2nd(X))

top(mark(X)) → top(proper(X))

Rewrite Strategy: FULL

As the TRS does not nest defined symbols, we have rc = irc.

The TRS R consists of the following rules:

top(ok(X)) → top(active(X))

from(ok(X)) → ok(from(X))

from(mark(X)) → mark(from(X))

s(ok(X)) → ok(s(X))

s(mark(X)) → mark(s(X))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

2nd(mark(X)) → mark(2nd(X))

cons(mark(X1), X2) → mark(cons(X1, X2))

2nd(ok(X)) → ok(2nd(X))

top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

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]

transitions:

ok0(0) → 0

active0(0) → 0

mark0(0) → 0

proper0(0) → 0

top0(0) → 1

from0(0) → 2

s0(0) → 3

cons0(0, 0) → 4

2nd0(0) → 5

active1(0) → 6

top1(6) → 1

from1(0) → 7

ok1(7) → 2

from1(0) → 8

mark1(8) → 2

s1(0) → 9

ok1(9) → 3

s1(0) → 10

mark1(10) → 3

cons1(0, 0) → 11

ok1(11) → 4

2nd1(0) → 12

mark1(12) → 5

cons1(0, 0) → 13

mark1(13) → 4

2nd1(0) → 14

ok1(14) → 5

proper1(0) → 15

top1(15) → 1

ok1(7) → 7

ok1(7) → 8

mark1(8) → 7

mark1(8) → 8

ok1(9) → 9

ok1(9) → 10

mark1(10) → 9

mark1(10) → 10

ok1(11) → 11

ok1(11) → 13

mark1(12) → 12

mark1(12) → 14

mark1(13) → 11

mark1(13) → 13

ok1(14) → 12

ok1(14) → 14