(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:
minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(X) → X
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of minus: s, 0, activate
The following defined symbols can occur below the 1th argument of minus: s, 0, activate
The following defined symbols can occur below the 0th argument of activate: s, 0, activate
The following defined symbols can occur below the 0th argument of geq: s, 0, activate
The following defined symbols can occur below the 1th argument of geq: s, 0, activate
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
(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:
minus(n__0, Y) → 0
activate(n__0) → 0
geq(X, n__0) → true
if(false, X, Y) → activate(Y)
0 → n__0
if(true, X, Y) → activate(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(X)
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(n__0, n__s(Y)) → false
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:
minus(n__0, Y) → 0
activate(n__0) → 0
geq(X, n__0) → true
if(false, X, Y) → activate(Y)
0 → n__0
if(true, X, Y) → activate(X)
s(X) → n__s(X)
activate(X) → X
activate(n__s(X)) → s(X)
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(n__0, n__s(Y)) → false
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 4.
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]
transitions:
n__00() → 0
true0() → 0
false0() → 0
n__s0(0) → 0
minus0(0, 0) → 1
activate0(0) → 2
geq0(0, 0) → 3
if0(0, 0, 0) → 4
00() → 5
s0(0) → 6
01() → 1
01() → 2
true1() → 3
activate1(0) → 4
n__01() → 5
n__s1(0) → 6
s1(0) → 2
activate1(0) → 7
activate1(0) → 8
geq1(7, 8) → 3
activate1(0) → 9
activate1(0) → 10
minus1(9, 10) → 1
false1() → 3
01() → 4
01() → 7
01() → 8
01() → 9
01() → 10
n__02() → 1
n__02() → 2
n__s2(0) → 2
s1(0) → 4
s1(0) → 7
s1(0) → 8
s1(0) → 9
s1(0) → 10
n__02() → 4
n__02() → 7
n__02() → 8
n__02() → 9
n__02() → 10
n__s2(0) → 4
n__s2(0) → 7
n__s2(0) → 8
n__s2(0) → 9
n__s2(0) → 10
02() → 1
true2() → 3
activate2(0) → 11
activate2(0) → 12
geq2(11, 12) → 3
activate2(0) → 13
activate2(0) → 14
minus2(13, 14) → 1
false2() → 3
01() → 11
01() → 12
01() → 13
01() → 14
n__03() → 1
s1(0) → 11
s1(0) → 12
s1(0) → 13
s1(0) → 14
n__02() → 11
n__02() → 12
n__02() → 13
n__02() → 14
n__s2(0) → 11
n__s2(0) → 12
n__s2(0) → 13
n__s2(0) → 14
03() → 1
true3() → 3
activate3(0) → 15
activate3(0) → 16
geq3(15, 16) → 3
activate3(0) → 17
activate3(0) → 18
minus3(17, 18) → 1
false3() → 3
01() → 15
01() → 16
01() → 17
01() → 18
n__04() → 1
s1(0) → 15
s1(0) → 16
s1(0) → 17
s1(0) → 18
n__02() → 15
n__02() → 16
n__02() → 17
n__02() → 18
n__s2(0) → 15
n__s2(0) → 16
n__s2(0) → 17
n__s2(0) → 18
0 → 2
0 → 4
0 → 7
0 → 8
0 → 9
0 → 10
0 → 11
0 → 12
0 → 13
0 → 14
0 → 15
0 → 16
0 → 17
0 → 18
(6) BOUNDS(1, n^1)