* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
div2(0()) -> 0()
div2(S(0())) -> 0()
div2(S(S(x))) -> +(S(0()),div2(x))
- Weak TRS:
+(x,S(0())) -> S(x)
+(S(0()),y) -> S(y)
- Signature:
{+/2,div2/1} / {0/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,div2} and constructors {0,S}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
div2(0()) -> 0()
div2(S(0())) -> 0()
div2(S(S(x))) -> +(S(0()),div2(x))
- Weak TRS:
+(x,S(0())) -> S(x)
+(S(0()),y) -> S(y)
- Signature:
{+/2,div2/1} / {0/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,div2} and constructors {0,S}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
div2(x){x -> S(S(x))} =
div2(S(S(x))) ->^+ +(S(0()),div2(x))
= C[div2(x) = div2(x){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div2(0()) -> 0()
div2(S(0())) -> 0()
div2(S(S(x))) -> +(S(0()),div2(x))
- Weak TRS:
+(x,S(0())) -> S(x)
+(S(0()),y) -> S(y)
- Signature:
{+/2,div2/1} / {0/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,div2} and constructors {0,S}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 1.
The enriched problem is compatible with follwoing automaton.
+_0(2,2) -> 1
+_1(3,4) -> 1
+_1(3,4) -> 4
0_0() -> 2
0_1() -> 1
0_1() -> 4
S_0(2) -> 1
S_0(2) -> 2
S_1(1) -> 3
S_1(3) -> 1
S_1(3) -> 4
S_1(4) -> 1
S_1(4) -> 4
div2_0(2) -> 1
div2_1(2) -> 4
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,S(0())) -> S(x)
+(S(0()),y) -> S(y)
div2(0()) -> 0()
div2(S(0())) -> 0()
div2(S(S(x))) -> +(S(0()),div2(x))
- Signature:
{+/2,div2/1} / {0/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,div2} and constructors {0,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))