* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
f(a(),a()) -> f(a(),b())
f(a(),b()) -> f(s(a()),c())
f(c(),c()) -> f(a(),a())
f(s(X),c()) -> f(X,c())
- Signature:
{f/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {a,b,c,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f(a(),a()) -> f(a(),b())
f(a(),b()) -> f(s(a()),c())
f(c(),c()) -> f(a(),a())
f(s(X),c()) -> f(X,c())
- Signature:
{f/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {a,b,c,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
f(x,c()){x -> s(x)} =
f(s(x),c()) ->^+ f(x,c())
= C[f(x,c()) = f(x,c()){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(a(),a()) -> f(a(),b())
f(a(),b()) -> f(s(a()),c())
f(c(),c()) -> f(a(),a())
f(s(X),c()) -> f(X,c())
- Signature:
{f/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {a,b,c,s}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 4.
The enriched problem is compatible with follwoing automaton.
a_0() -> 1
a_1() -> 6
a_2() -> 10
a_3() -> 13
b_0() -> 2
b_1() -> 7
b_2() -> 12
c_0() -> 3
c_1() -> 7
c_2() -> 9
c_3() -> 11
c_4() -> 14
f_0(1,1) -> 4
f_0(1,2) -> 4
f_0(1,3) -> 4
f_0(1,5) -> 4
f_0(2,1) -> 4
f_0(2,2) -> 4
f_0(2,3) -> 4
f_0(2,5) -> 4
f_0(3,1) -> 4
f_0(3,2) -> 4
f_0(3,3) -> 4
f_0(3,5) -> 4
f_0(5,1) -> 4
f_0(5,2) -> 4
f_0(5,3) -> 4
f_0(5,5) -> 4
f_1(1,7) -> 4
f_1(2,7) -> 4
f_1(3,7) -> 4
f_1(5,7) -> 4
f_1(6,6) -> 4
f_1(6,7) -> 4
f_2(6,9) -> 4
f_2(8,9) -> 4
f_2(10,12) -> 4
f_3(10,11) -> 4
f_4(13,14) -> 4
s_0(1) -> 5
s_0(2) -> 5
s_0(3) -> 5
s_0(5) -> 5
s_1(6) -> 6
s_2(10) -> 8
s_3(13) -> 10
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(a(),a()) -> f(a(),b())
f(a(),b()) -> f(s(a()),c())
f(c(),c()) -> f(a(),a())
f(s(X),c()) -> f(X,c())
- Signature:
{f/2} / {a/0,b/0,c/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {a,b,c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))