* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
f(a(x),a(y)) -> a(f(x,y))
f(b(x),b(y)) -> b(f(x,y))
- Signature:
{a/1,f/2} / {b/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,f} and constructors {b}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
f(a(x),a(y)) -> a(f(x,y))
f(b(x),b(y)) -> b(f(x,y))
- Signature:
{a/1,f/2} / {b/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,f} and constructors {b}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
f(x,y){x -> b(x),y -> b(y)} =
f(b(x),b(y)) ->^+ b(f(x,y))
= C[f(x,y) = f(x,y){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
f(a(x),a(y)) -> a(f(x,y))
f(b(x),b(y)) -> b(f(x,y))
- Signature:
{a/1,f/2} / {b/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,f} and constructors {b}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 1.
The enriched problem is compatible with follwoing automaton.
a_0(2) -> 1
b_0(2) -> 2
b_1(3) -> 1
b_1(3) -> 3
f_0(2,2) -> 1
f_1(2,2) -> 3
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
f(a(x),a(y)) -> a(f(x,y))
f(b(x),b(y)) -> b(f(x,y))
- Signature:
{a/1,f/2} / {b/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,f} and constructors {b}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))