* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,g(y,z)) -> g(f(x,y),z)
f(x,nil()) -> g(nil(),x)
norm(g(x,y)) -> s(norm(x))
norm(nil()) -> 0()
rem(g(x,y),0()) -> g(x,y)
rem(g(x,y),s(z)) -> rem(x,z)
rem(nil(),y) -> nil()
- Signature:
{f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,g(y,z)) -> g(f(x,y),z)
f(x,nil()) -> g(nil(),x)
norm(g(x,y)) -> s(norm(x))
norm(nil()) -> 0()
rem(g(x,y),0()) -> g(x,y)
rem(g(x,y),s(z)) -> rem(x,z)
rem(nil(),y) -> nil()
- Signature:
{f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,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_0() -> 2
0_1() -> 1
0_1() -> 4
f_0(2,2) -> 1
f_1(2,2) -> 3
g_0(2,2) -> 2
g_1(2,2) -> 1
g_1(3,2) -> 1
g_1(3,2) -> 3
nil_0() -> 2
nil_1() -> 1
nil_1() -> 3
norm_0(2) -> 1
norm_1(2) -> 4
rem_0(2,2) -> 1
rem_1(2,2) -> 1
s_0(2) -> 2
s_1(4) -> 1
s_1(4) -> 4
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,g(y,z)) -> g(f(x,y),z)
f(x,nil()) -> g(nil(),x)
norm(g(x,y)) -> s(norm(x))
norm(nil()) -> 0()
rem(g(x,y),0()) -> g(x,y)
rem(g(x,y),s(z)) -> rem(x,z)
rem(nil(),y) -> nil()
- Signature:
{f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))