* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
and(x,false()) -> false()
and(true(),true()) -> true()
f(0()) -> true()
f(s(x)) -> h(x)
g(s(x),s(y)) -> if(and(f(s(x)),f(s(y)))
,t(g(k(minus(m(x,y),n(x,y)),s(s(0()))),k(n(s(x),s(y)),s(s(0())))))
,g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
gt(0(),y) -> false()
gt(s(x),0()) -> true()
gt(s(x),s(y)) -> gt(x,y)
h(0()) -> false()
h(s(x)) -> f(x)
id(x) -> x
if(false(),x,y) -> y
if(true(),x,y) -> x
k(0(),s(y)) -> 0()
k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
m(x,0()) -> x
m(0(),y) -> y
m(s(x),s(y)) -> s(m(x,y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
n(x,0()) -> 0()
n(0(),y) -> 0()
n(s(x),s(y)) -> s(n(x,y))
not(x) -> if(x,false(),true())
p(0(),y) -> y
p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
t(x) -> p(x,x)
- Signature:
{and/2,f/1,g/2,gt/2,h/1,id/1,if/3,k/2,m/2,minus/2,n/2,not/1,p/2,t/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,f,g,gt,h,id,if,k,m,minus,n,not,p
,t} and constructors {0,false,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
and(x,false()) -> false()
and(true(),true()) -> true()
f(0()) -> true()
f(s(x)) -> h(x)
g(s(x),s(y)) -> if(and(f(s(x)),f(s(y)))
,t(g(k(minus(m(x,y),n(x,y)),s(s(0()))),k(n(s(x),s(y)),s(s(0())))))
,g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
gt(0(),y) -> false()
gt(s(x),0()) -> true()
gt(s(x),s(y)) -> gt(x,y)
h(0()) -> false()
h(s(x)) -> f(x)
id(x) -> x
if(false(),x,y) -> y
if(true(),x,y) -> x
k(0(),s(y)) -> 0()
k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
m(x,0()) -> x
m(0(),y) -> y
m(s(x),s(y)) -> s(m(x,y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
n(x,0()) -> 0()
n(0(),y) -> 0()
n(s(x),s(y)) -> s(n(x,y))
not(x) -> if(x,false(),true())
p(0(),y) -> y
p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
t(x) -> p(x,x)
- Signature:
{and/2,f/1,g/2,gt/2,h/1,id/1,if/3,k/2,m/2,minus/2,n/2,not/1,p/2,t/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,f,g,gt,h,id,if,k,m,minus,n,not,p
,t} and constructors {0,false,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
gt(x,y){x -> s(x),y -> s(y)} =
gt(s(x),s(y)) ->^+ gt(x,y)
= C[gt(x,y) = gt(x,y){}]
WORST_CASE(Omega(n^1),?)