* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
gt(0(),y) -> false()
gt(s(x),0()) -> true()
gt(s(x),s(y)) -> gt(x,y)
if(false(),x,y) -> 0()
if(true(),x,y) -> s(minus(x,y))
if1(false(),x,y) -> mod(minus(x,y),y)
if1(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if(gt(s(x),y),x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
- Signature:
{gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} 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:
gt(0(),y) -> false()
gt(s(x),0()) -> true()
gt(s(x),s(y)) -> gt(x,y)
if(false(),x,y) -> 0()
if(true(),x,y) -> s(minus(x,y))
if1(false(),x,y) -> mod(minus(x,y),y)
if1(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if(gt(s(x),y),x,y)
mod(x,0()) -> 0()
mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
- Signature:
{gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} 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),?)