* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
gt(s(x),s(y)) -> gt(x,y)
gt(s(x),zero()) -> true()
gt(zero(),y) -> false()
id(x) -> x
if(false(),x,y) -> y
if(true(),x,y) -> x
minus(x,0()) -> x
minus(minus(x,y),z) -> minus(x,plus(y,z))
minus(s(x),s(y)) -> minus(x,y)
not(x) -> if(x,false(),true())
plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
plus(zero(),y) -> y
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2} / {0/0,false/0,s/1,true/0,zero/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gt,id,if,minus,not,plus,quot} and constructors {0,false,s
,true,zero}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
gt(s(x),s(y)) -> gt(x,y)
gt(s(x),zero()) -> true()
gt(zero(),y) -> false()
id(x) -> x
if(false(),x,y) -> y
if(true(),x,y) -> x
minus(x,0()) -> x
minus(minus(x,y),z) -> minus(x,plus(y,z))
minus(s(x),s(y)) -> minus(x,y)
not(x) -> if(x,false(),true())
plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
plus(zero(),y) -> y
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2} / {0/0,false/0,s/1,true/0,zero/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gt,id,if,minus,not,plus,quot} and constructors {0,false,s
,true,zero}
+ 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),?)