* 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),?)