* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: and(false(),y) -> false() and(true(),y) -> y eq(apply(t,s),apply(t',s')) -> and(eq(t,t'),eq(s,s')) eq(apply(t,s),lambda(x,t)) -> false() eq(apply(t,s),var(l)) -> false() eq(cons(t,l),cons(t',l')) -> and(eq(t,t'),eq(l,l')) eq(cons(t,l),nil()) -> false() eq(lambda(x,t),apply(t,s)) -> false() eq(lambda(x,t),lambda(x',t')) -> and(eq(x,x'),eq(t,t')) eq(lambda(x,t),var(l)) -> false() eq(nil(),cons(t,l)) -> false() eq(nil(),nil()) -> true() eq(var(l),apply(t,s)) -> false() eq(var(l),lambda(x,t)) -> false() eq(var(l),var(l')) -> eq(l,l') if(false(),var(k),var(l')) -> var(l') if(true(),var(k),var(l')) -> var(k) ren(x,y,apply(t,s)) -> apply(ren(x,y,t),ren(x,y,s)) ren(x,y,lambda(z,t)) -> lambda(var(cons(x,cons(y,cons(lambda(z,t),nil())))) ,ren(x,y,ren(z,var(cons(x,cons(y,cons(lambda(z,t),nil())))),t))) ren(var(l),var(k),var(l')) -> if(eq(l,l'),var(k),var(l')) - Signature: {and/2,eq/2,if/3,ren/3} / {apply/2,cons/2,false/0,lambda/2,nil/0,true/0,var/1} - Obligation: innermost runtime complexity wrt. defined symbols {and,eq,if,ren} and constructors {apply,cons,false,lambda ,nil,true,var} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: and(false(),y) -> false() and(true(),y) -> y eq(apply(t,s),apply(t',s')) -> and(eq(t,t'),eq(s,s')) eq(apply(t,s),lambda(x,t)) -> false() eq(apply(t,s),var(l)) -> false() eq(cons(t,l),cons(t',l')) -> and(eq(t,t'),eq(l,l')) eq(cons(t,l),nil()) -> false() eq(lambda(x,t),apply(t,s)) -> false() eq(lambda(x,t),lambda(x',t')) -> and(eq(x,x'),eq(t,t')) eq(lambda(x,t),var(l)) -> false() eq(nil(),cons(t,l)) -> false() eq(nil(),nil()) -> true() eq(var(l),apply(t,s)) -> false() eq(var(l),lambda(x,t)) -> false() eq(var(l),var(l')) -> eq(l,l') if(false(),var(k),var(l')) -> var(l') if(true(),var(k),var(l')) -> var(k) ren(x,y,apply(t,s)) -> apply(ren(x,y,t),ren(x,y,s)) ren(x,y,lambda(z,t)) -> lambda(var(cons(x,cons(y,cons(lambda(z,t),nil())))) ,ren(x,y,ren(z,var(cons(x,cons(y,cons(lambda(z,t),nil())))),t))) ren(var(l),var(k),var(l')) -> if(eq(l,l'),var(k),var(l')) - Signature: {and/2,eq/2,if/3,ren/3} / {apply/2,cons/2,false/0,lambda/2,nil/0,true/0,var/1} - Obligation: innermost runtime complexity wrt. defined symbols {and,eq,if,ren} and constructors {apply,cons,false,lambda ,nil,true,var} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,z){x -> apply(x,y),z -> apply(z,u)} = eq(apply(x,y),apply(z,u)) ->^+ and(eq(x,z),eq(y,u)) = C[eq(x,z) = eq(x,z){}] WORST_CASE(Omega(n^1),?)