* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
and(false(),false()) -> false()
and(false(),true()) -> false()
and(true(),false()) -> false()
and(true(),true()) -> true()
eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
eq(apply(T,S),lambda(X,Tp)) -> false()
eq(apply(T,S),var(L)) -> false()
eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
eq(cons(T,L),nil()) -> false()
eq(lambda(X,T),apply(Tp,Sp)) -> false()
eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
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(Lp)) -> eq(L,Lp)
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(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
- 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(),false()) -> false()
and(false(),true()) -> false()
and(true(),false()) -> false()
and(true(),true()) -> true()
eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
eq(apply(T,S),lambda(X,Tp)) -> false()
eq(apply(T,S),var(L)) -> false()
eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
eq(cons(T,L),nil()) -> false()
eq(lambda(X,T),apply(Tp,Sp)) -> false()
eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
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(Lp)) -> eq(L,Lp)
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(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
- 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),?)