* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U21(mark(X1),X2,X3) -> mark(U21(X1,X2,X3))
U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3))
U31(mark(X)) -> mark(U31(X))
U31(ok(X)) -> ok(U31(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),N)) -> mark(N)
active(U21(X1,X2,X3)) -> U21(active(X1),X2,X3)
active(U21(tt(),M,N)) -> mark(s(plus(N,M)))
active(U31(X)) -> U31(active(X))
active(U31(tt())) -> mark(0())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),M,N)) -> mark(plus(x(N,M),N))
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
active(plus(N,0())) -> mark(U11(isNat(N),N))
active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
active(x(N,0())) -> mark(U31(isNat(N)))
active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
proper(0()) -> ok(0())
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
proper(x(X1,X2)) -> x(proper(X1),proper(X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U11/2,U21/3,U31/1,U41/3,active/1,and/2,isNat/1,plus/2,proper/1,s/1,top/1,x/2} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U21,U31,U41,active,and,isNat,plus,proper,s,top
,x} and constructors {0,mark,ok,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U21(mark(X1),X2,X3) -> mark(U21(X1,X2,X3))
U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3))
U31(mark(X)) -> mark(U31(X))
U31(ok(X)) -> ok(U31(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),N)) -> mark(N)
active(U21(X1,X2,X3)) -> U21(active(X1),X2,X3)
active(U21(tt(),M,N)) -> mark(s(plus(N,M)))
active(U31(X)) -> U31(active(X))
active(U31(tt())) -> mark(0())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),M,N)) -> mark(plus(x(N,M),N))
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isNat(0())) -> mark(tt())
active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
active(plus(N,0())) -> mark(U11(isNat(N),N))
active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
active(x(N,0())) -> mark(U31(isNat(N)))
active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N))
active(x(X1,X2)) -> x(X1,active(X2))
active(x(X1,X2)) -> x(active(X1),X2)
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
proper(0()) -> ok(0())
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
proper(x(X1,X2)) -> x(proper(X1),proper(X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U11/2,U21/3,U31/1,U41/3,active/1,and/2,isNat/1,plus/2,proper/1,s/1,top/1,x/2} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U21,U31,U41,active,and,isNat,plus,proper,s,top
,x} and constructors {0,mark,ok,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
U11(x,y){x -> mark(x)} =
U11(mark(x),y) ->^+ mark(U11(x,y))
= C[U11(x,y) = U11(x,y){}]
WORST_CASE(Omega(n^1),?)