WORST_CASE(?,O(n^2))
* Step 1: Ara WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
+ Applied Processor:
Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3}
+ Details:
Signatures used:
----------------
0 :: [] -(0)-> A(4, 4)
0 :: [] -(0)-> A(0, 4)
0 :: [] -(0)-> A(0, 0)
0 :: [] -(0)-> A(2, 0)
false :: [] -(0)-> A(9, 9)
false :: [] -(0)-> A(15, 15)
gcd :: [A(4, 4) x A(0, 4)] -(1)-> A(0, 0)
if_gcd :: [A(9, 9) x A(1, 4) x A(0, 4)] -(2)-> A(0, 0)
le :: [A(0, 0) x A(2, 0)] -(1)-> A(12, 12)
minus :: [A(4, 4) x A(0, 0)] -(1)-> A(4, 4)
s :: [A(8, 4)] -(4)-> A(4, 4)
s :: [A(4, 4)] -(0)-> A(0, 4)
s :: [A(5, 4)] -(1)-> A(1, 4)
s :: [A(0, 0)] -(0)-> A(0, 0)
s :: [A(2, 0)] -(2)-> A(2, 0)
s :: [A(2, 1)] -(1)-> A(1, 1)
true :: [] -(0)-> A(9, 9)
true :: [] -(0)-> A(15, 15)
Cost-free Signatures used:
--------------------------
0 :: [] -(0)-> A_cf(0, 0)
false :: [] -(0)-> A_cf(1, 1)
false :: [] -(0)-> A_cf(5, 3)
false :: [] -(0)-> A_cf(3, 15)
false :: [] -(0)-> A_cf(0, 0)
gcd :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
if_gcd :: [A_cf(1, 1) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
if_gcd :: [A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
le :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 3)
le :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 13)
le :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
le :: [A_cf(0, 0) x A_cf(0, 0)] -(1)-> A_cf(0, 0)
minus :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
minus :: [A_cf(4, 0) x A_cf(0, 0)] -(1)-> A_cf(0, 0)
s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
s :: [A_cf(4, 0)] -(4)-> A_cf(4, 0)
true :: [] -(0)-> A_cf(1, 1)
true :: [] -(0)-> A_cf(3, 3)
true :: [] -(0)-> A_cf(3, 15)
true :: [] -(0)-> A_cf(0, 0)
Base Constructor Signatures used:
---------------------------------
0_A :: [] -(0)-> A(1, 0)
0_A :: [] -(0)-> A(0, 1)
false_A :: [] -(0)-> A(1, 0)
false_A :: [] -(0)-> A(0, 1)
s_A :: [A(1, 0)] -(1)-> A(1, 0)
s_A :: [A(1, 1)] -(0)-> A(0, 1)
true_A :: [] -(0)-> A(1, 0)
true_A :: [] -(0)-> A(0, 1)
* Step 2: Open MAYBE
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
Following problems could not be solved:
- Strict TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
WORST_CASE(?,O(n^2))