WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ 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:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
le(x,y){x -> s(x),y -> s(y)} =
le(s(x),s(y)) ->^+ le(x,y)
= C[le(x,y) = le(x,y){}]
** Step 1.b:1: NaturalPI 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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = 2 + x1 + x2
p(if_gcd) = 2 + 4*x1 + x2 + x3
p(le) = 0
p(minus) = x1
p(s) = x1
p(true) = 0
Following rules are strictly oriented:
gcd(0(),y) = 2 + y
> y
= y
gcd(s(x),0()) = 2 + x
> x
= s(x)
Following rules are (at-least) weakly oriented:
gcd(s(x),s(y)) = 2 + x + y
>= 2 + x + y
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 2 + x + y
>= 2 + x + y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 2 + x + y
>= 2 + x + y
= gcd(minus(x,y),s(y))
le(0(),y) = 0
>= 0
= true()
le(s(x),0()) = 0
>= 0
= false()
le(s(x),s(y)) = 0
>= 0
= le(x,y)
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = x
>= x
= minus(x,y)
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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)
- Weak TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
- 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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = 4 + 4*x1 + 4*x2
p(if_gcd) = 4 + 4*x1 + 4*x2 + 4*x3
p(le) = 0
p(minus) = x1
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
if_gcd(false(),s(x),s(y)) = 12 + 4*x + 4*y
> 8 + 4*x + 4*y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 12 + 4*x + 4*y
> 8 + 4*x + 4*y
= gcd(minus(x,y),s(y))
minus(s(x),s(y)) = 1 + x
> x
= minus(x,y)
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 4 + 4*y
>= y
= y
gcd(s(x),0()) = 8 + 4*x
>= 1 + x
= s(x)
gcd(s(x),s(y)) = 12 + 4*x + 4*y
>= 12 + 4*x + 4*y
= if_gcd(le(y,x),s(x),s(y))
le(0(),y) = 0
>= 0
= true()
le(s(x),0()) = 0
>= 0
= false()
le(s(x),s(y)) = 0
>= 0
= le(x,y)
minus(x,0()) = x
>= x
= x
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
- Weak TRS:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
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))
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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 3
p(false) = 2
p(gcd) = 5 + x1 + x2
p(if_gcd) = 2*x1 + x2 + x3
p(le) = 2
p(minus) = x1
p(s) = 4 + x1
p(true) = 1
Following rules are strictly oriented:
gcd(s(x),s(y)) = 13 + x + y
> 12 + x + y
= if_gcd(le(y,x),s(x),s(y))
le(0(),y) = 2
> 1
= true()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 8 + y
>= y
= y
gcd(s(x),0()) = 12 + x
>= 4 + x
= s(x)
if_gcd(false(),s(x),s(y)) = 12 + x + y
>= 9 + x + y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 10 + x + y
>= 9 + x + y
= gcd(minus(x,y),s(y))
le(s(x),0()) = 2
>= 2
= false()
le(s(x),s(y)) = 2
>= 2
= le(x,y)
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 4 + x
>= x
= minus(x,y)
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
- Weak 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()
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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 2
p(gcd) = 5 + 4*x1 + 4*x2
p(if_gcd) = 1 + 2*x1 + 4*x2 + 4*x3
p(le) = 2
p(minus) = 1 + x1
p(s) = 1 + x1
p(true) = 2
Following rules are strictly oriented:
minus(x,0()) = 1 + x
> x
= x
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 5 + 4*y
>= y
= y
gcd(s(x),0()) = 9 + 4*x
>= 1 + x
= s(x)
gcd(s(x),s(y)) = 13 + 4*x + 4*y
>= 13 + 4*x + 4*y
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 13 + 4*x + 4*y
>= 13 + 4*x + 4*y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 13 + 4*x + 4*y
>= 13 + 4*x + 4*y
= gcd(minus(x,y),s(y))
le(0(),y) = 2
>= 2
= true()
le(s(x),0()) = 2
>= 2
= false()
le(s(x),s(y)) = 2
>= 2
= le(x,y)
minus(s(x),s(y)) = 2 + x
>= 1 + x
= minus(x,y)
** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
- Weak 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()
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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = 7 + x1 + x2
p(if_gcd) = 4 + x1 + x2 + x3
p(le) = 2
p(minus) = x1
p(s) = 4 + x1
p(true) = 0
Following rules are strictly oriented:
le(s(x),0()) = 2
> 0
= false()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 7 + y
>= y
= y
gcd(s(x),0()) = 11 + x
>= 4 + x
= s(x)
gcd(s(x),s(y)) = 15 + x + y
>= 14 + x + y
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 12 + x + y
>= 11 + x + y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 12 + x + y
>= 11 + x + y
= gcd(minus(x,y),s(y))
le(0(),y) = 2
>= 0
= true()
le(s(x),s(y)) = 2
>= 2
= le(x,y)
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 4 + x
>= x
= minus(x,y)
** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
le(s(x),s(y)) -> le(x,y)
- Weak 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()
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:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = 1
p(false) = 0
p(gcd) = 4*x1 + 5*x1*x2 + 2*x1^2 + 2*x2^2
p(if_gcd) = 1 + 4*x1 + 5*x2*x3 + 2*x2^2 + 2*x3^2
p(le) = x2
p(minus) = x1
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
le(s(x),s(y)) = 1 + y
> y
= le(x,y)
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 6 + 5*y + 2*y^2
>= y
= y
gcd(s(x),0()) = 13 + 13*x + 2*x^2
>= 1 + x
= s(x)
gcd(s(x),s(y)) = 13 + 13*x + 5*x*y + 2*x^2 + 9*y + 2*y^2
>= 10 + 13*x + 5*x*y + 2*x^2 + 9*y + 2*y^2
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 10 + 9*x + 5*x*y + 2*x^2 + 9*y + 2*y^2
>= 2 + 4*x + 5*x*y + 2*x^2 + 9*y + 2*y^2
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 10 + 9*x + 5*x*y + 2*x^2 + 9*y + 2*y^2
>= 2 + 9*x + 5*x*y + 2*x^2 + 4*y + 2*y^2
= gcd(minus(x,y),s(y))
le(0(),y) = y
>= 0
= true()
le(s(x),0()) = 1
>= 0
= false()
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 1 + x
>= x
= minus(x,y)
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak 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:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))