* 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: WeightGap 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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [0]
p(gcd) = [1] x1 + [1] x2 + [7]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [0]
p(minus) = [1] x1 + [3]
p(s) = [1] x1 + [2]
p(true) = [0]
Following rules are strictly oriented:
gcd(0(),y) = [1] y + [9]
> [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [11]
> [1] x + [2]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [11]
> [1] x + [1] y + [4]
= if_gcd(le(y,x),s(x),s(y))
minus(x,0()) = [1] x + [3]
> [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [5]
> [1] x + [3]
= minus(x,y)
Following rules are (at-least) weakly oriented:
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [4]
>= [1] x + [1] y + [12]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [4]
>= [1] x + [1] y + [12]
= 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)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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)
- 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))
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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [4]
p(gcd) = [1] x1 + [1] x2 + [2]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [1]
p(minus) = [1] x1 + [4]
p(s) = [1] x1 + [4]
p(true) = [4]
Following rules are strictly oriented:
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [12]
> [1] x + [1] y + [10]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [12]
> [1] x + [1] y + [10]
= gcd(minus(x,y),s(y))
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [2]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [6]
>= [1] x + [4]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [10]
>= [1] x + [1] y + [9]
= if_gcd(le(y,x),s(x),s(y))
le(0(),y) = [1]
>= [4]
= true()
le(s(x),0()) = [1]
>= [4]
= false()
le(s(x),s(y)) = [1]
>= [1]
= le(x,y)
minus(x,0()) = [1] x + [4]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [8]
>= [1] x + [4]
= minus(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
le(0(),y) -> true()
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))
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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(false) = [7]
p(gcd) = [1] x1 + [1] x2 + [4]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [4]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [3]
Following rules are strictly oriented:
le(0(),y) = [4]
> [3]
= true()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [8]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [9]
>= [1] x + [1]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [6]
>= [1] x + [1] y + [6]
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [9]
>= [1] x + [1] y + [5]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [5]
= gcd(minus(x,y),s(y))
le(s(x),0()) = [4]
>= [7]
= false()
le(s(x),s(y)) = [4]
>= [4]
= le(x,y)
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [1]
>= [1] x + [0]
= minus(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap 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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(false) = [4]
p(gcd) = [1] x1 + [1] x2 + [6]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [1]
p(le) = [5]
p(minus) = [1] x1 + [3]
p(s) = [1] x1 + [4]
p(true) = [4]
Following rules are strictly oriented:
le(s(x),0()) = [5]
> [4]
= false()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [11]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [15]
>= [1] x + [4]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [14]
>= [1] x + [1] y + [14]
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [13]
>= [1] x + [1] y + [13]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [13]
>= [1] x + [1] y + [13]
= gcd(minus(x,y),s(y))
le(0(),y) = [5]
>= [4]
= true()
le(s(x),s(y)) = [5]
>= [5]
= le(x,y)
minus(x,0()) = [1] x + [3]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [7]
>= [1] x + [3]
= minus(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: 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) = 0
p(false) = 0
p(gcd) = 3 + 4*x1 + 2*x1*x2 + 6*x2
p(if_gcd) = 5 + x1 + 4*x2 + 2*x2*x3 + 4*x3
p(le) = 2*x1
p(minus) = x1
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
le(s(x),s(y)) = 2 + 2*x
> 2*x
= le(x,y)
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 3 + 6*y
>= y
= y
gcd(s(x),0()) = 7 + 4*x
>= 1 + x
= s(x)
gcd(s(x),s(y)) = 15 + 6*x + 2*x*y + 8*y
>= 15 + 6*x + 2*x*y + 8*y
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 15 + 6*x + 2*x*y + 6*y
>= 9 + 6*x + 2*x*y + 6*y
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 15 + 6*x + 2*x*y + 6*y
>= 9 + 6*x + 2*x*y + 6*y
= gcd(minus(x,y),s(y))
le(0(),y) = 0
>= 0
= true()
le(s(x),0()) = 2 + 2*x
>= 0
= false()
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 1 + x
>= x
= minus(x,y)
** Step 1.b:6: 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))