```* Step 1: MI 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:
runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

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) = [9]
p(false) = [1]
p(gcd) = [1] x_1 + [1] x_2 + [13]
p(if_gcd) = [4] x_1 + [1] x_2 + [1] x_3 + [8]
p(le) = [1]
p(minus) = [1] x_1 + [0]
p(s) = [1] x_1 + [1]
p(true) = [1]

Following rules are strictly oriented:
gcd(0(),y) = [1] y + [22]
> [1] y + [0]
= y

gcd(s(x),0()) = [1] x + [23]
> [1] x + [1]
= s(x)

gcd(s(x),s(y)) = [1] x + [1] y + [15]
> [1] x + [1] y + [14]
= if_gcd(le(y,x),s(x),s(y))

minus(s(x),s(y)) = [1] x + [1]
> [1] x + [0]
= minus(x,y)

Following rules are (at-least) weakly oriented:
if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [14]
>= [1] x + [1] y + [14]
=  gcd(minus(y,x),s(x))

if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [14]
>= [1] x + [1] y + [14]
=  gcd(minus(x,y),s(y))

le(0(),y) =  [1]
>= [1]
=  true()

le(s(x),0()) =  [1]
>= [1]
=  false()

le(s(x),s(y)) =  [1]
>= [1]
=  le(x,y)

minus(x,0()) =  [1] x + [0]
>= [1] x + [0]
=  x

* Step 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)
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))
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:
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) = [6]
p(false) = [1]
p(gcd) = [1] x1 + [1] x2 + [7]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [1]
p(minus) = [1] x1 + [2]
p(s) = [1] x1 + [1]
p(true) = [4]

Following rules are strictly oriented:
minus(x,0()) = [1] x + [2]
> [1] x + [0]
= x

Following rules are (at-least) weakly oriented:
gcd(0(),y) =  [1] y + [13]
>= [1] y + [0]
=  y

gcd(s(x),0()) =  [1] x + [14]
>= [1] x + [1]
=  s(x)

gcd(s(x),s(y)) =  [1] x + [1] y + [9]
>= [1] x + [1] y + [3]
=  if_gcd(le(y,x),s(x),s(y))

if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [3]
>= [1] x + [1] y + [10]
=  gcd(minus(y,x),s(x))

if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [6]
>= [1] x + [1] y + [10]
=  gcd(minus(x,y),s(y))

le(0(),y) =  [1]
>= [4]
=  true()

le(s(x),0()) =  [1]
>= [1]
=  false()

le(s(x),s(y)) =  [1]
>= [1]
=  le(x,y)

minus(s(x),s(y)) =  [1] x + [3]
>= [1] x + [2]
=  minus(x,y)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: 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:
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 4: 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:
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 5: 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:
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 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:
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:
all
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(gcd) = x1 + 4*x1*x2 + 3*x1^2 + 4*x2 + 3*x2^2
p(if_gcd) = 3 + 4*x1 + 4*x2*x3 + 3*x2^2 + 3*x3^2
p(le) = x1
p(minus) = x1
p(s) = 1 + x1
p(true) = 0

Following rules are strictly oriented:
le(s(x),s(y)) = 1 + x
> x
= le(x,y)

Following rules are (at-least) weakly oriented:
gcd(0(),y) =  4*y + 3*y^2
>= y
=  y

gcd(s(x),0()) =  4 + 7*x + 3*x^2
>= 1 + x
=  s(x)

gcd(s(x),s(y)) =  15 + 11*x + 4*x*y + 3*x^2 + 14*y + 3*y^2
>= 13 + 10*x + 4*x*y + 3*x^2 + 14*y + 3*y^2
=  if_gcd(le(y,x),s(x),s(y))

if_gcd(false(),s(x),s(y)) =  13 + 10*x + 4*x*y + 3*x^2 + 10*y + 3*y^2
>= 7 + 10*x + 4*x*y + 3*x^2 + 5*y + 3*y^2
=  gcd(minus(y,x),s(x))

if_gcd(true(),s(x),s(y)) =  13 + 10*x + 4*x*y + 3*x^2 + 10*y + 3*y^2
>= 7 + 5*x + 4*x*y + 3*x^2 + 10*y + 3*y^2
=  gcd(minus(x,y),s(y))

le(0(),y) =  0
>= 0
=  true()

le(s(x),0()) =  1 + x
>= 0
=  false()

minus(x,0()) =  x
>= x
=  x

minus(s(x),s(y)) =  1 + x
>= x
=  minus(x,y)

* Step 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:
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(?,O(n^2))
```