* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if) = [1] x1 + [2] x2 + [1] x3 + [0]
p(le) = [0]
p(minus) = [6] x1 + [1] x2 + [0]
p(p) = [1] x1 + [5]
p(s) = [1] x1 + [0]
p(true) = [0]

Following rules are strictly oriented:
p(0()) = [5]
> [0]
= 0()

p(s(x)) = [1] x + [5]
> [1] x + [0]
= x

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [2] x + [1] y + [0]
>= [1] y + [0]
=  y

if(true(),x,y) =  [2] x + [1] y + [0]
>= [1] x + [0]
=  x

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()) =  [6] x + [0]
>= [1] x + [0]
=  x

minus(x,s(y)) =  [6] x + [1] y + [0]
>= [6] x + [1] y + [10]
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if) = [1] x1 + [2] x2 + [1] x3 + [0]
p(le) = [4]
p(minus) = [1] x1 + [1] x2 + [0]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]

Following rules are strictly oriented:
le(0(),y) = [4]
> [0]
= true()

le(s(x),0()) = [4]
> [0]
= false()

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [2] x + [1] y + [0]
>= [1] y + [0]
=  y

if(true(),x,y) =  [2] x + [1] y + [0]
>= [1] x + [0]
=  x

le(s(x),s(y)) =  [4]
>= [4]
=  le(x,y)

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

minus(x,s(y)) =  [1] x + [1] y + [0]
>= [1] x + [1] y + [4]
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  [0]
>= [0]
=  0()

p(s(x)) =  [1] x + [0]
>= [1] x + [0]
=  x

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if) = [1] x1 + [8] x2 + [1] x3 + [0]
p(le) = [0]
p(minus) = [8] x1 + [1] x2 + [1]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]

Following rules are strictly oriented:
minus(x,0()) = [8] x + [1]
> [1] x + [0]
= x

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [8] x + [1] y + [0]
>= [1] y + [0]
=  y

if(true(),x,y) =  [8] x + [1] y + [0]
>= [1] x + [0]
=  x

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,s(y)) =  [8] x + [1] y + [1]
>= [8] x + [1] y + [1]
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  [0]
>= [0]
=  0()

p(s(x)) =  [1] x + [0]
>= [1] x + [0]
=  x

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(s(x),s(y)) -> le(x,y)
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [1]
p(minus) = [8] x1 + [1] x2 + [0]
p(p) = [1] x1 + [8]
p(s) = [1] x1 + [8]
p(true) = [1]

Following rules are strictly oriented:
if(false(),x,y) = [1] x + [1] y + [1]
> [1] y + [0]
= y

if(true(),x,y) = [1] x + [1] y + [1]
> [1] x + [0]
= x

Following rules are (at-least) weakly oriented:
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()) =  [8] x + [0]
>= [1] x + [0]
=  x

minus(x,s(y)) =  [8] x + [1] y + [8]
>= [8] x + [1] y + [25]
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  [8]
>= [0]
=  0()

p(s(x)) =  [1] x + [16]
>= [1] x + [0]
=  x

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(x),s(y)) -> le(x,y)
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):

The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[0]
[0]
p(if) = [1 0 0]       [2 0 2]       [1 0 0]       [1]
[0 0 0] x_1 + [0 2 1] x_2 + [0 1 0] x_3 + [2]
[1 0 0]       [1 1 2]       [0 0 1]       [0]
p(le) = [0 0 0]       [0 0 0]       [0]
[2 1 0] x_1 + [0 0 2] x_2 + [3]
[1 0 0]       [0 0 2]       [0]
p(minus) = [3 2 0]       [2 0 1]       [3]
[3 2 0] x_1 + [2 1 3] x_2 + [1]
[3 2 1]       [3 0 3]       [0]
p(p) = [1 0 0]       [0]
[1 0 0] x_1 + [2]
[0 1 0]       [0]
p(s) = [1 1 0]       [0]
[0 0 1] x_1 + [0]
[0 0 1]       [2]
p(true) = [0]
[3]
[0]

Following rules are strictly oriented:
minus(x,s(y)) = [3 2 0]     [2 2 1]     [5]
[3 2 0] x + [2 2 4] y + [7]
[3 2 1]     [3 3 3]     [6]
> [3 2 0]     [2 2 1]     [4]
[3 2 0] x + [2 2 1] y + [7]
[3 2 0]     [3 3 3]     [3]
= if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [2 0 2]     [1 0 0]     [1]
[0 2 1] x + [0 1 0] y + [2]
[1 1 2]     [0 0 1]     [0]
>= [1 0 0]     [0]
[0 1 0] y + [0]
[0 0 1]     [0]
=  y

if(true(),x,y) =  [2 0 2]     [1 0 0]     [1]
[0 2 1] x + [0 1 0] y + [2]
[1 1 2]     [0 0 1]     [0]
>= [1 0 0]     [0]
[0 1 0] x + [0]
[0 0 1]     [0]
=  x

le(0(),y) =  [0 0 0]     [0]
[0 0 2] y + [3]
[0 0 2]     [0]
>= [0]
[3]
[0]
=  true()

le(s(x),0()) =  [0 0 0]     [0]
[2 2 1] x + [3]
[1 1 0]     [0]
>= [0]
[0]
[0]
=  false()

le(s(x),s(y)) =  [0 0 0]     [0 0 0]     [0]
[2 2 1] x + [0 0 2] y + [7]
[1 1 0]     [0 0 2]     [4]
>= [0 0 0]     [0 0 0]     [0]
[2 1 0] x + [0 0 2] y + [3]
[1 0 0]     [0 0 2]     [0]
=  le(x,y)

minus(x,0()) =  [3 2 0]     [3]
[3 2 0] x + [1]
[3 2 1]     [0]
>= [1 0 0]     [0]
[0 1 0] x + [0]
[0 0 1]     [0]
=  x

p(0()) =  [0]
[2]
[0]
>= [0]
[0]
[0]
=  0()

p(s(x)) =  [1 1 0]     [0]
[1 1 0] x + [2]
[0 0 1]     [0]
>= [1 0 0]     [0]
[0 1 0] x + [0]
[0 0 1]     [0]
=  x

* Step 6: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(x),s(y)) -> le(x,y)
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):

The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(false) = [0]
[0]
[1]
[0]
p(if) = [1 0 0 0]       [1 0 0 0]       [1 0 0 0]       [0]
[0 0 0 0] x_1 + [0 1 0 0] x_2 + [0 1 0 0] x_3 + [0]
[0 0 0 0]       [0 0 1 0]       [0 0 1 0]       [0]
[0 0 0 0]       [0 0 0 1]       [0 0 0 1]       [0]
p(le) = [0 0 0 0]       [0 0 0 1]       [0]
[1 0 0 1] x_1 + [0 0 0 0] x_2 + [1]
[0 0 0 0]       [0 0 1 1]       [1]
[0 0 1 0]       [0 0 0 1]       [1]
p(minus) = [1 1 0 0]       [1 0 1 0]       [0]
[1 1 1 0] x_1 + [1 0 1 0] x_2 + [1]
[1 1 1 0]       [1 0 1 0]       [0]
[1 1 1 1]       [1 1 0 0]       [0]
p(p) = [1 0 0 0]       [0]
[1 0 0 0] x_1 + [1]
[0 1 0 0]       [0]
[0 0 1 0]       [0]
p(s) = [1 1 0 1]       [1]
[0 0 1 0] x_1 + [0]
[0 0 1 1]       [1]
[0 0 0 1]       [1]
p(true) = [0]
[0]
[0]
[1]

Following rules are strictly oriented:
le(s(x),s(y)) = [0 0 0 0]     [0 0 0 1]     [1]
[1 1 0 2] x + [0 0 0 0] y + [3]
[0 0 0 0]     [0 0 1 2]     [3]
[0 0 1 1]     [0 0 0 1]     [3]
> [0 0 0 0]     [0 0 0 1]     [0]
[1 0 0 1] x + [0 0 0 0] y + [1]
[0 0 0 0]     [0 0 1 1]     [1]
[0 0 1 0]     [0 0 0 1]     [1]
= le(x,y)

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

if(true(),x,y) =  [1 0 0 0]     [1 0 0 0]     [0]
[0 1 0 0] x + [0 1 0 0] y + [0]
[0 0 1 0]     [0 0 1 0]     [0]
[0 0 0 1]     [0 0 0 1]     [0]
>= [1 0 0 0]     [0]
[0 1 0 0] x + [0]
[0 0 1 0]     [0]
[0 0 0 1]     [0]
=  x

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

le(s(x),0()) =  [0 0 0 0]     [0]
[1 1 0 2] x + [3]
[0 0 0 0]     [1]
[0 0 1 1]     [2]
>= [0]
[0]
[1]
[0]
=  false()

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

minus(x,s(y)) =  [1 1 0 0]     [1 1 1 2]     [2]
[1 1 1 0] x + [1 1 1 2] y + [3]
[1 1 1 0]     [1 1 1 2]     [2]
[1 1 1 1]     [1 1 1 1]     [1]
>= [1 1 0 0]     [1 1 1 2]     [2]
[1 1 0 0] x + [1 1 1 1] y + [2]
[1 1 1 0]     [1 1 1 1]     [2]
[1 1 1 0]     [1 1 1 1]     [1]
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  [0]
[1]
[0]
[0]
>= [0]
[0]
[0]
[0]
=  0()

p(s(x)) =  [1 1 0 1]     [1]
[1 1 0 1] x + [2]
[0 0 1 0]     [0]
[0 0 1 1]     [1]
>= [1 0 0 0]     [0]
[0 1 0 0] x + [0]
[0 0 1 0]     [0]
[0 0 0 1]     [0]
=  x

* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} 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^3))