* Step 1: Sum WORST_CASE(Omega(n^1),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:
innermost runtime complexity wrt. defined symbols {if,le,minus,p} 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:
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:
innermost runtime complexity wrt. defined symbols {if,le,minus,p} 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^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:
innermost 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 1.b: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:
innermost 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 1.b: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:
innermost 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 1.b: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:
innermost 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 1.b: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:
innermost 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:
{if,le,minus,p}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[1]
[1]
p(if) = [2 0 0] [2 0 0] [1 0 0] [0]
[1 1 0] x_1 + [0 1 0] x_2 + [0 1 0] x_3 + [0]
[0 0 1] [1 0 2] [0 0 1] [0]
p(le) = [0 0 0] [0]
[0 0 0] x_1 + [1]
[2 2 0] [1]
p(minus) = [1 0 0] [2 0 1] [0]
[1 1 0] x_1 + [2 1 0] x_2 + [1]
[3 3 1] [3 0 0] [3]
p(p) = [1 0 0] [0]
[1 0 0] x_1 + [0]
[0 1 0] [0]
p(s) = [1 1 0] [0]
[0 0 1] x_1 + [0]
[0 0 1] [1]
p(true) = [0]
[0]
[1]
Following rules are strictly oriented:
minus(x,s(y)) = [1 0 0] [2 2 1] [1]
[1 1 0] x + [2 2 1] y + [1]
[3 3 1] [3 3 0] [3]
> [1 0 0] [2 2 1] [0]
[1 0 0] x + [2 2 1] y + [1]
[3 3 0] [3 3 0] [2]
= 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 0] [1 0 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[1 0 2] [0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
if(true(),x,y) = [2 0 0] [1 0 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[1 0 2] [0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
le(0(),y) = [0]
[1]
[1]
>= [0]
[0]
[1]
= true()
le(s(x),0()) = [0 0 0] [0]
[0 0 0] x + [1]
[2 2 2] [1]
>= [0]
[1]
[1]
= false()
le(s(x),s(y)) = [0 0 0] [0]
[0 0 0] x + [1]
[2 2 2] [1]
>= [0 0 0] [0]
[0 0 0] x + [1]
[2 2 0] [1]
= le(x,y)
minus(x,0()) = [1 0 0] [0]
[1 1 0] x + [1]
[3 3 1] [3]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
p(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
p(s(x)) = [1 1 0] [0]
[1 1 0] x + [0]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
** Step 1.b: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:
innermost 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:
{if,le,minus,p}
TcT has computed the following interpretation:
p(0) = [0]
[1]
[1]
[0]
p(false) = [0]
[0]
[0]
[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 1 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]
[0 0 0 0] x_1 + [0 0 0 1] x_2 + [0]
[0 0 1 0] [0 0 1 0] [0]
[0 0 0 0] [0 0 1 0] [0]
p(minus) = [1 0 0 0] [1 0 1 0] [0]
[1 1 0 0] x_1 + [1 0 1 0] x_2 + [1]
[1 1 1 0] [1 0 1 1] [1]
[0 0 0 1] [0 0 1 0] [0]
p(p) = [1 0 0 0] [0]
[1 0 0 0] x_1 + [1]
[0 1 0 0] [0]
[0 0 0 1] [1]
p(s) = [1 1 1 0] [0]
[0 0 1 0] x_1 + [0]
[0 0 1 1] [1]
[0 0 0 1] [1]
p(true) = [0]
[0]
[0]
[0]
Following rules are strictly oriented:
le(s(x),s(y)) = [0 0 0 0] [0 0 0 1] [1]
[0 0 0 0] x + [0 0 0 1] y + [1]
[0 0 1 1] [0 0 1 1] [2]
[0 0 0 0] [0 0 1 1] [1]
> [0 0 0 0] [0 0 0 1] [0]
[0 0 0 0] x + [0 0 0 1] y + [0]
[0 0 1 0] [0 0 1 0] [0]
[0 0 0 0] [0 0 1 0] [0]
= 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 1 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 1 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 1] y + [0]
[0 0 1 0] [1]
[0 0 1 0] [0]
>= [0]
[0]
[0]
[0]
= true()
le(s(x),0()) = [0 0 0 0] [0]
[0 0 0 0] x + [0]
[0 0 1 1] [2]
[0 0 0 0] [1]
>= [0]
[0]
[0]
[0]
= false()
minus(x,0()) = [1 0 0 0] [1]
[1 1 0 0] x + [2]
[1 1 1 0] [2]
[0 0 0 1] [1]
>= [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 0 0 0] [1 1 2 1] [1]
[1 1 0 0] x + [1 1 2 1] y + [2]
[1 1 1 0] [1 1 2 2] [3]
[0 0 0 1] [0 0 1 1] [1]
>= [1 0 0 0] [1 1 2 1] [1]
[1 0 0 0] x + [1 1 2 0] y + [2]
[1 1 0 0] [1 1 2 0] [3]
[0 0 0 1] [0 0 1 0] [1]
= if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) = [0]
[1]
[1]
[1]
>= [0]
[1]
[1]
[0]
= 0()
p(s(x)) = [1 1 1 0] [0]
[1 1 1 0] x + [1]
[0 0 1 0] [0]
[0 0 0 1] [2]
>= [1 0 0 0] [0]
[0 1 0 0] x + [0]
[0 0 1 0] [0]
[0 0 0 1] [0]
= x
** Step 1.b: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:
innermost 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(Omega(n^1),O(n^3))