* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [7]
p(false) = [9]
p(ifMinus) = [1] x1 + [0]
p(le) = [0]
p(minus) = [5]
p(quot) = [1] x1 + [3] x2 + [0]
p(s) = [1] x1 + [3]
p(true) = [0]
Following rules are strictly oriented:
ifMinus(false(),s(X),Y) = [9]
> [8]
= s(minus(X,Y))
minus(s(X),Y) = [5]
> [0]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [3] Y + [16]
> [7]
= 0()
Following rules are (at-least) weakly oriented:
ifMinus(true(),s(X),Y) = [0]
>= [7]
= 0()
le(0(),Y) = [0]
>= [0]
= true()
le(s(X),0()) = [0]
>= [9]
= false()
le(s(X),s(Y)) = [0]
>= [0]
= le(X,Y)
minus(0(),Y) = [5]
>= [7]
= 0()
quot(s(X),s(Y)) = [1] X + [3] Y + [12]
>= [3] Y + [17]
= s(quot(minus(X,Y),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:
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(0(),Y) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [4]
p(ifMinus) = [1] x1 + [1]
p(le) = [2]
p(minus) = [4]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [5]
Following rules are strictly oriented:
ifMinus(true(),s(X),Y) = [6]
> [0]
= 0()
minus(0(),Y) = [4]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
ifMinus(false(),s(X),Y) = [5]
>= [4]
= s(minus(X,Y))
le(0(),Y) = [2]
>= [5]
= true()
le(s(X),0()) = [2]
>= [4]
= false()
le(s(X),s(Y)) = [2]
>= [2]
= le(X,Y)
minus(s(X),Y) = [4]
>= [3]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [0]
>= [0]
= 0()
quot(s(X),s(Y)) = [1] X + [0]
>= [4]
= s(quot(minus(X,Y),s(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^3))
+ Considered Problem:
- Strict TRS:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [15]
p(ifMinus) = [1] x1 + [0]
p(le) = [14]
p(minus) = [15]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
le(0(),Y) = [14]
> [0]
= true()
Following rules are (at-least) weakly oriented:
ifMinus(false(),s(X),Y) = [15]
>= [15]
= s(minus(X,Y))
ifMinus(true(),s(X),Y) = [0]
>= [0]
= 0()
le(s(X),0()) = [14]
>= [15]
= false()
le(s(X),s(Y)) = [14]
>= [14]
= le(X,Y)
minus(0(),Y) = [15]
>= [0]
= 0()
minus(s(X),Y) = [15]
>= [14]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [0]
>= [0]
= 0()
quot(s(X),s(Y)) = [1] X + [0]
>= [15]
= s(quot(minus(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{ifMinus,le,minus,quot}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(ifMinus) = [4] x_1 + [1] x_2 + [0]
p(le) = [0]
p(minus) = [1] x_1 + [0]
p(quot) = [2] x_1 + [8]
p(s) = [1] x_1 + [1]
p(true) = [0]
Following rules are strictly oriented:
quot(s(X),s(Y)) = [2] X + [10]
> [2] X + [9]
= s(quot(minus(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
ifMinus(false(),s(X),Y) = [1] X + [1]
>= [1] X + [1]
= s(minus(X,Y))
ifMinus(true(),s(X),Y) = [1] X + [1]
>= [0]
= 0()
le(0(),Y) = [0]
>= [0]
= true()
le(s(X),0()) = [0]
>= [0]
= false()
le(s(X),s(Y)) = [0]
>= [0]
= le(X,Y)
minus(0(),Y) = [0]
>= [0]
= 0()
minus(s(X),Y) = [1] X + [1]
>= [1] X + [1]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [8]
>= [0]
= 0()
** Step 1.b:5: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{ifMinus,le,minus,quot}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(false) = [0]
[1]
[3]
[0]
p(ifMinus) = [1 1 0 0] [1 1 0 0] [0 0 0 0] [0]
[0 0 2 0] x_1 + [0 1 0 0] x_2 + [1 0 3 0] x_3 + [0]
[0 0 0 0] [0 0 1 0] [0 0 0 0] [0]
[1 2 0 2] [0 2 0 0] [0 2 3 0] [0]
p(le) = [0 0 0 0] [1]
[0 0 0 0] x_2 + [2]
[1 0 0 0] [3]
[1 0 0 0] [0]
p(minus) = [1 0 1 0] [0 0 0 0] [1]
[1 3 1 0] x_1 + [3 2 3 0] x_2 + [3]
[0 0 1 0] [0 0 0 0] [0]
[0 0 2 0] [2 3 3 1] [1]
p(quot) = [2 0 2 0] [0]
[0 1 0 0] x_1 + [1]
[0 0 1 0] [0]
[0 0 0 0] [0]
p(s) = [1 0 2 0] [1]
[0 0 1 0] x_1 + [0]
[0 0 1 0] [2]
[0 0 0 0] [0]
p(true) = [1]
[0]
[0]
[0]
Following rules are strictly oriented:
le(s(X),0()) = [1]
[2]
[3]
[0]
> [0]
[1]
[3]
[0]
= false()
Following rules are (at-least) weakly oriented:
ifMinus(false(),s(X),Y) = [1 0 3 0] [0 0 0 0] [2]
[0 0 1 0] X + [1 0 3 0] Y + [6]
[0 0 1 0] [0 0 0 0] [2]
[0 0 2 0] [0 2 3 0] [2]
>= [1 0 3 0] [2]
[0 0 1 0] X + [0]
[0 0 1 0] [2]
[0 0 0 0] [0]
= s(minus(X,Y))
ifMinus(true(),s(X),Y) = [1 0 3 0] [0 0 0 0] [2]
[0 0 1 0] X + [1 0 3 0] Y + [0]
[0 0 1 0] [0 0 0 0] [2]
[0 0 2 0] [0 2 3 0] [1]
>= [0]
[0]
[0]
[0]
= 0()
le(0(),Y) = [0 0 0 0] [1]
[0 0 0 0] Y + [2]
[1 0 0 0] [3]
[1 0 0 0] [0]
>= [1]
[0]
[0]
[0]
= true()
le(s(X),s(Y)) = [0 0 0 0] [1]
[0 0 0 0] Y + [2]
[1 0 2 0] [4]
[1 0 2 0] [1]
>= [0 0 0 0] [1]
[0 0 0 0] Y + [2]
[1 0 0 0] [3]
[1 0 0 0] [0]
= le(X,Y)
minus(0(),Y) = [0 0 0 0] [1]
[3 2 3 0] Y + [3]
[0 0 0 0] [0]
[2 3 3 1] [1]
>= [0]
[0]
[0]
[0]
= 0()
minus(s(X),Y) = [1 0 3 0] [0 0 0 0] [4]
[1 0 6 0] X + [3 2 3 0] Y + [6]
[0 0 1 0] [0 0 0 0] [2]
[0 0 2 0] [2 3 3 1] [5]
>= [1 0 3 0] [0 0 0 0] [4]
[0 0 1 0] X + [3 0 3 0] Y + [6]
[0 0 1 0] [0 0 0 0] [2]
[0 0 2 0] [2 2 3 0] [5]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [0]
[1]
[0]
[0]
>= [0]
[0]
[0]
[0]
= 0()
quot(s(X),s(Y)) = [2 0 6 0] [6]
[0 0 1 0] X + [1]
[0 0 1 0] [2]
[0 0 0 0] [0]
>= [2 0 6 0] [3]
[0 0 1 0] X + [0]
[0 0 1 0] [2]
[0 0 0 0] [0]
= s(quot(minus(X,Y),s(Y)))
** Step 1.b:6: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(X),s(Y)) -> le(X,Y)
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{ifMinus,le,minus,quot}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
[0]
p(false) = [0]
[0]
[0]
[0]
p(ifMinus) = [1 0 0 0] [1 1 0 0] [0 0 0 0] [0]
[0 0 0 0] x_1 + [0 1 0 0] x_2 + [0 0 0 0] x_3 + [0]
[0 0 0 0] [0 0 1 0] [0 0 0 0] [0]
[0 0 1 2] [0 0 0 0] [0 0 0 3] [1]
p(le) = [0 0 2 0] [0 0 0 0] [0]
[0 1 2 0] x_1 + [0 0 0 0] x_2 + [0]
[0 0 0 0] [0 2 0 0] [0]
[0 1 0 0] [0 0 0 0] [3]
p(minus) = [1 1 2 0] [0 0 0 0] [0]
[0 1 0 0] x_1 + [0 0 0 0] x_2 + [0]
[0 0 1 0] [0 0 0 0] [0]
[0 2 2 0] [1 3 1 3] [3]
p(quot) = [2 2 1 0] [0 1 0 1] [2]
[0 1 0 0] x_1 + [0 0 0 0] x_2 + [1]
[0 0 1 0] [0 0 0 0] [0]
[2 1 0 0] [0 1 0 0] [1]
p(s) = [1 2 0 0] [0]
[0 1 2 0] x_1 + [0]
[0 0 1 0] [2]
[0 0 0 0] [1]
p(true) = [0]
[0]
[0]
[1]
Following rules are strictly oriented:
le(s(X),s(Y)) = [0 0 2 0] [0 0 0 0] [4]
[0 1 4 0] X + [0 0 0 0] Y + [4]
[0 0 0 0] [0 2 4 0] [0]
[0 1 2 0] [0 0 0 0] [3]
> [0 0 2 0] [0 0 0 0] [0]
[0 1 2 0] X + [0 0 0 0] Y + [0]
[0 0 0 0] [0 2 0 0] [0]
[0 1 0 0] [0 0 0 0] [3]
= le(X,Y)
Following rules are (at-least) weakly oriented:
ifMinus(false(),s(X),Y) = [1 3 2 0] [0 0 0 0] [0]
[0 1 2 0] X + [0 0 0 0] Y + [0]
[0 0 1 0] [0 0 0 0] [2]
[0 0 0 0] [0 0 0 3] [1]
>= [1 3 2 0] [0]
[0 1 2 0] X + [0]
[0 0 1 0] [2]
[0 0 0 0] [1]
= s(minus(X,Y))
ifMinus(true(),s(X),Y) = [1 3 2 0] [0 0 0 0] [0]
[0 1 2 0] X + [0 0 0 0] Y + [0]
[0 0 1 0] [0 0 0 0] [2]
[0 0 0 0] [0 0 0 3] [3]
>= [0]
[0]
[0]
[0]
= 0()
le(0(),Y) = [0 0 0 0] [0]
[0 0 0 0] Y + [0]
[0 2 0 0] [0]
[0 0 0 0] [3]
>= [0]
[0]
[0]
[1]
= true()
le(s(X),0()) = [0 0 2 0] [4]
[0 1 4 0] X + [4]
[0 0 0 0] [0]
[0 1 2 0] [3]
>= [0]
[0]
[0]
[0]
= false()
minus(0(),Y) = [0 0 0 0] [0]
[0 0 0 0] Y + [0]
[0 0 0 0] [0]
[1 3 1 3] [3]
>= [0]
[0]
[0]
[0]
= 0()
minus(s(X),Y) = [1 3 4 0] [0 0 0 0] [4]
[0 1 2 0] X + [0 0 0 0] Y + [0]
[0 0 1 0] [0 0 0 0] [2]
[0 2 6 0] [1 3 1 3] [7]
>= [1 3 4 0] [0 0 0 0] [4]
[0 1 2 0] X + [0 0 0 0] Y + [0]
[0 0 1 0] [0 0 0 0] [2]
[0 2 4 0] [0 2 0 3] [7]
= ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) = [0 1 2 0] [3]
[0 0 0 0] Y + [1]
[0 0 0 0] [0]
[0 1 2 0] [1]
>= [0]
[0]
[0]
[0]
= 0()
quot(s(X),s(Y)) = [2 6 5 0] [0 1 2 0] [5]
[0 1 2 0] X + [0 0 0 0] Y + [1]
[0 0 1 0] [0 0 0 0] [2]
[2 5 2 0] [0 1 2 0] [1]
>= [2 6 5 0] [0 1 2 0] [5]
[0 1 2 0] X + [0 0 0 0] Y + [1]
[0 0 1 0] [0 0 0 0] [2]
[0 0 0 0] [0 0 0 0] [1]
= s(quot(minus(X,Y),s(Y)))
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
ifMinus(false(),s(X),Y) -> s(minus(X,Y))
ifMinus(true(),s(X),Y) -> 0()
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
minus(0(),Y) -> 0()
minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
- Signature:
{ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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))