* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
min(x,y){x -> s(x),y -> s(y)} =
min(s(x),s(y)) ->^+ min(x,y)
= C[min(x,y) = min(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [4]
p(quot) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
min(X,0()) = [1] X + [4]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
log(s(0())) = [4]
>= [4]
= 0()
log(s(s(X))) = [1] X + [0]
>= [1] X + [4]
= s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) = [1] X + [4]
>= [1] X + [4]
= min(X,Y)
quot(0(),s(Y)) = [1] Y + [4]
>= [4]
= 0()
quot(s(X),s(Y)) = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [4]
= s(quot(min(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^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
min(X,0()) -> X
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [14]
p(log) = [1] x1 + [8]
p(min) = [1] x1 + [7]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
log(s(0())) = [22]
> [14]
= 0()
Following rules are (at-least) weakly oriented:
log(s(s(X))) = [1] X + [8]
>= [1] X + [8]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1] X + [7]
>= [1] X + [0]
= X
min(s(X),s(Y)) = [1] X + [7]
>= [1] X + [7]
= min(X,Y)
quot(0(),s(Y)) = [14]
>= [14]
= 0()
quot(s(X),s(Y)) = [1] X + [0]
>= [1] X + [7]
= s(quot(min(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^2))
+ Considered Problem:
- Strict TRS:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
min(X,0()) -> X
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [3]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [0]
p(quot) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
min(s(X),s(Y)) = [1] X + [1]
> [1] X + [0]
= min(X,Y)
quot(0(),s(Y)) = [1] Y + [4]
> [3]
= 0()
Following rules are (at-least) weakly oriented:
log(s(0())) = [4]
>= [3]
= 0()
log(s(s(X))) = [1] X + [2]
>= [1] X + [7]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
quot(s(X),s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= s(quot(min(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^2))
+ Considered Problem:
- Strict TRS:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{log,min,quot}
TcT has computed the following interpretation:
p(0) = [0]
p(log) = [3] x_1 + [2]
p(min) = [1] x_1 + [0]
p(quot) = [1] x_1 + [2]
p(s) = [1] x_1 + [4]
Following rules are strictly oriented:
log(s(s(X))) = [3] X + [26]
> [3] X + [24]
= s(log(s(quot(X,s(s(0()))))))
Following rules are (at-least) weakly oriented:
log(s(0())) = [14]
>= [0]
= 0()
min(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
min(s(X),s(Y)) = [1] X + [4]
>= [1] X + [0]
= min(X,Y)
quot(0(),s(Y)) = [2]
>= [0]
= 0()
quot(s(X),s(Y)) = [1] X + [6]
>= [1] X + [6]
= s(quot(min(X,Y),s(Y)))
** Step 1.b:5: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, 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(log) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{log,min,quot}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(log) = [2 0] x_1 + [2]
[0 1] [0]
p(min) = [1 0] x_1 + [0]
[0 1] [0]
p(quot) = [1 1] x_1 + [2]
[0 1] [0]
p(s) = [1 2] x_1 + [0]
[0 1] [2]
Following rules are strictly oriented:
quot(s(X),s(Y)) = [1 3] X + [4]
[0 1] [2]
> [1 3] X + [2]
[0 1] [2]
= s(quot(min(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
log(s(0())) = [2]
[2]
>= [0]
[0]
= 0()
log(s(s(X))) = [2 8] X + [10]
[0 1] [4]
>= [2 8] X + [10]
[0 1] [4]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
min(s(X),s(Y)) = [1 2] X + [0]
[0 1] [2]
>= [1 0] X + [0]
[0 1] [0]
= min(X,Y)
quot(0(),s(Y)) = [2]
[0]
>= [0]
[0]
= 0()
** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))