* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
f(x,y,z){z -> s(z)} =
f(x,y,s(z)) ->^+ s(f(0(),1(),z))
= C[f(0(),1(),z) = f(x,y,z){x -> 0(),y -> 1()}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,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(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [0]
p(f) = [0]
p(g) = [2] x1 + [2] x2 + [1]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
g(x,y) = [2] x + [2] y + [1]
> [1] x + [0]
= x
g(x,y) = [2] x + [2] y + [1]
> [1] y + [0]
= y
Following rules are (at-least) weakly oriented:
f(x,y,s(z)) = [0]
>= [0]
= s(f(0(),1(),z))
f(0(),1(),x) = [0]
>= [0]
= f(s(x),x,x)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
- Weak TRS:
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,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(s) = {1}
Following symbols are considered usable:
{f,g}
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [0]
p(f) = [5] x_3 + [0]
p(g) = [2] x_1 + [2] x_2 + [0]
p(s) = [1] x_1 + [1]
Following rules are strictly oriented:
f(x,y,s(z)) = [5] z + [5]
> [5] z + [1]
= s(f(0(),1(),z))
Following rules are (at-least) weakly oriented:
f(0(),1(),x) = [5] x + [0]
>= [5] x + [0]
= f(s(x),x,x)
g(x,y) = [2] x + [2] y + [0]
>= [1] x + [0]
= x
g(x,y) = [2] x + [2] y + [0]
>= [1] y + [0]
= y
** Step 1.b:3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(0(),1(),x) -> f(s(x),x,x)
- Weak TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))):
The following argument positions are considered usable:
uargs(s) = {1}
Following symbols are considered usable:
{f,g}
TcT has computed the following interpretation:
p(0) = [4]
[9]
p(1) = [0]
[0]
p(f) = [0 1] x_1 + [2 0] x_3 + [1]
[0 0] [0 2] [0]
p(g) = [2 1] x_1 + [2 1] x_2 + [0]
[1 2] [0 2] [1]
p(s) = [1 0] x_1 + [12]
[0 0] [2]
Following rules are strictly oriented:
f(0(),1(),x) = [2 0] x + [10]
[0 2] [0]
> [2 0] x + [3]
[0 2] [0]
= f(s(x),x,x)
Following rules are (at-least) weakly oriented:
f(x,y,s(z)) = [0 1] x + [2 0] z + [25]
[0 0] [0 0] [4]
>= [2 0] z + [22]
[0 0] [2]
= s(f(0(),1(),z))
g(x,y) = [2 1] x + [2 1] y + [0]
[1 2] [0 2] [1]
>= [1 0] x + [0]
[0 1] [0]
= x
g(x,y) = [2 1] x + [2 1] y + [0]
[1 2] [0 2] [1]
>= [1 0] y + [0]
[0 1] [0]
= y
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))