* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,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:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,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:
+(x,y){y -> s(y)} =
+(x,s(y)) ->^+ s(+(x,y))
= C[+(x,y) = +(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,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(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [0]
p(0) = [0]
p(1) = [0]
p(f) = [2] x1 + [0]
p(g) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [4]
Following rules are strictly oriented:
f(s(x)) = [2] x + [8]
> [2] x + [4]
= g(x,s(x))
g(s(x),y) = [1] x + [1] y + [4]
> [1] x + [1] y + [0]
= g(x,+(y,s(x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [0]
>= [1] x + [4]
= s(+(x,y))
f(0()) = [0]
>= [0]
= 1()
g(0(),y) = [1] y + [0]
>= [1] y + [0]
= y
g(s(x),y) = [1] x + [1] y + [4]
>= [1] x + [1] y + [4]
= g(x,s(+(y,x)))
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:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
g(0(),y) -> y
g(s(x),y) -> g(x,s(+(y,x)))
- Weak TRS:
f(s(x)) -> g(x,s(x))
g(s(x),y) -> g(x,+(y,s(x)))
- Signature:
{+/2,f/1,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(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [0]
p(0) = [3]
p(1) = [0]
p(f) = [2] x1 + [8]
p(g) = [1] x2 + [0]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
f(0()) = [14]
> [0]
= 1()
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [0]
>= [1] x + [0]
= s(+(x,y))
f(s(x)) = [2] x + [8]
>= [1] x + [0]
= g(x,s(x))
g(0(),y) = [1] y + [0]
>= [1] y + [0]
= y
g(s(x),y) = [1] y + [0]
>= [1] y + [0]
= g(x,+(y,s(x)))
g(s(x),y) = [1] y + [0]
>= [1] y + [0]
= g(x,s(+(y,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^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
g(0(),y) -> y
g(s(x),y) -> g(x,s(+(y,x)))
- Weak TRS:
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(s(x),y) -> g(x,+(y,s(x)))
- Signature:
{+/2,f/1,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(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [0]
p(0) = [0]
p(1) = [0]
p(f) = [4] x1 + [15]
p(g) = [1] x2 + [13]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
g(0(),y) = [1] y + [13]
> [1] y + [0]
= y
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [0]
>= [1] x + [2]
= s(+(x,y))
f(0()) = [15]
>= [0]
= 1()
f(s(x)) = [4] x + [23]
>= [1] x + [15]
= g(x,s(x))
g(s(x),y) = [1] y + [13]
>= [1] y + [13]
= g(x,+(y,s(x)))
g(s(x),y) = [1] y + [13]
>= [1] y + [15]
= g(x,s(+(y,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^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
g(s(x),y) -> g(x,s(+(y,x)))
- Weak TRS:
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
- Signature:
{+/2,f/1,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(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [1]
p(0) = [0]
p(1) = [0]
p(f) = [8] x1 + [2]
p(g) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
+(x,0()) = [1] x + [1]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
+(x,s(y)) = [1] x + [1]
>= [1] x + [3]
= s(+(x,y))
f(0()) = [2]
>= [0]
= 1()
f(s(x)) = [8] x + [18]
>= [2] x + [2]
= g(x,s(x))
g(0(),y) = [1] y + [0]
>= [1] y + [0]
= y
g(s(x),y) = [1] x + [1] y + [2]
>= [1] x + [1] y + [1]
= g(x,+(y,s(x)))
g(s(x),y) = [1] x + [1] y + [2]
>= [1] x + [1] y + [3]
= g(x,s(+(y,x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
g(s(x),y) -> g(x,s(+(y,x)))
- Weak TRS:
+(x,0()) -> x
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
- Signature:
{+/2,f/1,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(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [0]
p(0) = [1]
p(1) = [0]
p(f) = [5] x1 + [2]
p(g) = [4] x1 + [1] x2 + [10]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
g(s(x),y) = [4] x + [1] y + [18]
> [4] x + [1] y + [12]
= g(x,s(+(y,x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [0]
>= [1] x + [2]
= s(+(x,y))
f(0()) = [7]
>= [0]
= 1()
f(s(x)) = [5] x + [12]
>= [5] x + [12]
= g(x,s(x))
g(0(),y) = [1] y + [14]
>= [1] y + [0]
= y
g(s(x),y) = [4] x + [1] y + [18]
>= [4] x + [1] y + [10]
= g(x,+(y,s(x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
- Weak TRS:
+(x,0()) -> x
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(g) = {2},
uargs(s) = {1}
Following symbols are considered usable:
{+,f,g}
TcT has computed the following interpretation:
p(+) = 1 + x1 + 2*x2
p(0) = 1
p(1) = 0
p(f) = 1 + 4*x1 + 3*x1^2
p(g) = 1 + 3*x1^2 + x2
p(s) = 1 + x1
Following rules are strictly oriented:
+(x,s(y)) = 3 + x + 2*y
> 2 + x + 2*y
= s(+(x,y))
Following rules are (at-least) weakly oriented:
+(x,0()) = 3 + x
>= x
= x
f(0()) = 8
>= 0
= 1()
f(s(x)) = 8 + 10*x + 3*x^2
>= 2 + x + 3*x^2
= g(x,s(x))
g(0(),y) = 4 + y
>= y
= y
g(s(x),y) = 4 + 6*x + 3*x^2 + y
>= 4 + 2*x + 3*x^2 + y
= g(x,+(y,s(x)))
g(s(x),y) = 4 + 6*x + 3*x^2 + y
>= 3 + 2*x + 3*x^2 + y
= g(x,s(+(y,x)))
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,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^2))