*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {+,double,sqr}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [6]
p(0) = [4]
p(double) = [10]
p(s) = [1] x1 + [4]
p(sqr) = [5] x1 + [2]
Following rules are strictly oriented:
+(x,0()) = [1] x + [10]
> [1] x + [0]
= x
double(0()) = [10]
> [4]
= 0()
sqr(0()) = [22]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
+(x,s(y)) = [1] x + [1] y + [10]
>= [1] x + [1] y + [10]
= s(+(x,y))
double(s(x)) = [10]
>= [18]
= s(s(double(x)))
sqr(s(x)) = [5] x + [22]
>= [5] x + [22]
= +(sqr(x),s(double(x)))
sqr(s(x)) = [5] x + [22]
>= [5] x + [22]
= s(+(sqr(x),double(x)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+(x,s(y)) -> s(+(x,y))
double(s(x)) -> s(s(double(x)))
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
Weak DP Rules:
Weak TRS Rules:
+(x,0()) -> x
double(0()) -> 0()
sqr(0()) -> 0()
Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {+,double,sqr}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [0]
p(0) = [0]
p(double) = [1]
p(s) = [1] x1 + [2]
p(sqr) = [6] x1 + [1]
Following rules are strictly oriented:
sqr(s(x)) = [6] x + [13]
> [6] x + [4]
= +(sqr(x),s(double(x)))
sqr(s(x)) = [6] x + [13]
> [6] x + [4]
= s(+(sqr(x),double(x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [1] y + [2]
>= [1] x + [1] y + [2]
= s(+(x,y))
double(0()) = [1]
>= [0]
= 0()
double(s(x)) = [1]
>= [5]
= s(s(double(x)))
sqr(0()) = [1]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+(x,s(y)) -> s(+(x,y))
double(s(x)) -> s(s(double(x)))
Weak DP Rules:
Weak TRS Rules:
+(x,0()) -> x
double(0()) -> 0()
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {+,double,sqr}/{0,s}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{+,double,sqr}
TcT has computed the following interpretation:
p(+) = x1 + 2*x2
p(0) = 1
p(double) = 1 + 2*x1
p(s) = 1 + x1
p(sqr) = x1 + 3*x1^2
Following rules are strictly oriented:
+(x,s(y)) = 2 + x + 2*y
> 1 + x + 2*y
= s(+(x,y))
Following rules are (at-least) weakly oriented:
+(x,0()) = 2 + x
>= x
= x
double(0()) = 3
>= 1
= 0()
double(s(x)) = 3 + 2*x
>= 3 + 2*x
= s(s(double(x)))
sqr(0()) = 4
>= 1
= 0()
sqr(s(x)) = 4 + 7*x + 3*x^2
>= 4 + 5*x + 3*x^2
= +(sqr(x),s(double(x)))
sqr(s(x)) = 4 + 7*x + 3*x^2
>= 3 + 5*x + 3*x^2
= s(+(sqr(x),double(x)))
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
double(s(x)) -> s(s(double(x)))
Weak DP Rules:
Weak TRS Rules:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {+,double,sqr}/{0,s}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{+,double,sqr}
TcT has computed the following interpretation:
p(+) = 1 + x1 + 2*x2
p(0) = 0
p(double) = 3*x1
p(s) = 1 + x1
p(sqr) = 3*x1^2
Following rules are strictly oriented:
double(s(x)) = 3 + 3*x
> 2 + 3*x
= s(s(double(x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = 1 + x
>= x
= x
+(x,s(y)) = 3 + x + 2*y
>= 2 + x + 2*y
= s(+(x,y))
double(0()) = 0
>= 0
= 0()
sqr(0()) = 0
>= 0
= 0()
sqr(s(x)) = 3 + 6*x + 3*x^2
>= 3 + 6*x + 3*x^2
= +(sqr(x),s(double(x)))
sqr(s(x)) = 3 + 6*x + 3*x^2
>= 2 + 6*x + 3*x^2
= s(+(sqr(x),double(x)))
*** 1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {+,double,sqr}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).