*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Weak DP Rules:
Weak TRS Rules:
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
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(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [0]
p(gcd) = [1] x1 + [1] x2 + [7]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [0]
p(minus) = [1] x1 + [3]
p(s) = [1] x1 + [2]
p(true) = [0]
Following rules are strictly oriented:
gcd(0(),y) = [1] y + [9]
> [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [11]
> [1] x + [2]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [11]
> [1] x + [1] y + [4]
= if_gcd(le(y,x),s(x),s(y))
minus(x,0()) = [1] x + [3]
> [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [5]
> [1] x + [3]
= minus(x,y)
Following rules are (at-least) weakly oriented:
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [4]
>= [1] x + [1] y + [12]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [4]
>= [1] x + [1] y + [12]
= gcd(minus(x,y),s(y))
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
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:
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
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(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [4]
p(gcd) = [1] x1 + [1] x2 + [2]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [1]
p(minus) = [1] x1 + [4]
p(s) = [1] x1 + [4]
p(true) = [4]
Following rules are strictly oriented:
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [12]
> [1] x + [1] y + [10]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [12]
> [1] x + [1] y + [10]
= gcd(minus(x,y),s(y))
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [2]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [6]
>= [1] x + [4]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [10]
>= [1] x + [1] y + [9]
= if_gcd(le(y,x),s(x),s(y))
le(0(),y) = [1]
>= [4]
= true()
le(s(x),0()) = [1]
>= [4]
= false()
le(s(x),s(y)) = [1]
>= [1]
= le(x,y)
minus(x,0()) = [1] x + [4]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [8]
>= [1] x + [4]
= minus(x,y)
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:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
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(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(false) = [7]
p(gcd) = [1] x1 + [1] x2 + [4]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [4]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [3]
Following rules are strictly oriented:
le(0(),y) = [4]
> [3]
= true()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [8]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [9]
>= [1] x + [1]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [6]
>= [1] x + [1] y + [6]
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [9]
>= [1] x + [1] y + [5]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [5]
= gcd(minus(x,y),s(y))
le(s(x),0()) = [4]
>= [7]
= false()
le(s(x),s(y)) = [4]
>= [4]
= le(x,y)
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [1]
>= [1] x + [0]
= minus(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
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(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [5]
p(false) = [4]
p(gcd) = [1] x1 + [1] x2 + [6]
p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [1]
p(le) = [5]
p(minus) = [1] x1 + [3]
p(s) = [1] x1 + [4]
p(true) = [4]
Following rules are strictly oriented:
le(s(x),0()) = [5]
> [4]
= false()
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1] y + [11]
>= [1] y + [0]
= y
gcd(s(x),0()) = [1] x + [15]
>= [1] x + [4]
= s(x)
gcd(s(x),s(y)) = [1] x + [1] y + [14]
>= [1] x + [1] y + [14]
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [13]
>= [1] x + [1] y + [13]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [13]
>= [1] x + [1] y + [13]
= gcd(minus(x,y),s(y))
le(0(),y) = [5]
>= [4]
= true()
le(s(x),s(y)) = [5]
>= [5]
= le(x,y)
minus(x,0()) = [1] x + [3]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [7]
>= [1] x + [3]
= minus(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{gcd,if_gcd,le,minus}
TcT has computed the following interpretation:
p(0) = [0]
[1]
p(false) = [3]
[0]
p(gcd) = [1 1] x1 + [1 0] x2 + [5]
[0 1] [0 1] [2]
p(if_gcd) = [1 0] x1 + [1 0] x2 + [1
0] x3 + [2]
[0 0] [0 1] [0
1] [0]
p(le) = [0 1] x2 + [4]
[4 0] [0]
p(minus) = [1 0] x1 + [2]
[0 1] [1]
p(s) = [1 1] x1 + [3]
[0 1] [4]
p(true) = [3]
[0]
Following rules are strictly oriented:
le(s(x),s(y)) = [0 1] y + [8]
[4 4] [12]
> [0 1] y + [4]
[4 0] [0]
= le(x,y)
Following rules are (at-least) weakly oriented:
gcd(0(),y) = [1 0] y + [6]
[0 1] [3]
>= [1 0] y + [0]
[0 1] [0]
= y
gcd(s(x),0()) = [1 2] x + [12]
[0 1] [7]
>= [1 1] x + [3]
[0 1] [4]
= s(x)
gcd(s(x),s(y)) = [1 2] x + [1 1] y + [15]
[0 1] [0 1] [10]
>= [1 2] x + [1 1] y + [12]
[0 1] [0 1] [8]
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = [1 1] x + [1 1] y + [11]
[0 1] [0 1] [8]
>= [1 1] x + [1 1] y + [11]
[0 1] [0 1] [7]
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = [1 1] x + [1 1] y + [11]
[0 1] [0 1] [8]
>= [1 1] x + [1 1] y + [11]
[0 1] [0 1] [7]
= gcd(minus(x,y),s(y))
le(0(),y) = [0 1] y + [4]
[4 0] [0]
>= [3]
[0]
= true()
le(s(x),0()) = [5]
[0]
>= [3]
[0]
= false()
minus(x,0()) = [1 0] x + [2]
[0 1] [1]
>= [1 0] x + [0]
[0 1] [0]
= x
minus(s(x),s(y)) = [1 1] x + [5]
[0 1] [5]
>= [1 0] x + [2]
[0 1] [1]
= minus(x,y)
*** 1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
Signature:
{gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).