*** 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:
Full
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:
Full
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:
Full
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:
Full
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:
Full
basic terms: {gcd,if_gcd,le,minus}/{0,false,s,true}
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(gcd) = {1},
uargs(if_gcd) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = 1
p(false) = 1
p(gcd) = 2 + 2*x1 + 3*x1^2 + x2 + 3*x2^2
p(if_gcd) = 1 + x1 + x2 + 3*x2^2 + x3 + 3*x3^2
p(le) = x2
p(minus) = x1
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
le(s(x),s(y)) = 1 + y
> y
= le(x,y)
Following rules are (at-least) weakly oriented:
gcd(0(),y) = 7 + y + 3*y^2
>= y
= y
gcd(s(x),0()) = 11 + 8*x + 3*x^2
>= 1 + x
= s(x)
gcd(s(x),s(y)) = 11 + 8*x + 3*x^2 + 7*y + 3*y^2
>= 9 + 8*x + 3*x^2 + 7*y + 3*y^2
= if_gcd(le(y,x),s(x),s(y))
if_gcd(false(),s(x),s(y)) = 10 + 7*x + 3*x^2 + 7*y + 3*y^2
>= 6 + 7*x + 3*x^2 + 2*y + 3*y^2
= gcd(minus(y,x),s(x))
if_gcd(true(),s(x),s(y)) = 9 + 7*x + 3*x^2 + 7*y + 3*y^2
>= 6 + 2*x + 3*x^2 + 7*y + 3*y^2
= gcd(minus(x,y),s(y))
le(0(),y) = y
>= 0
= true()
le(s(x),0()) = 1
>= 1
= false()
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 1 + x
>= x
= 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:
Full
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).