*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
Weak DP Rules:
Weak TRS Rules:
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [1] x1 + [1]
p(0) = [4]
p(div) = [1] x1 + [1] x2 + [7]
p(false) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(lt) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
-(x,0()) = [1] x + [1]
> [1] x + [0]
= x
-(0(),s(y)) = [5]
> [4]
= 0()
div(x,0()) = [1] x + [11]
> [4]
= 0()
div(0(),y) = [1] y + [11]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
-(s(x),s(y)) = [1] x + [1]
>= [1] x + [1]
= -(x,y)
div(s(x),s(y)) = [1] x + [1] y + [7]
>= [1] x + [1] y + [12]
= if(lt(x,y)
,0()
,s(div(-(x,y),s(y))))
if(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [0]
= y
if(true(),x,y) = [1] x + [1] y + [0]
>= [1] x + [0]
= x
lt(x,0()) = [0]
>= [0]
= false()
lt(0(),s(y)) = [0]
>= [0]
= true()
lt(s(x),s(y)) = [0]
>= [0]
= lt(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:
-(s(x),s(y)) -> -(x,y)
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
Weak DP Rules:
Weak TRS Rules:
-(x,0()) -> x
-(0(),s(y)) -> 0()
div(x,0()) -> 0()
div(0(),y) -> 0()
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [1] x1 + [0]
p(0) = [0]
p(div) = [1] x1 + [8] x2 + [1]
p(false) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [9]
p(lt) = [8]
p(s) = [1] x1 + [0]
p(true) = [7]
Following rules are strictly oriented:
if(false(),x,y) = [1] x + [1] y + [9]
> [1] y + [0]
= y
if(true(),x,y) = [1] x + [1] y + [16]
> [1] x + [0]
= x
lt(x,0()) = [8]
> [0]
= false()
lt(0(),s(y)) = [8]
> [7]
= true()
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
-(0(),s(y)) = [0]
>= [0]
= 0()
-(s(x),s(y)) = [1] x + [0]
>= [1] x + [0]
= -(x,y)
div(x,0()) = [1] x + [1]
>= [0]
= 0()
div(0(),y) = [8] y + [1]
>= [0]
= 0()
div(s(x),s(y)) = [1] x + [8] y + [1]
>= [1] x + [8] y + [18]
= if(lt(x,y)
,0()
,s(div(-(x,y),s(y))))
lt(s(x),s(y)) = [8]
>= [8]
= lt(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:
-(s(x),s(y)) -> -(x,y)
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(s(x),s(y)) -> lt(x,y)
Weak DP Rules:
Weak TRS Rules:
-(x,0()) -> x
-(0(),s(y)) -> 0()
div(x,0()) -> 0()
div(0(),y) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [1] x1 + [8]
p(0) = [0]
p(div) = [1] x1 + [1] x2 + [1]
p(false) = [0]
p(if) = [1] x1 + [2] x2 + [1] x3 + [8]
p(lt) = [0]
p(s) = [1] x1 + [2]
p(true) = [0]
Following rules are strictly oriented:
-(s(x),s(y)) = [1] x + [10]
> [1] x + [8]
= -(x,y)
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [8]
>= [1] x + [0]
= x
-(0(),s(y)) = [8]
>= [0]
= 0()
div(x,0()) = [1] x + [1]
>= [0]
= 0()
div(0(),y) = [1] y + [1]
>= [0]
= 0()
div(s(x),s(y)) = [1] x + [1] y + [5]
>= [1] x + [1] y + [21]
= if(lt(x,y)
,0()
,s(div(-(x,y),s(y))))
if(false(),x,y) = [2] x + [1] y + [8]
>= [1] y + [0]
= y
if(true(),x,y) = [2] x + [1] y + [8]
>= [1] x + [0]
= x
lt(x,0()) = [0]
>= [0]
= false()
lt(0(),s(y)) = [0]
>= [0]
= true()
lt(s(x),s(y)) = [0]
>= [0]
= lt(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:
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(s(x),s(y)) -> lt(x,y)
Weak DP Rules:
Weak TRS Rules:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{0,false,s,true}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) = [1] x1 + [0]
p(0) = [0]
p(div) = [6] x1 + [0]
p(false) = [1]
p(if) = [2] x1 + [1] x2 + [1] x3 + [1]
p(lt) = [8]
p(s) = [1] x1 + [5]
p(true) = [8]
Following rules are strictly oriented:
div(s(x),s(y)) = [6] x + [30]
> [6] x + [22]
= if(lt(x,y)
,0()
,s(div(-(x,y),s(y))))
Following rules are (at-least) weakly oriented:
-(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
-(0(),s(y)) = [0]
>= [0]
= 0()
-(s(x),s(y)) = [1] x + [5]
>= [1] x + [0]
= -(x,y)
div(x,0()) = [6] x + [0]
>= [0]
= 0()
div(0(),y) = [0]
>= [0]
= 0()
if(false(),x,y) = [1] x + [1] y + [3]
>= [1] y + [0]
= y
if(true(),x,y) = [1] x + [1] y + [17]
>= [1] x + [0]
= x
lt(x,0()) = [8]
>= [1]
= false()
lt(0(),s(y)) = [8]
>= [8]
= true()
lt(s(x),s(y)) = [8]
>= [8]
= lt(x,y)
*** 1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
lt(s(x),s(y)) -> lt(x,y)
Weak DP Rules:
Weak TRS Rules:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) = x1
p(0) = 0
p(div) = x1 + x1^2
p(false) = 0
p(if) = x1 + x2 + x3
p(lt) = x1
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
lt(s(x),s(y)) = 1 + x
> x
= lt(x,y)
Following rules are (at-least) weakly oriented:
-(x,0()) = x
>= x
= x
-(0(),s(y)) = 0
>= 0
= 0()
-(s(x),s(y)) = 1 + x
>= x
= -(x,y)
div(x,0()) = x + x^2
>= 0
= 0()
div(0(),y) = 0
>= 0
= 0()
div(s(x),s(y)) = 2 + 3*x + x^2
>= 1 + 2*x + x^2
= if(lt(x,y)
,0()
,s(div(-(x,y),s(y))))
if(false(),x,y) = x + y
>= y
= y
if(true(),x,y) = x + y
>= x
= x
lt(x,0()) = x
>= 0
= false()
lt(0(),s(y)) = 0
>= 0
= true()
*** 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:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {-,div,if,lt}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).