*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
Weak DP Rules:
Weak TRS Rules:
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(div) = [1] x1 + [8] x2 + [1]
p(false) = [0]
p(geq) = [13]
p(if) = [1] x1 + [1] x2 + [4] x3 + [1]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
div(0(),s(Y)) = [8] Y + [10]
> [1]
= 0()
geq(X,0()) = [13]
> [0]
= true()
geq(0(),s(Y)) = [13]
> [0]
= false()
if(false(),X,Y) = [1] X + [4] Y + [1]
> [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [4] Y + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) = [1] X + [8] Y + [10]
>= [8] Y + [28]
= if(geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
geq(s(X),s(Y)) = [13]
>= [13]
= geq(X,Y)
minus(0(),Y) = [0]
>= [1]
= 0()
minus(s(X),s(Y)) = [0]
>= [0]
= minus(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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(s(X),s(Y)) -> geq(X,Y)
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [8] x2 + [0]
p(false) = [3]
p(geq) = [8]
p(if) = [1] x1 + [1] x2 + [8] x3 + [9]
p(minus) = [1]
p(s) = [1] x1 + [0]
p(true) = [1]
Following rules are strictly oriented:
minus(0(),Y) = [1]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [8] Y + [0]
>= [0]
= 0()
div(s(X),s(Y)) = [1] X + [8] Y + [0]
>= [8] Y + [18]
= if(geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
geq(X,0()) = [8]
>= [1]
= true()
geq(0(),s(Y)) = [8]
>= [3]
= false()
geq(s(X),s(Y)) = [8]
>= [8]
= geq(X,Y)
if(false(),X,Y) = [1] X + [8] Y + [12]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [8] Y + [10]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1]
>= [1]
= 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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(s(X),s(Y)) -> geq(X,Y)
minus(s(X),s(Y)) -> minus(X,Y)
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [8]
p(div) = [1] x1 + [1] x2 + [0]
p(false) = [10]
p(geq) = [1] x1 + [4]
p(if) = [1] x1 + [1] x2 + [1] x3 + [6]
p(minus) = [9]
p(s) = [1] x1 + [1]
p(true) = [2]
Following rules are strictly oriented:
geq(s(X),s(Y)) = [1] X + [5]
> [1] X + [4]
= geq(X,Y)
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1] Y + [9]
>= [8]
= 0()
div(s(X),s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [29]
= if(geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
geq(X,0()) = [1] X + [4]
>= [2]
= true()
geq(0(),s(Y)) = [12]
>= [10]
= false()
if(false(),X,Y) = [1] X + [1] Y + [16]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [1] Y + [8]
>= [1] X + [0]
= X
minus(0(),Y) = [9]
>= [8]
= 0()
minus(s(X),s(Y)) = [9]
>= [9]
= 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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
minus(s(X),s(Y)) -> minus(X,Y)
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [5]
p(div) = [1] x1 + [8]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [3] x3 + [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
minus(s(X),s(Y)) = [1] X + [8]
> [1] X + [0]
= minus(X,Y)
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [13]
>= [5]
= 0()
div(s(X),s(Y)) = [1] X + [16]
>= [1] X + [31]
= if(geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
geq(X,0()) = [0]
>= [0]
= true()
geq(0(),s(Y)) = [0]
>= [0]
= false()
geq(s(X),s(Y)) = [0]
>= [0]
= geq(X,Y)
if(false(),X,Y) = [1] X + [3] Y + [0]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [3] Y + [0]
>= [1] X + [0]
= X
minus(0(),Y) = [5]
>= [5]
= 0()
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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,minus}/{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) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{div,geq,if,minus}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [8] x1 + [1]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [4] x2 + [8] x3 + [0]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
div(s(X),s(Y)) = [8] X + [9]
> [8]
= if(geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1]
>= [0]
= 0()
geq(X,0()) = [0]
>= [0]
= true()
geq(0(),s(Y)) = [0]
>= [0]
= false()
geq(s(X),s(Y)) = [0]
>= [0]
= geq(X,Y)
if(false(),X,Y) = [4] X + [8] Y + [0]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [4] X + [8] Y + [0]
>= [1] X + [0]
= X
minus(0(),Y) = [0]
>= [0]
= 0()
minus(s(X),s(Y)) = [0]
>= [0]
= 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:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {div,geq,if,minus}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).