*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Weak DP Rules:
Weak TRS Rules:
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [3]
p(a__minus) = [0]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [0]
p(mark) = [0]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__div(X1,X2) = [1] X1 + [1]
> [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
> [0]
= 0()
a__if(X1,X2,X3) = [1] X1 + [3]
> [1] X1 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [3]
> [0]
= mark(Y)
a__if(true(),X,Y) = [3]
> [0]
= mark(X)
Following rules are (at-least) weakly oriented:
a__div(s(X),s(Y)) = [1] X + [1]
>= [3]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [0]
>= [0]
= true()
a__geq(X1,X2) = [0]
>= [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__geq(s(X),s(Y)) = [0]
>= [0]
= a__geq(X,Y)
a__minus(X1,X2) = [0]
>= [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [0]
>= [0]
= 0()
mark(div(X1,X2)) = [0]
>= [1]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [0]
>= [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0]
>= [3]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [0]
>= [0]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
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:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(a__div) = [1] x1 + [7]
p(a__geq) = [6]
p(a__if) = [1] x1 + [4]
p(a__minus) = [0]
p(div) = [1] x1 + [5]
p(false) = [7]
p(geq) = [0]
p(if) = [1] x1 + [1]
p(mark) = [2]
p(minus) = [1]
p(s) = [1] x1 + [0]
p(true) = [5]
Following rules are strictly oriented:
a__geq(X,0()) = [6]
> [5]
= true()
a__geq(X1,X2) = [6]
> [0]
= geq(X1,X2)
mark(minus(X1,X2)) = [2]
> [0]
= a__minus(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [7]
>= [1] X1 + [5]
= div(X1,X2)
a__div(0(),s(Y)) = [9]
>= [2]
= 0()
a__div(s(X),s(Y)) = [1] X + [7]
>= [10]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(0(),s(Y)) = [6]
>= [7]
= false()
a__geq(s(X),s(Y)) = [6]
>= [6]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [4]
>= [1] X1 + [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [11]
>= [2]
= mark(Y)
a__if(true(),X,Y) = [9]
>= [2]
= mark(X)
a__minus(X1,X2) = [0]
>= [1]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [2]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [2]
>= [2]
= 0()
mark(div(X1,X2)) = [2]
>= [9]
= a__div(mark(X1),X2)
mark(false()) = [2]
>= [7]
= false()
mark(geq(X1,X2)) = [2]
>= [6]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [2]
>= [6]
= a__if(mark(X1),X2,X3)
mark(s(X)) = [2]
>= [2]
= s(mark(X))
mark(true()) = [2]
>= [5]
= true()
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:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
mark(minus(X1,X2)) -> a__minus(X1,X2)
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [0]
p(a__geq) = [0]
p(a__if) = [1] x1 + [1]
p(a__minus) = [1]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1]
p(mark) = [1]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__minus(X1,X2) = [1]
> [0]
= minus(X1,X2)
a__minus(0(),Y) = [1]
> [0]
= 0()
mark(0()) = [1]
> [0]
= 0()
mark(false()) = [1]
> [0]
= false()
mark(geq(X1,X2)) = [1]
> [0]
= a__geq(X1,X2)
mark(true()) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [0]
>= [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [0]
>= [1]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [0]
>= [0]
= true()
a__geq(X1,X2) = [0]
>= [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__geq(s(X),s(Y)) = [0]
>= [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [1]
>= [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1]
>= [1]
= mark(Y)
a__if(true(),X,Y) = [1]
>= [1]
= mark(X)
a__minus(s(X),s(Y)) = [1]
>= [1]
= a__minus(X,Y)
mark(div(X1,X2)) = [1]
>= [1]
= a__div(mark(X1),X2)
mark(if(X1,X2,X3)) = [1]
>= [2]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
>= [1]
= a__minus(X1,X2)
mark(s(X)) = [1]
>= [1]
= s(mark(X))
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:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [0]
p(a__minus) = [0]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [0]
p(mark) = [0]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
a__div(s(X),s(Y)) = [1] X + [2]
> [0]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [1]
>= [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
>= [0]
= 0()
a__geq(X,0()) = [0]
>= [0]
= true()
a__geq(X1,X2) = [0]
>= [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__geq(s(X),s(Y)) = [0]
>= [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [0]
>= [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [0]
>= [0]
= mark(Y)
a__if(true(),X,Y) = [0]
>= [0]
= mark(X)
a__minus(X1,X2) = [0]
>= [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [0]
>= [0]
= 0()
mark(div(X1,X2)) = [0]
>= [1]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [0]
>= [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0]
>= [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [0]
>= [1]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
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:
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [7]
p(a__geq) = [1]
p(a__if) = [1] x1 + [2]
p(a__minus) = [1]
p(div) = [1] x1 + [7]
p(false) = [0]
p(geq) = [0]
p(if) = [0]
p(mark) = [1]
p(minus) = [1]
p(s) = [1] x1 + [0]
p(true) = [1]
Following rules are strictly oriented:
a__geq(0(),s(Y)) = [1]
> [0]
= false()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [7]
>= [1] X1 + [7]
= div(X1,X2)
a__div(0(),s(Y)) = [7]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [7]
>= [3]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [1]
>= [1]
= true()
a__geq(X1,X2) = [1]
>= [0]
= geq(X1,X2)
a__geq(s(X),s(Y)) = [1]
>= [1]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [2]
>= [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [2]
>= [1]
= mark(Y)
a__if(true(),X,Y) = [3]
>= [1]
= mark(X)
a__minus(X1,X2) = [1]
>= [1]
= minus(X1,X2)
a__minus(0(),Y) = [1]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [1]
>= [1]
= a__minus(X,Y)
mark(0()) = [1]
>= [0]
= 0()
mark(div(X1,X2)) = [1]
>= [8]
= a__div(mark(X1),X2)
mark(false()) = [1]
>= [0]
= false()
mark(geq(X1,X2)) = [1]
>= [1]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1]
>= [3]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
>= [1]
= a__minus(X1,X2)
mark(s(X)) = [1]
>= [1]
= s(mark(X))
mark(true()) = [1]
>= [1]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(s(X)) -> s(mark(X))
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [3]
p(a__geq) = [1] x1 + [0]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [0]
p(a__minus) = [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [1] x1 + [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
a__geq(s(X),s(Y)) = [1] X + [1]
> [1] X + [0]
= a__geq(X,Y)
mark(s(X)) = [4] X + [4]
> [4] X + [1]
= s(mark(X))
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [3]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [3]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [4]
>= [1] X + [4]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [1] X + [0]
>= [0]
= true()
a__geq(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [4] X + [4] Y + [0]
>= [4] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [4] X + [4] Y + [0]
>= [4] X + [0]
= mark(X)
a__minus(X1,X2) = [0]
>= [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [0]
>= [0]
= 0()
mark(div(X1,X2)) = [4] X1 + [0]
>= [4] X1 + [3]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [4] X1 + [0]
>= [1] X1 + [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0]
>= [4] X1 + [4] X2 + [4] X3 + [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [2]
p(a__geq) = [0]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [2]
p(a__minus) = [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [4]
> [4] X1 + [4] X2 + [4] X3 + [2]
= a__if(mark(X1),X2,X3)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [2]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [2]
>= [2]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [0]
>= [0]
= true()
a__geq(X1,X2) = [0]
>= [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__geq(s(X),s(Y)) = [0]
>= [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [4] X + [4] Y + [2]
>= [4] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [4] X + [4] Y + [2]
>= [4] X + [0]
= mark(X)
a__minus(X1,X2) = [0]
>= [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [0]
>= [0]
= 0()
mark(div(X1,X2)) = [4] X1 + [0]
>= [4] X1 + [2]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [0]
>= [0]
= a__geq(X1,X2)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [4] X + [0]
>= [4] X + [0]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__minus) = [1] x1 + [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [5]
p(true) = [0]
Following rules are strictly oriented:
a__minus(s(X),s(Y)) = [1] X + [5]
> [1] X + [0]
= a__minus(X,Y)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [1]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
>= [1]
= 0()
a__div(s(X),s(Y)) = [1] X + [6]
>= [1] X + [6]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [0]
>= [0]
= true()
a__geq(X1,X2) = [0]
>= [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__geq(s(X),s(Y)) = [0]
>= [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1] X + [1] Y + [0]
>= [1] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [0]
= mark(X)
a__minus(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= minus(X1,X2)
a__minus(0(),Y) = [1]
>= [1]
= 0()
mark(0()) = [1]
>= [1]
= 0()
mark(div(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [1]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [0]
>= [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= a__minus(X1,X2)
mark(s(X)) = [1] X + [5]
>= [1] X + [5]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
mark(div(X1,X2)) -> a__div(mark(X1),X2)
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__div) = [1 5] x1 + [2]
[0 0] [3]
p(a__geq) = [0]
[2]
p(a__if) = [1 1] x1 + [4 0] x2 + [4
2] x3 + [2]
[0 0] [0 1] [0
1] [1]
p(a__minus) = [0]
[0]
p(div) = [1 2] x1 + [2]
[0 0] [2]
p(false) = [0]
[0]
p(geq) = [0]
[2]
p(if) = [1 1] x1 + [1 0] x2 + [1
2] x3 + [1]
[0 0] [0 1] [0
1] [0]
p(mark) = [4 0] x1 + [0]
[0 1] [1]
p(minus) = [0]
[0]
p(s) = [1 0] x1 + [0]
[0 0] [2]
p(true) = [0]
[1]
Following rules are strictly oriented:
mark(div(X1,X2)) = [4 8] X1 + [8]
[0 0] [3]
> [4 5] X1 + [7]
[0 0] [3]
= a__div(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 5] X1 + [2]
[0 0] [3]
>= [1 2] X1 + [2]
[0 0] [2]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
[3]
>= [0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0] X + [12]
[0 0] [3]
>= [12]
[3]
= a__if(a__geq(X,Y)
,s(div(minus(X,Y),s(Y)))
,0())
a__geq(X,0()) = [0]
[2]
>= [0]
[1]
= true()
a__geq(X1,X2) = [0]
[2]
>= [0]
[2]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[2]
>= [0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[2]
>= [0]
[2]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 1] X1 + [4 0] X2 + [4
2] X3 + [2]
[0 0] [0 1] [0
1] [1]
>= [1 1] X1 + [1 0] X2 + [1
2] X3 + [1]
[0 0] [0 1] [0
1] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [4 0] X + [4 2] Y + [2]
[0 1] [0 1] [1]
>= [4 0] Y + [0]
[0 1] [1]
= mark(Y)
a__if(true(),X,Y) = [4 0] X + [4 2] Y + [3]
[0 1] [0 1] [1]
>= [4 0] X + [0]
[0 1] [1]
= mark(X)
a__minus(X1,X2) = [0]
[0]
>= [0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
>= [0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
>= [0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[1]
>= [0]
[0]
= 0()
mark(false()) = [0]
[1]
>= [0]
[0]
= false()
mark(geq(X1,X2)) = [0]
[3]
>= [0]
[2]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [4 4] X1 + [4 0] X2 + [4
8] X3 + [4]
[0 0] [0 1] [0
1] [1]
>= [4 1] X1 + [4 0] X2 + [4
2] X3 + [3]
[0 0] [0 1] [0
1] [1]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[1]
>= [0]
[0]
= a__minus(X1,X2)
mark(s(X)) = [4 0] X + [0]
[0 0] [3]
>= [4 0] X + [0]
[0 0] [2]
= s(mark(X))
mark(true()) = [0]
[2]
>= [0]
[1]
= true()
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
Obligation:
Innermost
basic terms: {a__div,a__geq,a__if,a__minus,mark}/{0,div,false,geq,if,minus,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).