*** 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)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
Weak DP Rules:
Weak TRS Rules:
Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {div,minus,p}/{0,s}
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(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [14] x2 + [0]
p(minus) = [1] x1 + [0]
p(p) = [1] x1 + [5]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
div(0(),s(Y)) = [14] Y + [28]
> [0]
= 0()
p(s(X)) = [1] X + [7]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) = [1] X + [14] Y + [30]
>= [1] X + [14] Y + [30]
= s(div(minus(X,Y),s(Y)))
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1] X + [2]
>= [1] X + [5]
= p(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)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
p(s(X)) -> X
Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {div,minus,p}/{0,s}
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(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [8]
p(div) = [1] x1 + [1] x2 + [8]
p(minus) = [1] x1 + [1]
p(p) = [1] x1 + [8]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
minus(X,0()) = [1] X + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1] Y + [16]
>= [8]
= 0()
div(s(X),s(Y)) = [1] X + [1] Y + [8]
>= [1] X + [1] Y + [9]
= s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) = [1] X + [1]
>= [1] X + [9]
= p(minus(X,Y))
p(s(X)) = [1] X + [8]
>= [1] X + [0]
= X
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)) -> s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) -> p(minus(X,Y))
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
p(s(X)) -> X
Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {div,minus,p}/{0,s}
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(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [2]
p(minus) = [1] x1 + [9]
p(p) = [1] x1 + [1]
p(s) = [1] x1 + [2]
Following rules are strictly oriented:
minus(s(X),s(Y)) = [1] X + [11]
> [1] X + [10]
= p(minus(X,Y))
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [2]
>= [0]
= 0()
div(s(X),s(Y)) = [1] X + [4]
>= [1] X + [13]
= s(div(minus(X,Y),s(Y)))
minus(X,0()) = [1] X + [9]
>= [1] X + [0]
= X
p(s(X)) = [1] X + [3]
>= [1] X + [0]
= 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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
Weak DP Rules:
Weak TRS Rules:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {div,minus,p}/{0,s}
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(p) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{div,minus,p}
TcT has computed the following interpretation:
p(0) = [1]
p(div) = [4] x1 + [6] x2 + [1]
p(minus) = [1] x1 + [0]
p(p) = [1] x1 + [3]
p(s) = [1] x1 + [3]
Following rules are strictly oriented:
div(s(X),s(Y)) = [4] X + [6] Y + [31]
> [4] X + [6] Y + [22]
= s(div(minus(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [6] Y + [23]
>= [1]
= 0()
minus(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
minus(s(X),s(Y)) = [1] X + [3]
>= [1] X + [3]
= p(minus(X,Y))
p(s(X)) = [1] X + [6]
>= [1] X + [0]
= X
*** 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)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {div,minus,p}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).