*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{minus/2,plus/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,plus,quot}/{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(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(minus) = [1] x1 + [2]
p(plus) = [6] x1 + [6] x2 + [0]
p(quot) = [1] x1 + [4] x2 + [4]
p(s) = [1] x1 + [4]
Following rules are strictly oriented:
minus(x,0()) = [1] x + [2]
> [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [6]
> [1] x + [2]
= minus(x,y)
plus(0(),y) = [6] y + [24]
> [1] y + [0]
= y
plus(s(x),y) = [6] x + [6] y + [24]
> [6] x + [6] y + [4]
= s(plus(x,y))
quot(0(),s(y)) = [4] y + [24]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
plus(minus(x,s(0())) = [6] x + [6] y + [24]
,minus(y,s(s(z))))
>= [6] x + [6] y + [24]
= plus(minus(y,s(s(z)))
,minus(x,s(0())))
quot(s(x),s(y)) = [1] x + [4] y + [24]
>= [1] x + [4] y + [26]
= s(quot(minus(x,y),s(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:
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
Signature:
{minus/2,plus/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,plus,quot}/{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(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{minus,plus,quot}
TcT has computed the following interpretation:
p(0) = [3]
p(minus) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(quot) = [2] x1 + [2]
p(s) = [1] x1 + [8]
Following rules are strictly oriented:
quot(s(x),s(y)) = [2] x + [18]
> [2] x + [10]
= s(quot(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [8]
>= [1] x + [0]
= minus(x,y)
plus(0(),y) = [1] y + [3]
>= [1] y + [0]
= y
plus(minus(x,s(0())) = [1] x + [1] y + [0]
,minus(y,s(s(z))))
>= [1] x + [1] y + [0]
= plus(minus(y,s(s(z)))
,minus(x,s(0())))
plus(s(x),y) = [1] x + [1] y + [8]
>= [1] x + [1] y + [8]
= s(plus(x,y))
quot(0(),s(y)) = [8]
>= [3]
= 0()
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Signature:
{minus/2,plus/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,plus,quot}/{0,s}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, 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(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{minus,plus,quot}
TcT has computed the following interpretation:
p(0) = [0]
[1]
[0]
p(minus) = [1 0 0] [0 0 0] [0]
[1 1 1] x1 + [0 0 0] x2 + [0]
[0 1 1] [1 0 0] [1]
p(plus) = [0 1 0] [1 0 1] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 1 0] [1 0 1] [0]
p(quot) = [1 0 0] [0]
[1 0 0] x1 + [1]
[0 0 0] [0]
p(s) = [1 0 0] [1]
[0 1 0] x1 + [1]
[0 0 1] [0]
Following rules are strictly oriented:
plus(minus(x,s(0())) = [1 1 1] [1 1 1] [1 0
,minus(y,s(s(z)))) 0] [3]
[1 1 1] x + [1 1 1] y + [0 0
0] z + [1]
[1 1 1] [1 1 1] [1 0
0] [3]
> [1 1 1] [1 1 1] [2]
[1 1 1] x + [1 1 1] y + [1]
[1 1 1] [1 1 1] [2]
= plus(minus(y,s(s(z)))
,minus(x,s(0())))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1 0 0] [0]
[1 1 1] x + [0]
[0 1 1] [1]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
minus(s(x),s(y)) = [1 0 0] [0 0 0] [1]
[1 1 1] x + [0 0 0] y + [2]
[0 1 1] [1 0 0] [3]
>= [1 0 0] [0 0 0] [0]
[1 1 1] x + [0 0 0] y + [0]
[0 1 1] [1 0 0] [1]
= minus(x,y)
plus(0(),y) = [1 0 1] [1]
[0 1 0] y + [2]
[1 0 1] [1]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= y
plus(s(x),y) = [0 1 0] [1 0 1] [1]
[0 1 0] x + [0 1 0] y + [2]
[0 1 0] [1 0 1] [1]
>= [0 1 0] [1 0 1] [1]
[0 1 0] x + [0 1 0] y + [2]
[0 1 0] [1 0 1] [0]
= s(plus(x,y))
quot(0(),s(y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= 0()
quot(s(x),s(y)) = [1 0 0] [1]
[1 0 0] x + [2]
[0 0 0] [0]
>= [1 0 0] [1]
[1 0 0] x + [2]
[0 0 0] [0]
= s(quot(minus(x,y),s(y)))
*** 1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
Signature:
{minus/2,plus/2,quot/2} / {0/0,s/1}
Obligation:
Innermost
basic terms: {minus,plus,quot}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).