*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Weak DP Rules:
Weak TRS Rules:
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{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(if_mod) = {1},
uargs(mod) = {1}
Following symbols are considered usable:
{if_mod,le,minus,mod}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_mod) = [2] x1 + [2] x2 + [1]
p(le) = [0]
p(minus) = [1] x1 + [0]
p(mod) = [2] x1 + [1]
p(s) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
if_mod(false(),s(x),s(y)) = [2] x + [17]
> [1] x + [8]
= s(x)
if_mod(true(),s(x),s(y)) = [2] x + [17]
> [2] x + [1]
= mod(minus(x,y),s(y))
minus(s(x),s(y)) = [1] x + [8]
> [1] x + [0]
= minus(x,y)
mod(0(),y) = [1]
> [0]
= 0()
mod(s(x),0()) = [2] x + [17]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
mod(s(x),s(y)) = [2] x + [17]
>= [2] x + [17]
= if_mod(le(y,x),s(x),s(y))
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Weak DP Rules:
Weak TRS Rules:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{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(if_mod) = {1},
uargs(mod) = {1}
Following symbols are considered usable:
{if_mod,le,minus,mod}
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [1]
p(if_mod) = [8] x1 + [4] x2 + [1] x3 + [1]
p(le) = [1]
p(minus) = [1] x1 + [1]
p(mod) = [4] x1 + [1] x2 + [13]
p(s) = [1] x1 + [2]
p(true) = [1]
Following rules are strictly oriented:
minus(x,0()) = [1] x + [1]
> [1] x + [0]
= x
mod(s(x),s(y)) = [4] x + [1] y + [23]
> [4] x + [1] y + [19]
= if_mod(le(y,x),s(x),s(y))
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = [4] x + [1] y + [19]
>= [1] x + [2]
= s(x)
if_mod(true(),s(x),s(y)) = [4] x + [1] y + [19]
>= [4] x + [1] y + [19]
= mod(minus(x,y),s(y))
le(0(),y) = [1]
>= [1]
= true()
le(s(x),0()) = [1]
>= [1]
= false()
le(s(x),s(y)) = [1]
>= [1]
= le(x,y)
minus(s(x),s(y)) = [1] x + [3]
>= [1] x + [1]
= minus(x,y)
mod(0(),y) = [1] y + [21]
>= [2]
= 0()
mod(s(x),0()) = [4] x + [23]
>= [2]
= 0()
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{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(if_mod) = {1},
uargs(mod) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [0]
p(if_mod) = [1] x1 + [1] x2 + [0]
p(le) = [2]
p(minus) = [1] x1 + [4]
p(mod) = [1] x1 + [2]
p(s) = [1] x1 + [0]
p(true) = [6]
Following rules are strictly oriented:
le(s(x),0()) = [2]
> [0]
= false()
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = [1] x + [0]
>= [1] x + [0]
= s(x)
if_mod(true(),s(x),s(y)) = [1] x + [6]
>= [1] x + [6]
= mod(minus(x,y),s(y))
le(0(),y) = [2]
>= [6]
= true()
le(s(x),s(y)) = [2]
>= [2]
= le(x,y)
minus(x,0()) = [1] x + [4]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [4]
>= [1] x + [4]
= minus(x,y)
mod(0(),y) = [4]
>= [2]
= 0()
mod(s(x),0()) = [1] x + [2]
>= [2]
= 0()
mod(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= if_mod(le(y,x),s(x),s(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:
le(0(),y) -> true()
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(s(x),0()) -> false()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{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(if_mod) = {1},
uargs(mod) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [1]
p(if_mod) = [1] x1 + [1] x2 + [4]
p(le) = [1]
p(minus) = [1] x1 + [0]
p(mod) = [1] x1 + [5]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
le(0(),y) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = [1] x + [6]
>= [1] x + [1]
= s(x)
if_mod(true(),s(x),s(y)) = [1] x + [5]
>= [1] x + [5]
= mod(minus(x,y),s(y))
le(s(x),0()) = [1]
>= [1]
= false()
le(s(x),s(y)) = [1]
>= [1]
= le(x,y)
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [1]
>= [1] x + [0]
= minus(x,y)
mod(0(),y) = [5]
>= [0]
= 0()
mod(s(x),0()) = [1] x + [6]
>= [0]
= 0()
mod(s(x),s(y)) = [1] x + [6]
>= [1] x + [6]
= if_mod(le(y,x),s(x),s(y))
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:
le(s(x),s(y)) -> le(x,y)
Weak DP Rules:
Weak TRS Rules:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{0,false,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(if_mod) = {1},
uargs(mod) = {1}
Following symbols are considered usable:
{if_mod,le,minus,mod}
TcT has computed the following interpretation:
p(0) = [0]
[1]
p(false) = [2]
[1]
p(if_mod) = [1 4] x1 + [1 0] x2 + [3
3] x3 + [3]
[0 2] [0 4] [0
0] [2]
p(le) = [0 1] x2 + [2]
[0 0] [1]
p(minus) = [1 0] x1 + [0]
[0 1] [0]
p(mod) = [1 1] x1 + [3 3] x2 + [7]
[0 4] [0 0] [5]
p(s) = [1 3] x1 + [0]
[0 1] [2]
p(true) = [0]
[1]
Following rules are strictly oriented:
le(s(x),s(y)) = [0 1] y + [4]
[0 0] [1]
> [0 1] y + [2]
[0 0] [1]
= le(x,y)
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = [1 3] x + [3 12] y + [15]
[0 4] [0 0] [12]
>= [1 3] x + [0]
[0 1] [2]
= s(x)
if_mod(true(),s(x),s(y)) = [1 3] x + [3 12] y + [13]
[0 4] [0 0] [12]
>= [1 1] x + [3 12] y + [13]
[0 4] [0 0] [5]
= mod(minus(x,y),s(y))
le(0(),y) = [0 1] y + [2]
[0 0] [1]
>= [0]
[1]
= true()
le(s(x),0()) = [3]
[1]
>= [2]
[1]
= false()
minus(x,0()) = [1 0] x + [0]
[0 1] [0]
>= [1 0] x + [0]
[0 1] [0]
= x
minus(s(x),s(y)) = [1 3] x + [0]
[0 1] [2]
>= [1 0] x + [0]
[0 1] [0]
= minus(x,y)
mod(0(),y) = [3 3] y + [8]
[0 0] [9]
>= [0]
[1]
= 0()
mod(s(x),0()) = [1 4] x + [12]
[0 4] [13]
>= [0]
[1]
= 0()
mod(s(x),s(y)) = [1 4] x + [3 12] y + [15]
[0 4] [0 0] [13]
>= [1 4] x + [3 12] y + [15]
[0 4] [0 0] [12]
= if_mod(le(y,x),s(x),s(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:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Innermost
basic terms: {if_mod,le,minus,mod}/{0,false,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).