*** 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:
Full
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) = [0]
p(if_mod) = [1] x1 + [1] x2 + [1] x3 + [0]
p(le) = [11]
p(minus) = [1] x1 + [1]
p(mod) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
if_mod(false(),s(x),s(y)) = [1] x + [1] y + [16]
> [1] x + [8]
= s(x)
if_mod(true(),s(x),s(y)) = [1] x + [1] y + [16]
> [1] x + [1] y + [10]
= mod(minus(x,y),s(y))
le(0(),y) = [11]
> [0]
= true()
le(s(x),0()) = [11]
> [0]
= false()
minus(x,0()) = [1] x + [1]
> [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [9]
> [1] x + [1]
= minus(x,y)
mod(0(),y) = [1] y + [1]
> [0]
= 0()
mod(s(x),0()) = [1] x + [9]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
le(s(x),s(y)) = [11]
>= [11]
= le(x,y)
mod(s(x),s(y)) = [1] x + [1] y + [17]
>= [1] x + [1] y + [27]
= 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 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
le(s(x),s(y)) -> le(x,y)
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))
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()
Signature:
{if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
Obligation:
Full
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) = [4]
p(false) = [0]
p(if_mod) = [1] x1 + [1] x2 + [0]
p(le) = [0]
p(minus) = [1] x1 + [1]
p(mod) = [1] x1 + [2]
p(s) = [1] x1 + [4]
p(true) = [0]
Following rules are strictly oriented:
mod(s(x),s(y)) = [1] x + [6]
> [1] x + [4]
= if_mod(le(y,x),s(x),s(y))
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = [1] x + [4]
>= [1] x + [4]
= s(x)
if_mod(true(),s(x),s(y)) = [1] x + [4]
>= [1] x + [3]
= mod(minus(x,y),s(y))
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 + [1]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [5]
>= [1] x + [1]
= minus(x,y)
mod(0(),y) = [6]
>= [4]
= 0()
mod(s(x),0()) = [1] x + [6]
>= [4]
= 0()
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:
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:
Full
basic terms: {if_mod,le,minus,mod}/{0,false,s,true}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
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) = 0
p(if_mod) = 1 + x1 + 7*x2 + 7*x2*x3
p(le) = x1
p(minus) = x1
p(mod) = 7*x1 + 7*x1*x2 + x2
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
le(s(x),s(y)) = 1 + x
> x
= le(x,y)
Following rules are (at-least) weakly oriented:
if_mod(false(),s(x),s(y)) = 15 + 14*x + 7*x*y + 7*y
>= 1 + x
= s(x)
if_mod(true(),s(x),s(y)) = 15 + 14*x + 7*x*y + 7*y
>= 1 + 14*x + 7*x*y + y
= mod(minus(x,y),s(y))
le(0(),y) = 0
>= 0
= true()
le(s(x),0()) = 1 + x
>= 0
= false()
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 1 + x
>= x
= minus(x,y)
mod(0(),y) = y
>= 0
= 0()
mod(s(x),0()) = 7 + 7*x
>= 0
= 0()
mod(s(x),s(y)) = 15 + 14*x + 7*x*y + 8*y
>= 15 + 14*x + 7*x*y + 8*y
= if_mod(le(y,x),s(x),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:
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:
Full
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).