MAYBE
* Step 1: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
fac(x) -> help(x,0())
help(x,c) -> if(lt(c,x),x,c)
if(false(),x,c) -> s(0())
if(true(),x,c) -> times(s(c),help(x,s(c)))
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{fac/1,help/2,if/3,lt/2} / {0/0,false/0,s/1,times/2,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,help,if,lt} and constructors {0,false,s,times,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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) = {1},
uargs(times) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(fac) = [2] x1 + [2]
p(false) = [15]
p(help) = [2] x1 + [4]
p(if) = [1] x1 + [2] x2 + [1]
p(lt) = [2]
p(s) = [0]
p(times) = [1] x1 + [1] x2 + [2]
p(true) = [7]
Following rules are strictly oriented:
help(x,c) = [2] x + [4]
> [2] x + [3]
= if(lt(c,x),x,c)
if(false(),x,c) = [2] x + [16]
> [0]
= s(0())
if(true(),x,c) = [2] x + [8]
> [2] x + [6]
= times(s(c),help(x,s(c)))
Following rules are (at-least) weakly oriented:
fac(x) = [2] x + [2]
>= [2] x + [4]
= help(x,0())
lt(x,0()) = [2]
>= [15]
= false()
lt(0(),s(x)) = [2]
>= [7]
= true()
lt(s(x),s(y)) = [2]
>= [2]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
fac(x) -> help(x,0())
lt(x,0()) -> false()
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
help(x,c) -> if(lt(c,x),x,c)
if(false(),x,c) -> s(0())
if(true(),x,c) -> times(s(c),help(x,s(c)))
- Signature:
{fac/1,help/2,if/3,lt/2} / {0/0,false/0,s/1,times/2,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,help,if,lt} and constructors {0,false,s,times,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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) = {1},
uargs(times) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [8]
p(fac) = [1] x1 + [8]
p(false) = [0]
p(help) = [1] x1 + [2] x2 + [12]
p(if) = [1] x1 + [1] x2 + [2] x3 + [10]
p(lt) = [2]
p(s) = [1] x1 + [2]
p(times) = [1] x2 + [0]
p(true) = [6]
Following rules are strictly oriented:
lt(x,0()) = [2]
> [0]
= false()
Following rules are (at-least) weakly oriented:
fac(x) = [1] x + [8]
>= [1] x + [28]
= help(x,0())
help(x,c) = [2] c + [1] x + [12]
>= [2] c + [1] x + [12]
= if(lt(c,x),x,c)
if(false(),x,c) = [2] c + [1] x + [10]
>= [10]
= s(0())
if(true(),x,c) = [2] c + [1] x + [16]
>= [2] c + [1] x + [16]
= times(s(c),help(x,s(c)))
lt(0(),s(x)) = [2]
>= [6]
= true()
lt(s(x),s(y)) = [2]
>= [2]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
fac(x) -> help(x,0())
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
help(x,c) -> if(lt(c,x),x,c)
if(false(),x,c) -> s(0())
if(true(),x,c) -> times(s(c),help(x,s(c)))
lt(x,0()) -> false()
- Signature:
{fac/1,help/2,if/3,lt/2} / {0/0,false/0,s/1,times/2,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,help,if,lt} and constructors {0,false,s,times,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
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) = {1},
uargs(times) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(fac) = [9] x1 + [9]
p(false) = [2]
p(help) = [2] x1 + [4] x2 + [8]
p(if) = [1] x1 + [2] x2 + [4] x3 + [1]
p(lt) = [2]
p(s) = [1] x1 + [1]
p(times) = [1] x2 + [2]
p(true) = [15]
Following rules are strictly oriented:
fac(x) = [9] x + [9]
> [2] x + [8]
= help(x,0())
Following rules are (at-least) weakly oriented:
help(x,c) = [4] c + [2] x + [8]
>= [4] c + [2] x + [3]
= if(lt(c,x),x,c)
if(false(),x,c) = [4] c + [2] x + [3]
>= [1]
= s(0())
if(true(),x,c) = [4] c + [2] x + [16]
>= [4] c + [2] x + [14]
= times(s(c),help(x,s(c)))
lt(x,0()) = [2]
>= [2]
= false()
lt(0(),s(x)) = [2]
>= [15]
= true()
lt(s(x),s(y)) = [2]
>= [2]
= lt(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: Failure MAYBE
+ Considered Problem:
- Strict TRS:
lt(0(),s(x)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
fac(x) -> help(x,0())
help(x,c) -> if(lt(c,x),x,c)
if(false(),x,c) -> s(0())
if(true(),x,c) -> times(s(c),help(x,s(c)))
lt(x,0()) -> false()
- Signature:
{fac/1,help/2,if/3,lt/2} / {0/0,false/0,s/1,times/2,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac,help,if,lt} and constructors {0,false,s,times,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE