* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
=(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
=(.(x,y),nil()) -> false()
=(nil(),.(y,z)) -> false()
=(nil(),nil()) -> true()
del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
f(false(),x,y,z) -> .(x,del(.(y,z)))
f(true(),x,y,z) -> del(.(y,z))
- Signature:
{=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
+ 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(.) = {2},
uargs(f) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x2 + [0]
p(=) = [0]
p(and) = [1] x1 + [1] x2 + [0]
p(del) = [0]
p(f) = [1] x1 + [5]
p(false) = [0]
p(nil) = [0]
p(true) = [0]
p(u) = [0]
p(v) = [0]
Following rules are strictly oriented:
f(false(),x,y,z) = [5]
> [0]
= .(x,del(.(y,z)))
f(true(),x,y,z) = [5]
> [0]
= del(.(y,z))
Following rules are (at-least) weakly oriented:
=(.(x,y),.(u(),v())) = [0]
>= [0]
= and(=(x,u()),=(y,v()))
=(.(x,y),nil()) = [0]
>= [0]
= false()
=(nil(),.(y,z)) = [0]
>= [0]
= false()
=(nil(),nil()) = [0]
>= [0]
= true()
del(.(x,.(y,z))) = [0]
>= [5]
= f(=(x,y),x,y,z)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
=(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
=(.(x,y),nil()) -> false()
=(nil(),.(y,z)) -> false()
=(nil(),nil()) -> true()
del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
- Weak TRS:
f(false(),x,y,z) -> .(x,del(.(y,z)))
f(true(),x,y,z) -> del(.(y,z))
- Signature:
{=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
+ 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(.) = {2},
uargs(f) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [4]
p(=) = [1] x1 + [0]
p(and) = [0]
p(del) = [2] x1 + [1]
p(f) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [9]
p(false) = [4]
p(nil) = [8]
p(true) = [0]
p(u) = [8]
p(v) = [0]
Following rules are strictly oriented:
=(.(x,y),.(u(),v())) = [1] x + [1] y + [4]
> [0]
= and(=(x,u()),=(y,v()))
=(nil(),.(y,z)) = [8]
> [4]
= false()
=(nil(),nil()) = [8]
> [0]
= true()
del(.(x,.(y,z))) = [2] x + [2] y + [2] z + [17]
> [2] x + [2] y + [2] z + [9]
= f(=(x,y),x,y,z)
Following rules are (at-least) weakly oriented:
=(.(x,y),nil()) = [1] x + [1] y + [4]
>= [4]
= false()
f(false(),x,y,z) = [1] x + [2] y + [2] z + [13]
>= [1] x + [2] y + [2] z + [13]
= .(x,del(.(y,z)))
f(true(),x,y,z) = [1] x + [2] y + [2] z + [9]
>= [2] y + [2] z + [9]
= del(.(y,z))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
=(.(x,y),nil()) -> false()
- Weak TRS:
=(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
=(nil(),.(y,z)) -> false()
=(nil(),nil()) -> true()
del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
f(false(),x,y,z) -> .(x,del(.(y,z)))
f(true(),x,y,z) -> del(.(y,z))
- Signature:
{=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
+ 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(.) = {2},
uargs(f) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [1]
p(=) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [1]
p(del) = [2] x1 + [2]
p(f) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [6]
p(false) = [0]
p(nil) = [0]
p(true) = [0]
p(u) = [1]
p(v) = [0]
Following rules are strictly oriented:
=(.(x,y),nil()) = [1] x + [1] y + [1]
> [0]
= false()
Following rules are (at-least) weakly oriented:
=(.(x,y),.(u(),v())) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= and(=(x,u()),=(y,v()))
=(nil(),.(y,z)) = [0]
>= [0]
= false()
=(nil(),nil()) = [0]
>= [0]
= true()
del(.(x,.(y,z))) = [2] x + [2] y + [2] z + [6]
>= [2] x + [2] y + [2] z + [6]
= f(=(x,y),x,y,z)
f(false(),x,y,z) = [1] x + [2] y + [2] z + [6]
>= [1] x + [2] y + [2] z + [5]
= .(x,del(.(y,z)))
f(true(),x,y,z) = [1] x + [2] y + [2] z + [6]
>= [2] y + [2] z + [4]
= del(.(y,z))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
=(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
=(.(x,y),nil()) -> false()
=(nil(),.(y,z)) -> false()
=(nil(),nil()) -> true()
del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
f(false(),x,y,z) -> .(x,del(.(y,z)))
f(true(),x,y,z) -> del(.(y,z))
- Signature:
{=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))