WORST_CASE(?,O(n^1))
* Step 1: NaturalPI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
plus_x#1(0(),x6) -> x6
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(S) = {1},
uargs(comp_f_g) = {2},
uargs(comp_f_g#1) = {2},
uargs(foldr_f#3) = {1},
uargs(plus_x#1) = {2}
Following symbols are considered usable:
{comp_f_g#1,foldr#3,foldr_f#3,main,map#2,plus_x#1}
TcT has computed the following interpretation:
p(0) = 8
p(Cons) = 8 + x2
p(Nil) = 5
p(S) = x1
p(comp_f_g) = x2
p(comp_f_g#1) = 13 + 4*x2
p(foldr#3) = 2
p(foldr_f#3) = 6 + 2*x1 + x2
p(id) = 1
p(main) = 15 + 8*x1
p(map#2) = 2*x1
p(plus_x) = 0
p(plus_x#1) = x2
Following rules are strictly oriented:
comp_f_g#1(plus_x(x3),id(),0()) = 17
> 8
= plus_x#1(x3,0())
foldr#3(Nil()) = 2
> 1
= id()
foldr_f#3(Cons(x16,x5),x24) = 22 + x24 + 2*x5
> 21
= comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) = 24
> 8
= 0()
main(x3) = 15 + 8*x3
> 14 + 4*x3
= foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) = 16 + 2*x6
> 8 + 2*x6
= Cons(plus_x(x16),map#2(x6))
map#2(Nil()) = 10
> 5
= Nil()
Following rules are (at-least) weakly oriented:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) = 13 + 4*x2
>= 13 + 4*x2
= plus_x#1(x3,comp_f_g#1(x1,x2,0()))
foldr#3(Cons(x32,x6)) = 2
>= 2
= comp_f_g(x32,foldr#3(x6))
plus_x#1(0(),x6) = x6
>= x6
= x6
plus_x#1(S(x8),x10) = x10
>= x10
= S(plus_x#1(x8,x10))
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
plus_x#1(0(),x6) -> x6
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Weak TRS:
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(S) = {1},
uargs(comp_f_g) = {2},
uargs(comp_f_g#1) = {2},
uargs(foldr_f#3) = {1},
uargs(plus_x#1) = {2}
Following symbols are considered usable:
{comp_f_g#1,foldr#3,foldr_f#3,main,map#2,plus_x#1}
TcT has computed the following interpretation:
p(0) = [2]
p(Cons) = [1] x2 + [4]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(comp_f_g) = [1] x2 + [0]
p(comp_f_g#1) = [2] x2 + [1] x3 + [14]
p(foldr#3) = [2] x1 + [1]
p(foldr_f#3) = [4] x1 + [1] x2 + [0]
p(id) = [1]
p(main) = [8] x1 + [2]
p(map#2) = [2] x1 + [0]
p(plus_x) = [2]
p(plus_x#1) = [1] x2 + [0]
Following rules are strictly oriented:
foldr#3(Cons(x32,x6)) = [2] x6 + [9]
> [2] x6 + [1]
= comp_f_g(x32,foldr#3(x6))
Following rules are (at-least) weakly oriented:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) = [2] x2 + [16]
>= [2] x2 + [16]
= plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) = [18]
>= [2]
= plus_x#1(x3,0())
foldr#3(Nil()) = [1]
>= [1]
= id()
foldr_f#3(Cons(x16,x5),x24) = [1] x24 + [4] x5 + [16]
>= [1] x24 + [4] x5 + [16]
= comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) = [2]
>= [2]
= 0()
main(x3) = [8] x3 + [2]
>= [8] x3 + [2]
= foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) = [2] x6 + [8]
>= [2] x6 + [4]
= Cons(plus_x(x16),map#2(x6))
map#2(Nil()) = [0]
>= [0]
= Nil()
plus_x#1(0(),x6) = [1] x6 + [0]
>= [1] x6 + [0]
= x6
plus_x#1(S(x8),x10) = [1] x10 + [0]
>= [1] x10 + [0]
= S(plus_x#1(x8,x10))
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
plus_x#1(0(),x6) -> x6
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Weak TRS:
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(S) = {1},
uargs(comp_f_g) = {2},
uargs(comp_f_g#1) = {2},
uargs(foldr_f#3) = {1},
uargs(plus_x#1) = {2}
Following symbols are considered usable:
{comp_f_g#1,foldr#3,foldr_f#3,main,map#2,plus_x#1}
TcT has computed the following interpretation:
p(0) = [1]
p(Cons) = [1] x2 + [2]
p(Nil) = [4]
p(S) = [1] x1 + [0]
p(comp_f_g) = [1] x2 + [2]
p(comp_f_g#1) = [1] x2 + [0]
p(foldr#3) = [4] x1 + [0]
p(foldr_f#3) = [6] x1 + [1]
p(id) = [4]
p(main) = [12] x1 + [1]
p(map#2) = [2] x1 + [0]
p(plus_x) = [1] x1 + [0]
p(plus_x#1) = [1] x2 + [0]
Following rules are strictly oriented:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) = [1] x2 + [2]
> [1] x2 + [0]
= plus_x#1(x3,comp_f_g#1(x1,x2,0()))
Following rules are (at-least) weakly oriented:
comp_f_g#1(plus_x(x3),id(),0()) = [4]
>= [1]
= plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) = [4] x6 + [8]
>= [4] x6 + [2]
= comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) = [16]
>= [4]
= id()
foldr_f#3(Cons(x16,x5),x24) = [6] x5 + [13]
>= [4] x5 + [0]
= comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) = [25]
>= [1]
= 0()
main(x3) = [12] x3 + [1]
>= [12] x3 + [1]
= foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) = [2] x6 + [4]
>= [2] x6 + [2]
= Cons(plus_x(x16),map#2(x6))
map#2(Nil()) = [8]
>= [4]
= Nil()
plus_x#1(0(),x6) = [1] x6 + [0]
>= [1] x6 + [0]
= x6
plus_x#1(S(x8),x10) = [1] x10 + [0]
>= [1] x10 + [0]
= S(plus_x#1(x8,x10))
* Step 4: NaturalPI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
plus_x#1(0(),x6) -> x6
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Weak TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(S) = {1},
uargs(comp_f_g) = {2},
uargs(comp_f_g#1) = {2},
uargs(foldr_f#3) = {1},
uargs(plus_x#1) = {2}
Following symbols are considered usable:
{comp_f_g#1,foldr#3,foldr_f#3,main,map#2,plus_x#1}
TcT has computed the following interpretation:
p(0) = 2
p(Cons) = 8 + x2
p(Nil) = 4
p(S) = x1
p(comp_f_g) = 8 + x2
p(comp_f_g#1) = 12 + x2 + x3
p(foldr#3) = 1 + 2*x1
p(foldr_f#3) = 3*x1 + x2
p(id) = 1
p(main) = 7 + 4*x1
p(map#2) = 1 + x1
p(plus_x) = 1 + x1
p(plus_x#1) = 4 + x2
Following rules are strictly oriented:
plus_x#1(0(),x6) = 4 + x6
> x6
= x6
Following rules are (at-least) weakly oriented:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) = 22 + x2
>= 18 + x2
= plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) = 15
>= 6
= plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) = 17 + 2*x6
>= 9 + 2*x6
= comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) = 9
>= 1
= id()
foldr_f#3(Cons(x16,x5),x24) = 24 + x24 + 3*x5
>= 13 + x24 + 2*x5
= comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) = 14
>= 2
= 0()
main(x3) = 7 + 4*x3
>= 5 + 3*x3
= foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) = 9 + x6
>= 9 + x6
= Cons(plus_x(x16),map#2(x6))
map#2(Nil()) = 5
>= 4
= Nil()
plus_x#1(S(x8),x10) = 4 + x10
>= 4 + x10
= S(plus_x#1(x8,x10))
* Step 5: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Weak TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
plus_x#1(0(),x6) -> x6
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(S) = {1},
uargs(comp_f_g) = {2},
uargs(comp_f_g#1) = {2},
uargs(foldr_f#3) = {1},
uargs(plus_x#1) = {2}
Following symbols are considered usable:
{comp_f_g#1,foldr#3,foldr_f#3,main,map#2,plus_x#1}
TcT has computed the following interpretation:
p(0) = [2]
p(Cons) = [1] x1 + [1] x2 + [4]
p(Nil) = [2]
p(S) = [1] x1 + [4]
p(comp_f_g) = [1] x1 + [1] x2 + [0]
p(comp_f_g#1) = [2] x1 + [2] x2 + [2] x3 + [3]
p(foldr#3) = [2] x1 + [7]
p(foldr_f#3) = [4] x1 + [2] x2 + [1]
p(id) = [0]
p(main) = [8] x1 + [9]
p(map#2) = [2] x1 + [1]
p(plus_x) = [1] x1 + [4]
p(plus_x#1) = [2] x1 + [1] x2 + [0]
Following rules are strictly oriented:
plus_x#1(S(x8),x10) = [1] x10 + [2] x8 + [8]
> [1] x10 + [2] x8 + [4]
= S(plus_x#1(x8,x10))
Following rules are (at-least) weakly oriented:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) = [2] x1 + [2] x2 + [2] x3 + [15]
>= [2] x1 + [2] x2 + [2] x3 + [7]
= plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) = [2] x3 + [15]
>= [2] x3 + [2]
= plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) = [2] x32 + [2] x6 + [15]
>= [1] x32 + [2] x6 + [7]
= comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) = [11]
>= [0]
= id()
foldr_f#3(Cons(x16,x5),x24) = [4] x16 + [2] x24 + [4] x5 + [17]
>= [2] x16 + [2] x24 + [4] x5 + [17]
= comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) = [13]
>= [2]
= 0()
main(x3) = [8] x3 + [9]
>= [8] x3 + [9]
= foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) = [2] x16 + [2] x6 + [9]
>= [1] x16 + [2] x6 + [9]
= Cons(plus_x(x16),map#2(x6))
map#2(Nil()) = [5]
>= [2]
= Nil()
plus_x#1(0(),x6) = [1] x6 + [4]
>= [1] x6 + [0]
= x6
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
comp_f_g#1(plus_x(x3),comp_f_g(x1,x2),0()) -> plus_x#1(x3,comp_f_g#1(x1,x2,0()))
comp_f_g#1(plus_x(x3),id(),0()) -> plus_x#1(x3,0())
foldr#3(Cons(x32,x6)) -> comp_f_g(x32,foldr#3(x6))
foldr#3(Nil()) -> id()
foldr_f#3(Cons(x16,x5),x24) -> comp_f_g#1(x16,foldr#3(x5),x24)
foldr_f#3(Nil(),0()) -> 0()
main(x3) -> foldr_f#3(map#2(x3),0())
map#2(Cons(x16,x6)) -> Cons(plus_x(x16),map#2(x6))
map#2(Nil()) -> Nil()
plus_x#1(0(),x6) -> x6
plus_x#1(S(x8),x10) -> S(plus_x#1(x8,x10))
- Signature:
{comp_f_g#1/3,foldr#3/1,foldr_f#3/2,main/1,map#2/1,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,comp_f_g/2,id/0
,plus_x/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {comp_f_g#1,foldr#3,foldr_f#3,main,map#2
,plus_x#1} and constructors {0,Cons,Nil,S,comp_f_g,id,plus_x}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))