C7X's Structural Array Notations

Epsilon Array Notation Edit

Definition Edit

Use the operator :=:= to represent assignment of equality, and :∈:∈ to represent assignment of membership (i.e., assigning the property of set membership to an object)

Create a set ArAr: 0:∈Ar[A]:∈Ar⟺A∈Ar[A1,A2]:∈Ar⟺A1,A2∈Ar0:∈Ar[A]:∈Ar⟺A∈Ar[A1,A2]:∈Ar⟺A1,A2∈Ar

Create a set Ar2Ar2: [0]:∈Ar2[A]:∈Ar2⟺A∈Ar2[A1,A2]:∈Ar2⟺A1,A2∈Ar2[0]:∈Ar2[A]:∈Ar2⟺A∈Ar2[A1,A2]:∈Ar2⟺A1,A2∈Ar2

Create a set BrBr: [A]:∈Br⟺A∈Ar2[A1,A2]:∈Br⟺A1,A2∈Br[A]:∈Br⟺A∈Ar2[A1,A2]:∈Br⟺A1,A2∈Br

Define ∗(A,n)∗(A,n): ∗(A,1):=A∗(A,n+1):=[∗(A,n),A]∗(A,1):=A∗(A,n+1):=[∗(A,n),A]

Define Reduce(A,n)Reduce(A,n): Reduce([A1,A2],n):=Reduce([A1,Reduce(A2,n)],n)⟺A1,A2∈BrReduce([[A,[0]]],n):=Reduce(∗([A],n),n)⟺A∈ArReduce([A],n):=Reduce([Reduce(A,n)],n)⟺A∈BrReduce(A,n):=A⟺A∈Ar∧¬A∈BrReduce([A1,A2],n):=Reduce([A1,Reduce(A2,n)],n)⟺A1,A2∈BrReduce([[A,[0]]],n):=Reduce(∗([A],n),n)⟺A∈ArReduce([A],n):=Reduce([Reduce(A,n)],n)⟺A∈BrReduce(A,n):=A⟺A∈Ar∧¬A∈Br

Define εm(A,n)εm(A,n): ε1(0,n):=n+1ε1([A,[0]],n):=εn(A,n)⟺A∈Arε1(A,n):=ε1(Reduce(A,n),n)⟺A∈Brεm+1(A,n):=ε1(A,εm(A,n))ε1(0,n):=n+1ε1([A,[0]],n):=εn(A,n)⟺A∈Arε1(A,n):=ε1(Reduce(A,n),n)⟺A∈Brεm+1(A,n):=ε1(A,εm(A,n))

ε(A,n):=ε1(A,n)

Explanation Edit

Explanation here later

Analysis Edit

ε(0,n)≈f(0,n)ε([0],n)≈f(1,n)ε([[0],[0]],n)≈f(2,n)ε([[[0],[0]],[0]],n)≈f(3,n)ε([[0]],n)≈f(ω,n)ε([[[0]],[0]],n)≈f(ω+1,n)ε([[[[0]],[0]],[0]],n)≈f(ω+2,n)ε([[[[[0]],[0]],[0]],[0]],n)≈f(ω+3,n)ε([[[0]],[[0]]],n)≈f(ω2,n)ε([[[[0]],[[0]]],[0]],n)≈f(ω2+1,n)ε([[[[[0]],[[0]]],[0]],[0]],n)≈f(ω2+2,n)ε([[[[0]],[[0]]],[[0]]],n)≈f(ω3,n)ε([[[[[0]],[[0]]],[[0]]],[0]],n)≈f(ω4,n)ε([[[0],[0]]],n)≈f(ω2,n)ε([[[[0],[0]],[0]]],n)≈f(ω3,n)ε([[[0]]],n)≈f(ωω,n)ε([[[[0]]],[0]],n)≈f(ωω+1,n)ε([[[[0]]],[[0]]],n)≈f(ωω+ω,n)ε([[[[0]]],[[[0]]]],n)≈f(ωω2,n)ε([[[[0]]]],n)≈f(ωωω,n)ε([[[[[0]]]]],n)≈f(ωωωω,n)