C7X's Structural Array Notations

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)

Epsilon Array Notation Edit

Definition Edit

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

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

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

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

Define Reduce(A,n):Ar,N↦Ar: 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∈Br

Define εm(A,n):Ar,N↦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

WIP

Gamma Array Notation Edit

Definition Edit

WIP

Create a set Ar: 0:∈Ar[A1,A2]:∈Ar⟺A1,A2∈Ar{A1,A2}:∈Ar⟺A1,A2∈Ar

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

Create a set Br: [A1,A2]:∈Br⟺A1,A2∈Br{0,A}:∈Br⟺A∈Ar2{A1,A2}:∈Br⟺A1∈Ar2∧A2∈Ar

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

Define Γm(A,n): Γ1(0,n):=n+1Γ1([A,[0]],n):=Γn(A,n)⟺A∈ArΓ1(A,n):=Γ1(Reduce(A,n),n)⟺A∈BrΓm+1(A,n):=Γ1(A,Γm(A,n))

Γ(A,n):=Γ1(A,n)

Explanation Edit

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Analysis Edit

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