Simplification of the exixtential graph system of Charles Sanders Peirce

2. PEIRCE EXISTENTIAL GRAPH SYSTEM
One axiom
P0 VOID : VOID,
Five rules of inference
P1 even deletion: (gx)->(x)
when (..) is even number of nested [..]
P2 odd insertion: <x>-><gx>
when <..> is odd number of nested [..]
P3 iteration: g[x]->g[gx] (half of GSB generation)
P4 deiteration: g[gx]->g[x] (half of GSB generation)
P5 double cut: [[x]]=x (GSB reflection)

3. SIMPLE EXISTENTIAL GRAPH SYSTEM
One axiom: Consistency
M0 : [x[ x ]]
One rule of inference: Iteration
M1: g[x]->g[gx] .

4. THEOREM P3 ITERATION G[X]->G[GX]
P3 is equal to N1

5. THEOREM P4 DEITERATION
Proof
[g[x][g[ x]]] N0 iconsistency of g[x]
[g[gx][g[x]]] N1 iteration of g
g[gx]->g[x] D1 definition of ->

6. THEOREM P5 DOUBLE CUT [[X]]=X
P5a. [[x]]->x
Proof
[[x] [ ]] N0 consistency of [x]
D1 definition of
P5b. x->[[x]]
proof:
[x [ x ]] N0 indifference of x
[x [ x[x[x]]]] N0 indifference of x
[x [ [ [x]]]] P4 deiteration of x
x->[[x]] D1 definition of ->

7. LEMMA: INVERSION
(X->Y)->([Y]->[X])
Proof
[[x[y]] [y] x ] N0 consistency of x[y]
[[x[y]] [y][[x]] ] P5 double cut
[[x[y]][[[y][[x]]]]] P5 double cut
(x->y)->([y]->[x]) definition

8. LEMMA 2: ADDITION
[[gx[gy]] gx[gy] ] N0 consistency of gx[gx]
[[x[y]] gx[gy] ] P4 deiteration of g
[[x[y]][[gx[gy]]]] P5 double cut
[x[y]]->[gx[gy]] definition

9. 0-DEPTH DELETION
(GX->Y)->(X->Y)
proof
[gx[y][gx[y]]] N0 consistency of gx[y]
[gx[y][ x[y]]] deiteration of g
(gx->y)->(x->y) definition

10. 1-DEPTH INSERTION
[X]->[GX]
proof
gx->x 0-depth deletion
[x]->[gx]] inversion

11. 2-DEPTH DELETION
[X[GY]]->[X[Y]]
proof
[y]->[gy] 1-depth insertion
x[y]->x[gy] addition
[x[gy]]->[x[y]] inversion

12. 3-DEPTH INSERTION
[X[Y[Z]]->[X[Y[GZ]]]
proof
[y[gz]]-> [y[z]] 2-depth deletion
x[y[gz]]->x[y[z]] addition
[x[y[z]]]->[x[y[gz]]] inversion

13. P1 EVEN-DEPTH DELETION
P2 ODD-DEPTH INSERTION
In general
(2n+1)-depth insertion can be proved by inverting of added 2n-depth deletion
2n-depth deletion can be proved by inverting of added (2n-1)-depth insertion

14. THEOREM P0
VOID
Proof
[x[]]=
=[[]] substitution
= VOID double cut