Table 2. Definitions of Zsyntax formalization.
Name | Formalized Form | Description |
---|---|---|
Elimination of Z-Conjunction Rule |
⊢ ∀ l x. zsyn_conjun_elimin l x = if MEM x l then [x] else l |
|
Introduction of Z-Conjunction and Z-Interaction |
⊢ ∀ l x y. zsyn_conjun_intro l x y = CONS (FLAT [EL x l; EL y l]) l |
|
Reactants Deletion |
⊢ ∀ l x y. zsyn_delet l x y =
if
x > y then delet (delet l x) y else delet (delet l y) x |
|
Element Deletion |
⊢ ∀ l. delet l 0 = TL l ∧ ∀ l y. delet l (y + 1) = CONS (HD l) ( delet (TL l) y) |
|
EVF Matching |
⊢ ∀ l e x y. zsyn_EVF l e 0 x y = if FST (EL 0 e) = HD l then (T,zsyn_delet (APPEND (TL l) (SND (EL 0 e))) x y) else (F,TL l) ∧ ∀ l e p x y. zsyn_EVF l e (p + 1) x y = if FST (EL (p + 1) e) = HD l then (T,zsyn_delet (APPEND (TL l) (SND (EL (SUC p) e))) x y) else zsyn_EVF l e p x y |
|
Recursive Function to model the argument y in function zsyn_EVF |
⊢ ∀ l e x. zsyn_recurs1 l e x 0 = zsyn_EVF (zsyn_conjun_intro l x 0) e (LENGTH e - 1) x 0 ∧ ∀ l e x y. zsyn_recurs1 l e x (y + 1) = if FST (zsyn_EVF (zsyn_conjun_intro l x (y + 1)) e (LENGTH e - 1) x (y + 1)) ⇔ T then zsyn_EVF (zsyn_conjun_intro l x (y + 1)) e (LENGTH e - 1) x (y + 1) else zsyn_recurs1 l e x y |
|
Recursive Function to model the argument x in function zsyn_EVF |
⊢ ∀ l e y. zsyn_recurs2 l e 0 y = if FST (zsyn_recurs1 l e 0 y) ⇔ T then (T,SND (zsyn_recurs1 l e 0 y)) else (F,SND (zsyn_recurs1 l e 0 y)) ∧ ∀ l e x y. zsyn_recurs2 l e (x + 1) y = if FST (zsyn_recurs1 l e (x + 1) y) ⇔ T then (T,SND (zsyn_recurs1 l e (x + 1) y)) else zsyn_recurs2 l e x (LENGTH l - 1) |
|
Final Recursion Function for Zsyntax |
⊢ ∀ l e x y. zsyn_deduct_recurs l e x y 0 = (T,l) ∧ ∀ l e x y q. zsyn_deduct_recurs l e x y (q + 1) = if FST (zsyn_recurs2 l e x y) ⇔ T then zsyn_deduct_recurs (SND (zsyn_recurs2 l e x y)) e (LENGTH (SND (zsyn_recurs2 l e x y))—1) (LENGTH (SND (zsyn_recurs2 l e x y))—1) q else (T,SND (zsyn_recurs2 l e (LENGTH l - 1) (LENGTH l- 1))) |
|
Final Deduction Function for Zsyntax |
⊢ ∀ l e. zsyn_deduct l e = SND (zsyn_deduct_recurs l e (LENGTH l- 1) (LENGTH l - 1) LENGTH e) |
|