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	MEM x l: True if x is a member of list 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	FLAT l: Flatten a list of lists l to a single list EL y l: yth element of list l CONS: Adds a new element to the top of the list
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	delet l x: Deletes the element at index x of the list l
Element Deletion	⊢ ∀ l. delet l 0 = TL l ∧   ∀ l y. delet l (y + 1) =     CONS (HD l) ( delet (TL l) y)	HD l: Head element of list l TL l: Tail of list l
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	FST: First component of a pair SND: Second component of a pair APPEND: Merges two lists zsyn_delet: Reactants deletion
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	LENGTH e: Length of list e zsyn_EVF: EVF Matching zsyn_conjun_intro: Introduction of Z-Conjunction and Z-Interaction
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)	zsyn_recurs1: Recursive function to model the augment y in zsyn_EVF
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)))	zsyn_recurs2: Recursive function to model the augment x in zsyn_EVF
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)	zsyn_deduct_recurs: Recursive Function for calling zsyn_EVF