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Extra resources for Proof Theory and Intuitionistic Systems
Example text
W(~R),R(q), free finite by restricting ning of ~_ rule to i n t u i t i o n i s t i e system which special P' by systems or l e s s to the n e w proofs: (x) of the of m o r e . By a d d i n g apply occur more attention tain R(y), can again to a n y to be d e n o t e d is f r e e fied with ZT/I) a series ZT can new use form not q rules to g e n e r a t e a proof does (and where P quite system in that we old ones to of the n e w obtain ~g. ZT one not contai- if w e combine rules we any obtain in a selfexplanatory w a y ZT*/IN, ZTi*/I~ etc.
T! of f o r m u l a s is that one d e s c r i b e d by 1 ~l-formula is a f o r m u l a of the f o r m where as f o l l o w s : contain without " class a) (O~)(Ex)R(~(x)) termined contain for of f o r m u l a s function In o t h e r ~ R is w e l l f o u n d e d . g. as a b b r e v i a t i o n (that for ~Ry R) ~(~R) in p l a c e x parameters° to c o n t a i n N( ~ W(~R) that ~ R ) By %Rg~(x)). The both serves as a b b r e v i a t i o n s by ~R R(x) for ~R~(X)); y I x/R(x)~ (x) ~ R Y A ( x do not for is clear: expression which as a b b r e v i a t i o n x ~ Definition 8: properties: 2) if S/S' S ; x C~ is P, 3) if does is there fied with as Remark: is the does the and free 8 . On 3) P' that P the other of def. S" hand, if P certain S" is free S" ; 4) a proof as if x x S in quanti- P ~ then as quantiS'/S" . ]), which then occurs of inference transform variables if c~ with below infe- endsequent, 5) induction below S/S' , t h e n endsequent satisfies always inference S~ with S it; a quantifier below the in induction SI/S 2 always can of an following sequent P in and any in below an the bound the is or and in not inference has not S2 proof we but S1,/2 variable 8,then replacing sequent S) in variable P it a quantifier but below variable of free of in if free variable inference S2 both occur any S in normal variable not a quantifier induction by occurs quantified in such S and called induction a sequent A pure 2), proof x either x def.