# How to build a proof using structural induction? [duplicate]

I am requested to build a proof for the following I am requested to prove the following statement, however I have no idea on how to begin... Does any of you know how can I do such induction on the structure? Maybe you could give me a link with some similar approach, or give me a text book on this theme...

A predicate logic formula $$\phi$$ belongs to the negative fragment if $$\phi$$ can be constructed from the following grammar, where $$t_1, t_2,..., t_n (n>0)$$ are terms:

$$\phi::= \hspace{2 mm}\neg p(t_1, t_2,..., t_n)\hspace{2 mm} ||\hspace{2 mm}\bot\hspace{2 mm}||\hspace{2 mm}(\neg\phi) \hspace{2 mm} ||\hspace{2 mm}(\phi\land\phi)\hspace{2 mm}||\hspace{2 mm}(\phi\rightarrow\phi)\hspace{2 mm}||\hspace{2 mm}(\forall_x \phi)$$

Prove in minimal logic that $$\neg\neg\theta \vdash_m \theta$$ for any formula $$\theta$$ belonging to the negative fragment. Use induction on the structure of $$\theta$$.

• You can see thus post Apr 26 at 6:03
• @MauroALLEGRANZA, thanks for helping me... I have built a proof which is in the following link math.stackexchange.com/questions/4436851/… Could you please help me further asserting if it is correct? Apr 26 at 16:13