Hi guys

I am a beginner at FOL, trying to prove sentence1 implies sentence 2. I have followed some steps, but I am not able to formally write a proof. Please see and correct.

sentence 1

$\displaystyle \forall $ CPlusPlus program C$\displaystyle \exists $ Turning Machine Mc$\displaystyle \forall$ Instance I Mc(I)=C(I)

sentence 2

$\displaystyle \forall $ Problem P [

$\displaystyle \exists $ CPlusPlus program Cp$\displaystyle \forall$ Instance I Cp(I)=P(I)

$\displaystyle \implies$

$\displaystyle \exists $ Turning Machine Mp$\displaystyle \forall$ Instance I Mp(I)=P(I) ]

So first I assume sentence 1 is right. My first question is what should I note after this assumption? How to state the facts?

Now I say

for arbitrary problem P, it is true that

$\displaystyle \exists $ CPlusPlus program Cp$\displaystyle \forall$ Instance I Cp(I)=P(I)

and prove

$\displaystyle \exists $ Turning Machine Mp$\displaystyle \forall$ Instance I Mp(I)=P(I)

So I say something like

Let Mp1 be a Turning machine for arbitrary instance I. then we have assumed Cp(I)=P(I) from sentence 2 part 1, and from sentence 1 I know Cp(I)=Mp1(I) therefore we have Mp1(I)=P(I)

Could you help me write this formally?

Thank you so mcuh.

Jungar

Thanks

Jungar