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So, I am trying to learn Coq using the "Introduction to Computational Logic" script and I have been given an excercise. It is to prove the following: "forall a b, a + S b = S (a + b)". I am given a definition of "nat_ind":

  (p : nat -> Prop)
  (basis : p 0)
  (step : forall n, p n -> p (S n)) :
    forall n, p n := fix f n :=
      match n return p n with
      | 0 => basis
      | S n => step n (f n)
      end.

I have had an attempt at this and solved the issue using this method and it worked:

intros a b. revert a.
  apply (nat_ind(fun a => a + S b = S (a+b))); simpl.
  -reflexivity.
  -intros. f_equal. exact H.

The problem comes now that i have to solve the same issue, but I need to apply the induction lemma immediatelly and can't use intros or induction tactics before i have done that. How do I go about this?

I have tried to remove the first line from my first attempt, but it throws an error: "The reference b was not found in the current environment."

Update: I have gotten somewhere. This is my current code: 

Goal forall a b, a + S b = S (a + b).
Proof. 
  apply nat_ind.
  - intros a b. revert a. 
    apply (nat_ind (fun a => a + S b = S (a + b)));simpl.
      + reflexivity.
      + intros.  f_equal. exact H.
  -intros. revert a. apply (nat_ind (fun a => a + S b = S (a + b))); simpl.
    + reflexivity.
    + intros. f_equal. exact H0.

The last subgoal is just nat, and I don't know how to go about it at all.

Karlo Kampić
  • 11
  • 4

1 Answers1

0

This is the final solution: 

Goal forall a b, a + S b = S (a + b).
Proof. 
  apply nat_ind.
  - intros a b. revert a. 
    apply (nat_ind (fun a => a + S b = S (a + b)));simpl.
      + reflexivity.
      + intros.  f_equal. exact H.
  -intros. revert a. apply (nat_ind (fun a => a + S b = S (a + b))); simpl.
    + reflexivity.
    + intros. f_equal. exact H0.
  - exact O. 
Qed.
Karlo Kampić
  • 11
  • 4