How do multiple rewrites expand into with?

Question

I am a beginner in PLFA, when I read the Induction section, I accidentally wrote a +-swap proof that I can't understand:

+-suc': ∀ (m n: ℕ) → m + suc n ≡ suc (m + n)
+-suc' zero n = refl
+-suc' (suc m) n rewrite +-suc' m n = refl

+-swap: ∀ (m n p: ℕ) → m + (n + p) ≡ n + (m + p)
+-swap zero n p = refl
+-swap (suc m) n p rewrite +-suc' n (m + p) | +-swap m n p = refl

I don't know why this proof is right, so I try to prove it by a chain of equations (which is wrong):

+-swap (suc m) n p =
  begin
    (suc m) + (n + p)
  ≡⟨⟩ n + (suc (m + p))
  ≡⟨ +-suc' n (m + p)⟩
    suc (n + (m + p))
  ≡⟨ cong suc (+-swap m n p)⟩
    n + ((suc m) + p)
  ∎

I know I really don't understand how rewrite works. I learn from the following document that rewrite will expand into with: https://agda.readthedocs.io/en/v2.6.2/language/with-abstraction.html#with-rewrite

But I don't find how rewrite containing | expands in the document. I guess the | in rewrite is also a kind of syntactic sugar:

+-swap (suc m) n p rewrite +-suc' n (m + p) | +-swap m n p = refl

will expand into:

+-swap (suc m) n p rewrite +-suc' n (m + p) rewrite +-swap m n p = refl

I tried to replace the second rewrite with with, no problem:

+-swap (suc m) n p rewrite +-suc' n (m + p)
                   with m + (n + p) | +-swap m n p
...                | .(n + (m + p)) | refl = refl

But if I replaced the first rewrite with with, it gives an error:

+-swap (suc m) n p with n + (suc (m + p)) | +-suc' n (m + p)
...                  | .(suc (n + m + p)) | refl
                   rewrite +-swap m n p = refl

+-swap (suc m) n p with n + (suc (m + p)) | +-suc' n (m + p)
...                  | .(suc (n + m + p)) | refl
                   with m + (n + p)   | +-swap m n p
...                  | .(n + (m + p)) | refl = refl

Error message:

n + m != n of type ℕ
when checking that the given dot pattern suc (n + m + p) matches
the inferred value suc (n + (m + p))

How do multiple rewrites expand into with? How can this proof be accomplished with an equation chain?

You should probably try switching the order of the two `with`s. — Trebor, Oct 15 '21 at 06:15
Also, there's no need to include tags in the title; we can see them below the question anyway. — Trebor, Oct 15 '21 at 06:16
@Trebor Do you mean this way: ``` +-swap (suc m) n p with m + (n + p) | +-swap m n p ... | .(n + (m + p)) | refl with n + (suc (m + p)) | +-suc' n (m + p) ... | .(suc (n + m + p)) | refl = refl ``` But this code can't work neither. — Noah Ma, Oct 15 '21 at 10:46
You missed a pair of parentheses. `n+m+p` should be right-associated instead. — Trebor, Oct 15 '21 at 11:15

How do multiple rewrites expand into with?

0 Answers0