171 lines
4.5 KiB
Markdown
171 lines
4.5 KiB
Markdown
---
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title : "Formally proving true ≢ false in cubical Agda"
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slug : "proving-true-from-false"
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date : 2023-04-21
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tags : ["type-theory", "agda"]
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math : true
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---
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<details>
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<summary>Imports</summary>
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These are some imports that are required for code on this page to work properly.
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```agda
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{-# OPTIONS --cubical #-}
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open import Cubical.Foundations.Prelude
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open import Data.Bool
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open import Data.Unit
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open import Data.Empty
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¬_ : Set → Set
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¬ A = A → ⊥
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infix 4 _≢_
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_≢_ : ∀ {A : Set} → A → A → Set
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x ≢ y = ¬ (x ≡ y)
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```
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</details>
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The other day, I was trying to prove `true ≢ false` in Agda. I would write the
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statement like this:
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```
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true≢false : true ≢ false
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```
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For many "obvious" statements, it suffices to just write `refl` since the two
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sides are trivially true via rewriting. For example:
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```
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open import Data.Nat
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1+2≡3 : 1 + 2 ≡ 3
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1+2≡3 = refl
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```
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This is saying that using the way addition is defined, we can just rewrite the
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left side so it becomes judgmentally equal to the right:
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```
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-- For convenience, here's the definition of addition:
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-- _+_ : Nat → Nat → Nat
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-- zero + m = m
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-- suc n + m = suc (n + m)
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```
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- 1 + 2
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- suc zero + suc (suc zero)
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- suc (zero + suc (suc zero))
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- suc (suc (suc zero))
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- 3
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However, in cubical Agda, naively using `refl` with the inverse statement
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doesn't work. I've commented it out so the code on this page can continue to
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compile.
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```
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-- true≢false = refl
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```
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It looks like it's not obvious to the interpreter that this statement is
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actually true. Why is that
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## Intuition
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Well, in constructive logic / constructive type theory, proving something is
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false is actually a bit different. You see, the definition of the `not`
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operator, or $\neg$, was:
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```
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-- ¬_ : Set → Set
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-- ¬ A = A → ⊥
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```
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> The code is commented out because it was already defined at the top of the
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> page in order for the code to compile.
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This roughly translates to, "give me any proof of A, and I'll produce a value of
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the bottom type." Since the bottom type $\bot$ is a type without values, being
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able to produce a value represents logical falsehood. So we're looking for a way
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to ensure that any proof of `true ≢ false` must lead to $\bot$.
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The strategy here is we define some kind of "type-map". Every time we see
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`true`, we'll map it to some arbitrary inhabited type, and every time we see
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`false`, we'll map it to empty.
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```
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bool-map : Bool → Type
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bool-map true = ⊤
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bool-map false = ⊥
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```
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This way, we can use the fact that transporting
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over a path (the path supposedly given to us as the witness that true ≢ false)
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will produce a function from the inhabited type we chose to the empty type!
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```
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true≢false p = transport (λ i → bool-map (p i)) tt
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```
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I used `true` here, but I could equally have just used anything else:
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```
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bool-map2 : Bool → Type
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bool-map2 true = 1 ≡ 1
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bool-map2 false = ⊥
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true≢false2 : true ≢ false
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true≢false2 p = transport (λ i → bool-map2 (p i)) refl
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```
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## Note on proving divergence on equivalent values
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Let's make sure this isn't broken by trying to apply this to something that's
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actually true:
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```
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data NotBool : Type where
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true1 : NotBool
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true2 : NotBool
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same : true1 ≡ true2
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```
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In this data type, we have a path over `true1` and `true2` that is a part of the
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definition of the `NotBool` type. Since this is an intrinsic equality, we can't
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map `true1` and `true2` to divergent types. Let's see what happens:
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```
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notbool-map : NotBool → Type
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notbool-map true1 = ⊤
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notbool-map true2 = ⊥
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```
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Ok, we've defined the same thing that we did before, but Agda gives us this
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message:
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```text
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Errors:
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Incomplete pattern matching for notbool-map. Missing cases:
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notbool-map (same i)
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when checking the definition of notbool-map
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```
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Agda helpfully notes that we still have another case in the inductive type to
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pattern match on. Let's just go ahead and give it some value:
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```text
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notbool-map (same i) = ⊤
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```
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If you give it `⊤`, it will complain that `⊥ != ⊤ of type Type`, but if you give
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it `⊥`, it will also complain! Because pattern matching on higher inductive
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types requires a functor over the path, we must provide a function that
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satisfies the equality `notbool-map true1 ≡ notbool-map true2`, which unless we
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have provided the same type to both, will not be possible.
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So in the end, this type `NotBool → Type` is only possible to write if the two
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types we mapped `true1` and `true2` can be proven equivalent. But this also
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means we can't use it to prove `true1 ≢ true2`, which is exactly the property we
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wanted to begin with.
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