[Agda] Agda with the excluded middle is inconsistent ?

[Chung Kil Hur]

Hi everyone,

I proved the absurdity in Agda assuming the excluded middle.
Is it a well-known fact ?
It seems that Agda's set theory is weird.

This comes from the injectivity of inductive type constructors.

The proof sketch is as follows.

Define a family of inductive type

data I : (Set -> Set) -> Set where

with no constructors.

By injectivity of type constructors, I can show that I : (Set -> Set) -> Set is injective.

As you may see, there is a size problem with this injectivity.

So, I just used the cantor's diagonalization to derive absurdity in a classical way.

Does anyone know whether cantor's diagonalization essentially needs the classical axiom, or can be done intuitionistically ?
If the latter is true, then the Agda system is inconsistent.

Please have a look at the Agda code below, and let me know if there's any mistakes.

All comments are wellcome.

Thanks,
Chung-Kil Hur


============== Agda code ===============

module cantor where

  data Empty : Set where

  data One : Set where
    one : One

  data coprod (A : Set1) (B : Set1) : Set1 where
    inl : ∀ (a : A) -> coprod A B
    inr : ∀ (b : B) -> coprod A B 

  postulate exmid : ∀ (A : Set1) -> coprod A (A -> Empty)

  data Eq1 {A : Set1} (x : A) : A -> Set1 where
    refleq1 : Eq1 x x

  cast : ∀ { A B } -> Eq1 A B -> A -> B
  cast refleq1 a = a

  Eq1cong : ∀ {f g : Set -> Set} a -> Eq1 f g -> Eq1 (f a) (g a)
  Eq1cong a refleq1 = refleq1

  Eq1sym : ∀ {A : Set1} { x y : A } -> Eq1 x y -> Eq1 y x
  Eq1sym refleq1 = refleq1

  Eq1trans : ∀ {A : Set1} { x y z : A } -> Eq1 x y -> Eq1 y z -> Eq1 x z
  Eq1trans refleq1 refleq1 = refleq1

  data I : (Set -> Set) -> Set where

  Iinj : ∀ { x y : Set -> Set } -> Eq1 (I x) (I y) -> Eq1 x y 
  Iinj refleq1 = refleq1

  data invimageI (a : Set) : Set1 where
    invelmtI : forall x -> (Eq1 (I x) a) -> invimageI a

  J : Set -> (Set -> Set)
  J a with exmid (invimageI a)
  J a | inl (invelmtI x y) = x
  J a | inr b = λ x → Empty

  data invimageJ (x : Set -> Set) : Set1 where
    invelmtJ : forall a -> (Eq1 (J a) x) -> invimageJ x

  IJIeqI : ∀ x -> Eq1 (I (J (I x))) (I x)
  IJIeqI x with exmid (invimageI (I x))
  IJIeqI x | inl (invelmtI x' y) = y
  IJIeqI x | inr b with b (invelmtI x refleq1)
  IJIeqI x | inr b | ()

  J_srj : ∀ (x : Set -> Set) -> invimageJ x
  J_srj x = invelmtJ (I x) (Iinj (IJIeqI x))

  cantor : Set -> Set
  cantor a with exmid (Eq1 (J a a) Empty)
  cantor a | inl a' = One
  cantor a | inr b = Empty

  OneNeqEmpty : Eq1 One Empty -> Empty
  OneNeqEmpty p = cast p one

  cantorone : ∀ a -> Eq1 (J a a) Empty -> Eq1 (cantor a) One 
  cantorone a p with exmid (Eq1 (J a a) Empty)
  cantorone a p | inl a' = refleq1
  cantorone a p | inr b with b p
  cantorone a p | inr b | ()

  cantorempty : ∀ a -> (Eq1 (J a a) Empty -> Empty) -> Eq1 (cantor a) Empty
  cantorempty a p with exmid (Eq1 (J a a) Empty)
  cantorempty a p | inl a' with p a'
  cantorempty a p | inl a' | ()
  cantorempty a p | inr b = refleq1

  cantorcase : ∀ a -> Eq1 cantor (J a) -> Empty
  cantorcase a pf with exmid (Eq1 (J a a) Empty)
  cantorcase a pf | inl a' = OneNeqEmpty (Eq1trans (Eq1trans (Eq1sym (cantorone a a')) (Eq1cong a pf)) a')
  cantorcase a pf | inr b = b (Eq1trans (Eq1sym (Eq1cong a pf)) (cantorempty a b))

  absurd : Empty
  absurd with (J_srj cantor)
  absurd | invelmtJ a y = cantorcase a (Eq1sym y)

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