module halt where

open import Level renaming ( zero to Zero ; suc to Suc )
open import Data.Nat
open import Data.Maybe
open import Data.List hiding ([_])
open import Data.Nat.Properties
open import Relation.Nullary
open import Data.Empty
open import Data.Unit
open import  Relation.Binary.Core hiding (_⇔_)
open import  Relation.Binary.Definitions
open import Relation.Binary.PropositionalEquality

open import logic

record HBijection {n m : Level} (R : Set n) (S : Set m) : Set (n Level.⊔ m)  where
   field
       fun←  :  S → R
       fun→  :  R → S
       fiso← : (x : R)  → fun← ( fun→ x )  ≡ x 
--  normal bijection required below, but we don't need this to show the inconsistency
--     fiso→ : (x : S ) → fun→ ( fun← x )  ≡ x 

injection :  {n m : Level} (R : Set n) (S : Set m) (f : R → S ) → Set (n Level.⊔ m)
injection R S f = (x y : R) → f x ≡ f y → x ≡ y

open HBijection 

diag : {S : Set }  (b : HBijection  ( S → Bool ) S) → S → Bool
diag b n = not (fun← b n n)

diagonal : { S : Set } → ¬ HBijection  ( S → Bool ) S
diagonal {S} b = diagn1 (fun→ b (diag b) ) refl where
    diagn1 : (n : S ) → ¬ (fun→ b (diag b) ≡ n ) 
    diagn1 n dn = ¬t=f (diag b n ) ( begin
           not (diag b n)
        ≡⟨⟩
           not (not fun← b n n)
        ≡⟨ cong (λ k → not (k  n) ) (sym (fiso← b _)) ⟩
           not (fun← b (fun→ b (diag b))  n)
        ≡⟨ cong (λ k → not (fun← b k n) ) dn ⟩
           not (fun← b n n)
        ≡⟨⟩
           diag b n 
        ∎ ) where open ≡-Reasoning

postulate
   utm         : List Bool → List Bool → Maybe Bool

record TM : Set where
   field
      tm : List Bool → Maybe Bool
      encode : List Bool
      is-tm : utm encode ≡ tm

open TM

Halt1 : TM → List Bool  → Bool -- ℕ → ( ℕ → Bool )
Halt1 = {!!}

record Halt1-is-tm : Set where
   field
       htm : TM
       htm-is-Halt1 : (t : TM ) → (x : List Bool) → (Halt1 t x ≡ true) ⇔ ((tm htm (encode t ++ x)) ≡ just true)

Halt2 : (tm : TM ) →  List Bool -- ( ℕ → Bool ) → ℕ  
Halt2 tm = encode tm

not-halt : {!!}
not-halt = {!!}

open _∧_

open ≡-Reasoning


open import Axiom.Extensionality.Propositional
postulate f-extensionality : { n : Level}  → Axiom.Extensionality.Propositional.Extensionality n n 
open import Relation.Binary.HeterogeneousEquality as HE using (_≅_;refl ) renaming ( sym to ≅-sym ; trans to ≅-trans ; cong to ≅-cong ) 

record Halt : Set where
   field
      htm : TM
      is-halt :          (t : TM ) → (x : List Bool ) → (tm htm  (encode t ++ x) ≡ just true) ⇔ ((tm t x ≡ just true) ∨ (tm t x ≡ just false)) 
      nhtm : TM
      nhtm-is-negation : (t : TM ) → (x : List Bool ) → (tm htm (encode t ++ x) ≡ just true) ⇔ (tm nhtm (encode t ++ x) ≡ nothing )

open Halt

TNℕ : Halt → HBijection (TM → Bool) TM
TNℕ = {!!}

tm-cong : {x y : TM} → tm x ≡ tm y → encode x ≡ encode y → is-tm x ≅ is-tm y → x ≡ y
tm-cong refl refl refl  = refl

tm-bij :  HBijection TM (List Bool)
tm-bij = record {
       fun←  = λ x → record { tm = utm x ; encode = x ; is-tm = refl }
     ; fun→  = λ tm → encode tm
     ; fiso← = tmb1
    }  where
         tmb1 : (x : TM ) →  record { tm = utm (encode x) ; encode = encode x ; is-tm = refl } ≡ x
         tmb1 x with is-tm x | inspect is-tm x
         ... | refl | record { eq = refl } = tm-cong (is-tm x) refl refl

--  Halt1 (Halt2 x) ≡ x
--  Halt2 (Halt2 y) ≡ y

halting : ¬ Halt
halting h with tm (htm h) (encode (nhtm h) ++ encode (nhtm h))
... | just true = ¬t=f {!!} {!!}
... | nothing = {!!}
... | just false = {!!}