{-# OPTIONS --allow-unsolved-metas #-}

open import Level hiding (suc ; zero )
open import Ordinals
module cardinal {n : Level } (O : Ordinals {n}) where

open import logic
-- import OD
import OD hiding ( _⊆_ )

import ODC
open import Data.Nat 
open import Relation.Binary.PropositionalEquality
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.Core

open inOrdinal O
open OD O
open OD.OD
open ODAxiom odAxiom
open import ZProduct O

import OrdUtil
import ODUtil
open Ordinals.Ordinals  O
open Ordinals.IsOrdinals isOrdinal
-- open Ordinals.IsNext isNext
open OrdUtil O
open ODUtil O

_⊆_ : ( A B : HOD ) → Set n
_⊆_ A B = {x : Ordinal } → odef A x → odef B x


open _∧_
open _∨_
open Bool
open _==_

open HOD

record OrdBijection (A B : Ordinal ) : Set n where
   field
       fun←  : (x : Ordinal ) → odef (* A)  x → Ordinal
       fun→  : (x : Ordinal ) → odef (* B)  x → Ordinal
       funB  : (x : Ordinal ) → ( lt : odef (* A)  x ) → odef (* B) ( fun← x lt )
       funA  : (x : Ordinal ) → ( lt : odef (* B)  x ) → odef (* A) ( fun→ x lt )
       fiso← : (x : Ordinal ) → ( lt : odef (* B)  x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x
       fiso→ : (x : Ordinal ) → ( lt : odef (* A)  x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x

ordbij-refl : { a b : Ordinal } → a ≡ b → OrdBijection a b
ordbij-refl {a} refl = record {
       fun←  = λ x _ → x 
     ; fun→  = λ x _ → x 
     ; funB  = λ x lt → lt
     ; funA  = λ x lt → lt
     ; fiso← = λ x lt → refl
     ; fiso→ = λ x lt → refl
    }

open Injection
open OrdBijection

record IsImage (a b : Ordinal) (iab : Injection a b ) (x : Ordinal ) : Set n where
   field
      y : Ordinal 
      ay : odef (* a) y
      x=fy : x ≡ i→ iab _ ay

Image : (a : Ordinal) { b : Ordinal } → Injection a b → HOD
Image a {b} iab = record { od = record { def = λ x → IsImage a b iab x } ; odmax = b ; <odmax = im00  } where
    im00 : {x : Ordinal } → IsImage a b iab x → x o< b
    im00 {x} record { y = y ; ay = ay ; x=fy = x=fy } = subst₂ ( λ j k → j o< k) (trans &iso (sym x=fy)) &iso 
         (c<→o< (subst ( λ k → odef (* b) k) (sym &iso) (iB iab y ay)) )

record IsInverseImage (a b : Ordinal) (iab : Injection a b ) (x y : Ordinal ) : Set n where
   field
      ax : odef (* a) x
      x=fy : y ≡ i→ iab x ax

InverseImage : {a : Ordinal} ( b : Ordinal ) → Injection a b → (y : Ordinal ) → HOD
InverseImage {a} b iab y = record { od = record { def = λ x → IsInverseImage a b iab x y } ; odmax = & (* a) ; <odmax = im00  } where
    im00 : {x : Ordinal } → IsInverseImage a b iab x y → x o< & (* a)
    im00 {x} record { ax = ax ; x=fy = x=fy } = odef< ax

Image⊆b : { a b : Ordinal } → (iab : Injection a b) → Image a iab ⊆ * b 
Image⊆b {a} {b} iab {x} record { y = y ; ay = ay ; x=fy = x=fy } = subst (λ k → odef (* b) k) (sym x=fy) (iB iab y ay)

_=c=_ : ( A B : HOD ) → Set n
A =c= B = OrdBijection ( & A ) ( & B )

c=→≡ : {A B : HOD} → A =c= B → (A ≡ ?) ∧ (B ≡ ?)
c=→≡ = ?

≡→c= : {A B : HOD} → A ≡ B → A =c= B
≡→c= = ?

open import BAlgebra O

_-_ : (a b : Ordinal ) → Ordinal 
a - b = & ( (* a) ＼ (* b) )

-→<  : (a b : Ordinal ) → (a - b) o≤ a
-→< a b = subst₂ (λ j k → j o≤ k) &iso &iso ( ⊆→o≤ ( λ {x} a-b → proj1 (subst ( λ k → odef k x) *iso a-b) ) )

b-a⊆b : {a b x : Ordinal } → odef (* (b - a)) x → odef (* b) x
b-a⊆b {a} {b} {x} lt with subst (λ k → odef k x) *iso lt
... | ⟪ bx , ¬ax ⟫ = bx

open Data.Nat

Injection-⊆ : {a b c : Ordinal } → * c ⊆ * a → Injection a b → Injection c b
Injection-⊆ {a} {b} {c} le f = record { i→ = λ x cx → i→ f x (le cx) ; iB = λ x cx → iB f x (le cx) 
        ; inject = λ x y ix iy eq → inject f x y (le ix) (le iy) eq  } 

Injection-∙ : {a b c : Ordinal } → Injection a b → Injection b c → Injection a c
Injection-∙ {a} {b} {c} f g = record { i→ = λ x ax → i→ g (i→ f x ax) (iB f x ax) ; iB = λ x ax → iB g (i→ f x ax) (iB f x ax)
        ; inject = λ x y ix iy eq → inject f x y ix iy (inject g (i→ f x ix) (i→ f y iy) (iB f x ix) (iB f y iy) eq)   } 


Bernstein : {a b : Ordinal } → Injection a b → Injection b a → OrdBijection a b
Bernstein {a} {b} iab iba = be00 where
    be05 : {a b : Ordinal } → a o< b → Injection a b → Injection b a → ⊥ 
    be05 {a} {b} a<b iab iba = TransFinite0 {λ x → (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥  } 
          ind a b a<b iab iba where
       ind :(x : Ordinal) →
            ((y : Ordinal) → y o< x → (b : Ordinal) → y o< b → Injection y b → Injection b y → ⊥ ) →
            (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥ 
       ind a prev b x<b (f @ record { i→ = fab ; iB = b∋fab ; inject = fab-inject }) 
                       ( g @ record { i→ = fba ; iB = a∋fba ; inject = fba-inject })= prev _ ? _ ? Uf fU where
 
          gf : Injection a a
          gf = record { i→ = λ x ax → fba (fab x ax) (b∋fab x ax) ; iB = λ x ax → a∋fba _ (b∋fab x ax) 
             ; inject = λ x y ax ay eq → fab-inject _ _ ax ay ( fba-inject _ _ (b∋fab _ ax) (b∋fab _ ay) eq) } 

          data gfImage : (i : ℕ) (x : Ordinal) → Set n where
              a-g : {x : Ordinal} → (ax : odef (* a) x ) → (¬ib : ¬ ( IsImage b a g x )) → gfImage 0 x
              next-gf : {x y : Ordinal} → {i : ℕ} → (gfiy : gfImage i y )→  (ix : IsImage a a gf x) → gfImage (suc i) x

          a∋gfImage : (i : ℕ) → {x : Ordinal } → gfImage i x → odef (* a) x
          a∋gfImage 0 {x} (a-g ax ¬ib) = ax
          a∋gfImage (suc i) {x} (next-gf lt record { y = y ; ay = ay ; x=fy = x=fy }) = subst (λ k → odef (* a) k ) (sym x=fy) (a∋fba _ (b∋fab y ay) )

          C : ℕ → HOD                                               --  Image {& (C i)} {a} (gf i)  does not work
          C i = record { od = record { def = gfImage i } ; odmax = & (* a) ; <odmax = λ lt → odef< (a∋gfImage i lt) } 

          record CN (x : Ordinal) : Set n where
              field 
                 i : ℕ
                 gfix : gfImage i x

          UC : HOD
          UC = record { od = record { def = λ x → CN x } ; odmax = & (* a) ; <odmax = λ lt → odef< (a∋gfImage (CN.i lt) (CN.gfix lt))  }

          b-UC : HOD
          b-UC = record { od = record { def = λ x → odef (* b) x ∧ (¬ CN x) } ; odmax = & (* b) ; <odmax = λ lt → odef< (proj1 lt)  }
          
          --  UC ⊆ * a
          --     f : UC → Image f UC    is injection
          --     g : Image f UC → UC    is injection

          UC⊆a : * (& UC) ⊆ * a
          UC⊆a {x} lt = a∋gfImage (CN.i be02) (CN.gfix be02)  where
                be02 : CN x
                be02 = subst (λ k → odef k x) *iso lt

          b-UC⊆b : * (& b-UC) ⊆ * b
          b-UC⊆b {x} lt = proj1 ( subst (λ k → odef k x) *iso lt )

          open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )

          fab-refl : {x : Ordinal } → {ax ax1 : odef (* a) x}  → fab x ax ≡ fab x ax1
          fab-refl {x} {ax} {ax1} = cong (λ k → fab x k) ( HE.≅-to-≡ ( ∋-irr {* a} ax ax1 ))  

          fba-refl : {x : Ordinal } → {bx bx1 : odef (* b) x}  → fba x bx ≡ fba x bx1
          fba-refl {x} {bx} {bx1} = cong (λ k → fba x k) ( HE.≅-to-≡ ( ∋-irr {* b} bx bx1 ))  

          be10 : Injection (& b-UC) (& (Image (& b-UC) {a} (Injection-⊆ b-UC⊆b g) ))
          be10 = record { i→ = be12 ; iB = be13 ; inject = ? } where
              be12 : (x : Ordinal) → odef (* (& b-UC)) x → Ordinal
              be12 x lt = i→ g x (proj1 be02) where
                    be02 : odef (* b) x ∧ ( ¬ CN x )
                    be02 = subst (λ k → odef k x) *iso lt
              be13 : (x : Ordinal) (lt : odef (* (& b-UC)) x) → odef (* (& (Image (& b-UC) (Injection-⊆ b-UC⊆b g)))) (be12 x lt)
              be13 x lt = subst (λ k → odef k (be12 x lt)) (sym *iso) record { y = x ; ay = subst (λ k → odef k x) (sym *iso) be02 ; x=fy = fba-refl } where
                    be02 : odef (* b) x ∧ ( ¬ CN x )
                    be02 = subst (λ k → odef k x) *iso lt


          be11 : Injection (& (Image (& b-UC) {a} (Injection-⊆ b-UC⊆b g) )) (& b-UC)
          be11 = record { i→ = be13 ; iB = ? ; inject = ? } where
              be13 : (x : Ordinal) → odef (* (& (Image (& b-UC) (Injection-⊆ b-UC⊆b g)))) x → Ordinal
              be13 x lt with subst (λ k → odef k x) *iso lt 
              ... | record { y = y ; ay = ay ; x=fy = x=fy } = y where -- x=fy : x ≡ fba y (proj1 (subst (λ k → OD.def (od k) y) *iso ay))
                    be02 : odef (* b) y ∧ ( ¬ CN y )
                    be02 = subst (λ k → odef k y) *iso ay

          fUC = & (Image (& UC) {b} (Injection-⊆ UC⊆a  f) )
          --   C n → f (C n) 
          fU : Injection (& UC) (& (Image (& UC) {b} (Injection-⊆ UC⊆a  f) ))
          fU = record { i→ = be00 ; iB = λ x lt →  be50 x lt ; inject = be51 } where
                be00 : (x : Ordinal) (lt : odef (* (& UC)) x) → Ordinal
                be00 x lt = be03 where
                    be02 : CN x
                    be02 = subst (λ k → odef k x) *iso lt
                    be03 : Ordinal
                    be03 with CN.i be02 | CN.gfix be02
                    ... | zero | a-g {x} ax ¬ib = fab x ax
                    ... | suc i | next-gf {x} gfiy record { y = y ; ay = ay ; x=fy = x=fy } 
                           = fab x (subst (λ k → odef (* a) k) (sym x=fy) (a∋fba _ (b∋fab y ay) ))
                be50 : (x : Ordinal) (lt : odef (* (& UC)) x) 
                     → odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f )))) (be00 x lt)
                be50 x lt1 =  subst (λ k → odef k (be00 x lt1 )) (sym *iso) be03 where
                    be02 : CN x
                    be02 = subst (λ k → odef k x) *iso lt1
                    be03 : odef (Image (& UC) (Injection-⊆ UC⊆a f )) (be00 x lt1 )
                    be03 with CN.i be02 | CN.gfix be02
                    ... | zero | a-g {x} ax ¬ib = record { y = x ; ay = lt1 ; x=fy = fab-refl } 
                    ... | suc i | next-gf {x} gfiy record { y = y ; ay = ay ; x=fy = x=fy } = record { y = _ ; ay = lt1 ; x=fy = fab-refl } 

                be51 : (x y : Ordinal) (ltx : odef (* (& UC)) x) (lty : odef (* (& UC)) y) → be00 x ltx ≡ be00 y lty → x ≡ y
                be51 x y ltx lty eq = be04 where
                    be0x : CN x
                    be0x = subst (λ k → odef k x) *iso ltx
                    be0y : CN y
                    be0y = subst (λ k → odef k y) *iso lty
                    be04 : x ≡ y
                    be04 with CN.i be0x | CN.gfix be0x | CN.i be0y | CN.gfix be0y 
                    ... | 0 | a-g ax ¬ib | 0 | a-g ax₁ ¬ib₁ = fab-inject _ _ ax ax₁ eq 
                    ... | 0 | a-g ax ¬ib | suc j | next-gf gfyi iy = fab-inject _ _ ax ay eq where
                        ay : odef (* a) y
                        ay = a∋gfImage (suc j) (next-gf gfyi iy )
                    ... | suc i | next-gf gfxi ix | 0 | a-g ay ¬ib = fab-inject _ _ ax ay eq where
                        ax : odef (* a) x
                        ax = a∋gfImage (suc i) (next-gf gfxi ix )
                    ... | suc i | next-gf gfxi ix | suc j | next-gf gfyi iy = fab-inject _ _ ax ay eq where
                        ax : odef (* a) x
                        ax = a∋gfImage (suc i) (next-gf gfxi ix )
                        ay : odef (* a) y
                        ay = a∋gfImage (suc j) (next-gf gfyi iy )
                        

          Uf : Injection (& (Image (& UC) {b} (Injection-⊆ UC⊆a  f))) (& UC) 
          Uf = record { i→ = ? ; iB = λ x lt →  ? ; inject = ? } where
                 be00 : (x : Ordinal) → odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x → Ordinal
                 be00 = ?

    be00 : OrdBijection a b
    be00 with trio< a b
    ... | tri< a ¬b ¬c = ⊥-elim ( be05 a iab iba )
    ... | tri≈ ¬a b ¬c = ordbij-refl b
    ... | tri> ¬a ¬b c = ⊥-elim ( be05 c iba iab )

_c<_ : ( A B : HOD ) → Set n
A c< B = ¬ ( Injection (& A)  (& B) )

Card : OD
Card = record { def = λ x → (a : Ordinal) → a o< x → ¬ OrdBijection a x }

record Cardinal (a : Ordinal ) : Set (Level.suc n) where
   field
       card : Ordinal
       ciso : OrdBijection a card
       cmax : (x : Ordinal) → card o< x → ¬ OrdBijection a x

Cardinal∈ : { s : HOD } → { t : Ordinal } → Ord t ∋ s →   s c< Ord t
Cardinal∈ = {!!}

Cardinal⊆ : { s t : HOD } → s ⊆ t →  ( s c< t ) ∨ ( s =c= t )
Cardinal⊆ = {!!}

Cantor1 : { u : HOD } → u c< Power u
Cantor1 = {!!}

Cantor2 : { u : HOD } → ¬ ( u =c=  Power u )
Cantor2 = {!!}