{-# OPTIONS --allow-unsolved-metas #-}
open import Level hiding ( suc ; zero )
open import Ordinals
import OD 
module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) where

open import zf
open import logic
-- open import partfunc {n} O

open import Relation.Nullary 
open import Relation.Binary 
open import Data.Empty 
open import Relation.Binary
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality
import BAlgbra 


open inOrdinal O
open OD O
open OD.OD
open ODAxiom odAxiom
import OrdUtil
import ODUtil
open Ordinals.Ordinals  O
open Ordinals.IsOrdinals isOrdinal
open Ordinals.IsNext isNext
open OrdUtil O
open ODUtil O


import ODC


open _∧_
open _∨_
open Bool


open HOD

_≤_ : (x y : HOD) → Set (Level.suc n)
x ≤ y = ( x ≡ y ) ∨ ( x < y )

record Element (A : HOD) : Set (Level.suc n) where
    field
       elm : HOD
       is-elm : A ∋ elm

open Element

_<A_ : {A : HOD} → (x y : Element A ) → Set n
x <A y = elm x < elm y
_≡A_ : {A : HOD} → (x y : Element A ) → Set (Level.suc n)
x ≡A y = elm x ≡ elm y

IsPartialOrderSet : ( A : HOD ) → Set (Level.suc n)
IsPartialOrderSet A = IsStrictPartialOrder (_≡A_ {A}) _<A_  

open _==_
open _⊆_

isA : { A B  : HOD } → B ⊆ A → (x : Element B) → Element A
isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) }

⊆-IsPartialOrderSet : { A B  : HOD } → B ⊆ A → IsPartialOrderSet A → IsPartialOrderSet B 
⊆-IsPartialOrderSet {A} {B} B⊆A  PA = record {
       isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ;  trans = λ {x} {y} {z} → trans1 {x} {y} {z} 
     ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; <-resp-≈ = resp0 
   } where
   _<B_ : (x y : Element B ) → Set n
   x <B y = elm x < elm y
   trans1 : {x y z : Element B} → x <B y → y <B z → x <B z 
   trans1 {x} {y} {z} x<y y<z  = IsStrictPartialOrder.trans PA {isA B⊆A x} {isA B⊆A y} {isA B⊆A z} x<y y<z 
   irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y  )
   irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y 
   open import Data.Product
   resp0 : ({x y z : Element B} → elm y ≡ elm z → x <B y → x <B z) × ({x y z : Element B} → elm y ≡ elm z → y <B x → z <B x) 
   resp0 = Data.Product._,_  (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) 
                             (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })

-- open import Relation.Binary.Properties.Poset as Poset

IsTotalOrderSet : ( A : HOD ) →  Set (Level.suc n)
IsTotalOrderSet A = IsStrictTotalOrder  (_≡A_ {A}) _<A_ 

me : { A a : HOD } → A ∋ a → Element A
me {A} {a} lt = record { elm = a ; is-elm = lt }

A∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (A ∋ x) ≡ (A ∋ y )
A∋x-irr A {x} {y} refl = refl

me-elm-refl : (A : HOD) → (x : Element A) → elm (me {A} (d→∋ A (is-elm x))) ≡ elm x
me-elm-refl A record { elm = ex ; is-elm = ax } = *iso 

-- <-induction : (A : HOD) { ψ : (x : HOD) → A ∋ x → Set (Level.suc n)}
--    →  IsPartialOrderSet A
--    → ( {x : HOD } → A ∋ x → ({ y : HOD } → A ∋ y →  y < x → ψ y ) → ψ x )
--    → {x0 x : HOD } → A ∋ x0 → A ∋ x → x0 < x → ψ x
-- <-induction A {ψ} PO ind ax0 ax x0<a = subst (λ k → ψ k ) *iso (<-induction-ord (osuc (& x)) <-osuc )  where
--      -- y < * ox → & y o< ox
--      induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
--      induction ox prev = ind  ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) {!!})) 
--      <-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
--      <-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy


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

-- Don't use Element other than Order, you'll be in a trouble
-- postulate   -- may be proved by transfinite induction and functional extentionality
--   ∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (ax : A ∋ x) (ay : A ∋ y ) → ax ≅ ay 
--   odef-irr : (A : OD) {x y : Ordinal} → x ≡ y → (ax : def A x) (ay : def A y ) → ax ≅ ay 

-- is-elm-irr : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y 
-- is-elm-irr A {x} {y} eq = ∋x-irr A eq (is-elm x) (is-elm y )

El-irr2 : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y → x ≡ y
El-irr2  A {x} {y} refl HE.refl = refl

-- El-irr : {A : HOD} → {x y : Element A } → elm x ≡ elm y → x ≡ y
-- El-irr {A} {x} {y} eq = El-irr2 A eq ( is-elm-irr A eq ) 

record _Set≈_ (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← lt )
       funA  : {x : Ordinal } → ( lt : odef (* B)  x ) → odef (* A) ( fun→ lt )
       fiso← : {x : Ordinal } → ( lt : odef (* B)  x ) → fun←  ( funA lt ) ≡ x
       fiso→ : {x : Ordinal } → ( lt : odef (* A)  x ) → fun→  ( funB lt ) ≡ x

open _Set≈_
record _OS≈_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where
   field
      iso : (& A ) Set≈  (& B)
      fmap : {x y : Ordinal} → (ax : odef A x) → (ay : odef A y) → * x < * y
          → * (fun← iso (subst (λ k → odef k x) (sym *iso) ax)) < * (fun← iso (subst (λ k → odef k y) (sym *iso) ay))

Cut< : ( A x : HOD )  → HOD
Cut<  A x = record { od = record { def = λ y → ( odef A y ) ∧ ( x < * y ) } ; odmax = & A
    ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (proj1 lt))) }

Cut<T : {A : HOD}   → (TA : IsTotalOrderSet A ) ( x : HOD )→  IsTotalOrderSet ( Cut< A x )
Cut<T {A} TA x =  record { isEquivalence = record { refl = refl ; trans = trans ; sym = sym }
   ; trans = λ {x} {y} {z} → IsStrictTotalOrder.trans TA {me (proj1 (is-elm x))} {me (proj1 (is-elm y))} {me (proj1 (is-elm z))} ;
         compare = λ x y → IsStrictTotalOrder.compare TA (me (proj1 (is-elm x))) (me (proj1 (is-elm y)))  }

record _OS<_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where
   field
      x : HOD
      iso : TA OS≈ (Cut<T TA x) 

-- OS<-cmp : {x : HOD} → Trichotomous {_} {IsTotalOrderSet x} _OS≈_ _OS<_ 
-- OS<-cmp A B = {!!}

      
record Maximal ( A : HOD )  : Set (Level.suc n) where
   field
      maximal : HOD
      A∋maximal : A ∋ maximal
      ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x       -- A is Partial, use negative


--
-- inductive maxmum tree from x
-- tree structure
--

≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n)
≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧  odef A (f x )

record Indirect< (A : HOD) {x y : Ordinal } (xa : odef A x) (ya : odef A y) (z : Ordinal)  : Set n where
   field
      az : odef A z
      x<z : * x < * z 
      z<y : * z < * y 

record Prev< (A : HOD) {x : Ordinal } (xa : odef A x)  ( f : Ordinal → Ordinal )  : Set n where
   field
      y : Ordinal
      ay : odef A y
      x=fy :  x ≡ f y 

record SUP ( A B : HOD )  : Set (Level.suc n) where
   field
      sup : HOD
      A∋maximal : A ∋ sup
      x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup )   -- B is Total, use positive

SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n)
SupCond A B _ _ = SUP A B  

record ZChain ( A : HOD ) {x : Ordinal} (ax : A ∋ * x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
         (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) (z : Ordinal)  : Set (Level.suc n) where
   field
      chain : HOD
      chain⊆A : chain ⊆ A
      f-total : IsTotalOrderSet chain 
      f-next : {a : Ordinal } → odef chain a → odef chain (f a)
      is-max :  {a b : Ordinal } → (ca : odef chain a ) → odef A b → a o< z
          → ( Prev< A (incl chain⊆A (subst (λ k → odef chain k ) (sym &iso) ca)) f
               ∨ (sup (& chain) (subst (λ k → k  ⊆ A) (sym *iso) chain⊆A)  (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ))
          → * a < * b  → odef chain b

Zorn-lemma : { A : HOD } 
    → o∅ o< & A 
    → IsPartialOrderSet A 
    → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B   ) -- SUP condition
    → Maximal A 
Zorn-lemma {A}  0<A PO supP = zorn00 where
     supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal
     supO C C⊆A TC = & ( SUP.sup ( supP (* C)  C⊆A TC ))
     z01 : {a b : HOD} → A ∋ a → A ∋ b  → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
     z01 {a} {b} A∋a A∋b (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋a} (sym a=b) b<a
     z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl
          (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b}  b<a a<b)
     z07 :   {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A
     z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
     s : HOD
     s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) 
     sa : A ∋ * ( & s  )
     sa =  subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))  )
     HasMaximal : HOD
     HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) →  odef A m → ¬ (* x < * m)) }  ; odmax = & A ; <odmax = z07 } 
     no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) →  odef A m →  odef A x ∧ (¬ (* x < * m) )) →  ⊥
     no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ )  
     Gtx : { x : HOD} → A ∋ x → HOD
     Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } 
     z08  : ¬ Maximal A →  HasMaximal =h= od∅
     z08 nmx  = record { eq→  = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt)
         ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← =  λ {y} lt → ⊥-elim ( ¬x<0 lt )}
     x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m))
     x-is-maximal nmx {x} ax nogt m am  = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) ,  ¬x<m  ⟫ where
        ¬x<m :  ¬ (* x < * m)
        ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫  (≡o∅→=od∅ nogt) 
     cf : ¬ Maximal A → Ordinal → Ordinal
     cf  nmx x with ODC.∋-p O A (* x)
     ... | no _ = o∅
     ... | yes ax with is-o∅ (& ( Gtx ax ))
     ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) -- no larger element, so it is maximal
     ... | no not =  & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)))
     is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) )
     is-cf nmx {x} ax with ODC.∋-p O A (* x)
     ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax ))
     ... | yes ax with is-o∅ (& ( Gtx ax ))
     ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ )
     ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))
     cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) →  odef A x → ( * x < * (cf nmx x) ) ∧  odef A (cf nmx x )
     cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫
     cf-is-≤-monotonic : (nmx : ¬ Maximal A ) →  ≤-monotonic-f A ( cf nmx )
     cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫
     zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) →  (zc : ZChain A sa f mf supO (& A)) → SUP A  (ZChain.chain zc) 
     zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc  )   
     -- zsup zc f mf = & ( SUP.sup (supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc f mf )  ) )
     A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) 
        →  A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ))
     A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )
     z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) → f (& ( SUP.sup (zsup f mf zc ))) ≡ & (SUP.sup (zsup f mf zc ))
     z03 f mf zc = {!!}
     z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥
     z04 nmx zc = z01  {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso)
           (proj1 (is-cf nmx (SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc )))))
           (A∋zsup nmx zc ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc )))
           (proj1 (cf-is-<-monotonic nmx c ((subst λ k → odef A k ) &iso (A∋zsup nmx zc )))) where
          c = & (SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ))
     -- ZChain is not compatible with the SUP condition
     ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x →  ZChain A sa f mf supO y )
         →  ZChain A sa f mf supO x 
     ind f mf x prev with Oprev-p x
     ... | yes op with ODC.∋-p O A (* x)
     ... | no ¬Ax = zc1 where
          -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
          px = Oprev.oprev op
          zc0 : ZChain A sa f mf supO (Oprev.oprev op) 
          zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
          zc1 : ZChain A sa f mf supO x 
          zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 ; is-max = {!!} }
     ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x
          px = Oprev.oprev op
          zc0 : ZChain A sa f mf supO (Oprev.oprev op) 
          zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
          --   x is in the previous chain, use the same
          --   x has some y which y < x ∧ f y ≡ x
          --   x has no y which y < x 
          zc4 : ZChain A sa f mf supO x
          zc4 = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; is-max = {!!} }
     ind f mf x prev | no ¬ox with trio< (& A) x   --- limit ordinal case
     ... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
              ; is-max = {!!} } where
          zc0 = prev (& A) a
     ... | tri≈ ¬a b ¬c = {!!}
     ... | tri> ¬a ¬b c = {!!}
     zorn00 : Maximal A 
     zorn00 with is-o∅ ( & HasMaximal )  -- we have no Level (suc n) LEM 
     ... | no not = record { maximal = ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x  = zorn02 } where
         -- yes we have the maximal
         zorn03 :  odef HasMaximal ( & ( ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ) )
         zorn03 =  ODC.x∋minimal  O HasMaximal  (λ eq → not (=od∅→≡o∅ eq))
         zorn01 :  A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
         zorn01  = proj1  zorn03  
         zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
         zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
     ... | yes ¬Maximal = ⊥-elim ( z04 nmx  (zorn03 (cf nmx) (cf-is-≤-monotonic nmx))) where
         -- if we have no maximal, make ZChain, which contradict SUP condition
         nmx : ¬ Maximal A 
         nmx mx =  ∅< {HasMaximal} zc5 ( ≡o∅→=od∅  ¬Maximal ) where
              zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) →  odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) 
              zc5 = ⟪  Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫
         zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → ZChain A sa f mf supO (& A)
         zorn03 f mf = TransFinite (ind f mf)  (& A) 

-- usage (see filter.agda )
--
-- _⊆'_ : ( A B : HOD ) → Set n
-- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x

-- MaximumSubset : {L P : HOD} 
--        → o∅ o< & L →  o∅ o< & P → P ⊆ L
--        → IsPartialOrderSet P _⊆'_
--        → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
--        → Maximal P (_⊆'_)
-- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP  = Zorn-lemma {P} {_⊆'_} 0<P PO SP