{-# OPTIONS --allow-unsolved-metas #-}
open import Level
open import Ordinals
module zorn {n : Level } (O : Ordinals {n})   where

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

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

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

open Element

IsPartialOrderSet : ( A : HOD ) →  (_<_ : (x y : HOD) → Set n )  → Set (suc n)
IsPartialOrderSet A _<_ = IsStrictPartialOrder _≡A_ _<A_  where
   _<A_ : (x y : Element A ) → Set n
   x <A y = elm x < elm y
   _≡A_ : (x y : Element A ) → Set (suc n)
   x ≡A y = elm x ≡ elm y

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 →  {_<_ : (x y : HOD) → Set n }  → IsPartialOrderSet A _<_ → IsPartialOrderSet B _<_
⊆-IsPartialOrderSet {A} {B} B⊆A {_<_} PA = record {
       isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = λ eq → case1 eq ; trans = λ {x} {y} {z} → trans1 {x} {y} {z} 
     ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; trans = trans1 ; <-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 })

IsTotalOrderSet : ( A : HOD ) →  (_<_ : (x y : HOD) → Set n )  → Set (suc n)
IsTotalOrderSet A _<_ = IsTotalOrder  _≡A_ _≤A_ where
   _≤A_ : (x y : Element A ) → Set (suc n)
   x ≤A y = (elm x ≡ elm y) ∨ (elm x < elm y)
   _≡A_ : (x y : Element A ) → Set (suc n)
   x ≡A y = elm x ≡ elm y

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

record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (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

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


record ZChain ( A : HOD ) (y : Ordinal)  (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
   field
      fb : (x : Ordinal ) → HOD
      A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox 
      total : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ( ox ≡ oz ) ∨ ( fb ox < fb oz ) ∨ ( fb oz < fb ox  )
      monotonic : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ox o< oz → fb ox < fb oz 

Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
    → 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
     someA : HOD
     someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) 
     isSomeA : A ∋ someA
     isSomeA =  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 y → odef A y ∧ ((m : Ordinal) →  odef A m → ¬ (* y < * m))} ; odmax = & A ; <odmax = z08 } where
         z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) →  odef A m → ¬ (* y < * m)))  → y o< & A
         z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
     no-maximal : HasMaximal =h= od∅ → (y : Ordinal) →  (odef A y ∧ ((m : Ordinal) →  odef A m → ¬ (* y < * m))) →  ⊥
     no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ ) 
     Gtx : { x : HOD} → A ∋ x → HOD
     Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 }  where
         z09 : {y : Ordinal} → (odef A y ∧ (x < (* y)))  → y o< & A
         z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
     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 = {!!} -- proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b<a
     z01 {a} {b} A∋a A∋b (case2 a<b) b<a = {!!} -- proj1 (proj1 (PO (me A∋b) (me A∋a)) b<a) a<b
     -- ZChain is not compatible with the SUP condition
     record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where
        field
            bx : Ordinal
            bx<y : bx o< y
            is-fb : x ≡ & (fb bx )
     bx<A : (z : ZChain A (& A) _<_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z ))  → BX.bx bx o< & A
     bx<A z {x} bx = BX.bx<y bx
     B :  (z : ZChain A (& A) _<_ ) → HOD
     B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z )  } ; odmax = & A ; <odmax = {!!} }
     z11 :  (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡  ZChain.fb z (BX.bx (is-elm x)) 
     z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
     obx : (z : ZChain A (& A) _<_ ) → {x : HOD} → B z ∋ x → Ordinal
     obx z {x} bx = BX.bx bx
     obx=fb : (z : ZChain A (& A) _<_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z (BX.bx bx) 
     obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx)) 
     B⊆A : (z : ZChain A (& A) _<_ ) → B z ⊆ A
     B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) (ZChain.A∋fb z (BX.bx bx) (BX.bx<y bx) ) }
     PO-B : (z : ZChain A (& A) _<_ ) → IsPartialOrderSet (B z) _<_
     PO-B z = subst₂ (λ j k → IsStrictPartialOrder j k ) {!!} {!!} {!!} where
           _<B_ = {!!}
           _≡B_ = {!!}
        -- a b = {!!} -- PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) }  record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) }  
     bx-monotonic : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → obx z (is-elm x) o< obx z (is-elm y) → elm x < elm y 
     bx-monotonic z {x} {y} a = subst₂ (λ j k → j < k ) (sym (z11 z x)) (sym (z11 z y)) (ZChain.monotonic z (bx<A z (is-elm x)) (bx<A z (is-elm y)) a ) 
     open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
     z12 : (z : ZChain A (& A) _<_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z))  (y : BX (& b) (& A) (ZChain.fb z))
          → obx z x ≡ obx z y → bx<A z x ≅ bx<A z y
     z12 z {a} {b} x y eq = {!!}
     bx-inject :  (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y
     bx-inject z {x} {y} eq = begin
            elm x ≡⟨  {!!}   ⟩
            {!!} ≡⟨ cong (λ k → {!!} ) {!!}  ⟩
            {!!} ≡⟨ {!!}   ⟩
            elm y ∎ where open ≡-Reasoning
     B-is-total :  (z : ZChain A (& A) _<_ ) → IsTotalOrderSet (B z) _<_ 
     B-is-total = {!!}
     B-Tri : (z : ZChain A (& A) _<_ ) → Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x < elm y )
     B-Tri z x y with trio< (obx z {!!}) (obx z {!!})
     ... | tri< a ¬b ¬c = {!!} where -- tri< z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where
          z10 : elm x < elm y
          z10 = {!!} -- bx-monotonic z {x} {y} a
     ... | tri≈ ¬a b ¬c = {!!} -- tri≈ {!!} (bx-inject z {x} {y} b) {!!}
     ... | tri> ¬a ¬b c = {!!} -- tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c)
     ZChain→¬SUP :  (z : ZChain A (& A) _<_ ) →  ¬ (SUP A (B z) _<_ )
     ZChain→¬SUP z sp = ⊥-elim {!!} where
         z03 : & (SUP.sup sp) o< osuc (& A)
         z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
         z02 :  (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥
         z02 x xe s<x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x<sup sp xe) s<x 
     ind :  HasMaximal =h= od∅
         → (x : Ordinal) → ((y : Ordinal) → y o< x →  ZChain A y _<_ )
         →  ZChain A x _<_
     ind nomx x prev with Oprev-p x
     ... | yes op with ODC.∋-p O A (* x)
     ... | no ¬Ax = {!!} where
          -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
          px = Oprev.oprev op
          zc1 : ZChain A px _<_
          zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
          z04 :  {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as
          z04 sup as s<x with trio< (& sup) x
          ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )  
          ... | tri< a ¬b ¬c  = {!!} -- ZChain.¬x<sup zc1 _ as ( subst (λ k → & sup o< k ) (sym (Oprev.oprev=x op)) a )
          ... | tri> ¬a ¬b c with  osuc-≡< s<x
          ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )  
          ... | case2 lt = ⊥-elim (¬a lt )
     ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
          px = Oprev.oprev op
          zc1 : ZChain A px _<_
          zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
          z06 : ZChain A x _<_
          z06 with is-o∅ (& (Gtx ax))
          ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
              x-is-maximal :  (m : Ordinal) → odef A m → ¬ (* x < * m)
              x-is-maximal m am  =  ¬x<m   where
                 ¬x<m :  ¬ (* x < * m)
                 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫  (≡o∅→=od∅ nogt) 
          ... | no not = {!!} where
     ind nomx x prev | no ¬ox with trio< (& A) x   --- limit ordinal case
     ... | tri< a ¬b ¬c = {!!} where
          zc1 : ZChain A (& A) _<_
          zc1 = prev (& A) a 
     ... | tri≈ ¬a b ¬c = {!!} where
     ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
     ... | no ¬Ax = {!!} where
     ... | yes ax with is-o∅ (& (Gtx ax))
     ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
              x-is-maximal :  (m : Ordinal) → odef A m → ¬ (* x < * m)
              x-is-maximal m am  =  ¬x<m   where
                 ¬x<m :  ¬ (* x < * m)
                 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫  (≡o∅→=od∅ nogt) 
     ... | no not = {!!} where
     zorn00 : Maximal A _<_
     zorn00 with is-o∅ ( & HasMaximal )
     ... | 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
         hasm :  odef HasMaximal ( & ( ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ) )
         hasm =  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 hasm
         zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
         zorn02 {x} ax m<x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
     ... | yes ¬Maximal = ⊥-elim {!!} where
         -- if we have no maximal, make ZChain, which contradict SUP condition
         z : (x : Ordinal) → HasMaximal =h= od∅  → ZChain A x _<_ 
         z x nomx = TransFinite (ind nomx) x

_⊆'_ : ( 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