--- a/src/zorn.agda	Sat Apr 23 18:39:07 2022 +0900
+++ b/src/zorn.agda	Sun Apr 24 08:04:42 2022 +0900
@@ -94,6 +94,18 @@
 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
@@ -204,30 +216,39 @@
      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 } where
-         z07 :   {y : Ordinal} → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m)) → 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 )))
-     no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → ((m : Ordinal) →  odef A m →  odef A x ∧ (¬ (* x < * m) )) →  ⊥
-     no-maximum nomx x P = ¬x<0 (eq→ nomx {x} {!!} ) 
+     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 = {!!} } 
+     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 (≡o∅→=od∅ {!!} ) x x-is-maximal ) where -- no larger element, so it is maximal
-          x-is-maximal :  (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m))
-          x-is-maximal m am  = ⟪ subst (λ k → odef A k) &iso ax ,  ¬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) 
+     ... | 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 = ⟪ {!!} , {!!} ⟫
+     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) 
@@ -237,9 +258,11 @@
         →  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 = {!!}
+     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} {!!} (A∋zsup nmx zc ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc )))
+     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