# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1650999870 -32400
# Node ID f0b45446c7ea00396ec174507ee1c5d0904f56ec
# Parent  3826221c61a65f1ed42025073101e8f32e7be36f
...

diff -r 3826221c61a6 -r f0b45446c7ea src/zorn.agda
--- a/src/zorn.agda	Tue Apr 26 16:10:30 2022 +0900
+++ b/src/zorn.agda	Wed Apr 27 04:04:30 2022 +0900
@@ -162,7 +162,6 @@
       A∋maximal : A ∋ maximal
       ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x       -- A is Partial, use negative
 
-
 --
 -- inductive maxmum tree from x
 -- tree structure
@@ -198,6 +197,7 @@
       chain : HOD
       chain⊆A : chain ⊆ A
       chain∋x : odef chain x
+      ¬chain∋x>z : { a : Ordinal } → z o< osuc a → ¬ odef chain a
       f-total : IsTotalOrderSet chain 
       f-next : {a : Ordinal } → odef chain a → a o< z  → odef chain (f a)
       f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) )
@@ -237,6 +237,8 @@
      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) 
+
+     -- Uncountable acending chain by axiom of choice
      cf : ¬ Maximal A → Ordinal → Ordinal
      cf  nmx x with ODC.∋-p O A (* x)
      ... | no _ = o∅
@@ -254,6 +256,7 @@
      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  )   
      A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) 
@@ -262,6 +265,12 @@
      sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) → SUP A (* (& (ZChain.chain zc)))
      sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc))
                (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) )
+     zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
+     zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y)
+
+     ---
+     --- sup is fix point in maximum chain
+     ---
      z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A))
             → f (& (SUP.sup (sp0 f mf zc  ))) ≡ & (SUP.sup (sp0 f mf zc  ))
      z03 f mf zc = z14 where
@@ -272,6 +281,8 @@
                    ∨  (supO (& chain) (subst (λ k → k  ⊆ A) (sym *iso) (ZChain.chain⊆A zc))  (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc)) ≡ b )
               → * a < * b  → odef chain b
            z10 = ZChain.is-max zc
+           z11 : & (SUP.sup sp1) o< & A
+           z11 = c<→o< ( SUP.A∋maximal sp1)
            z12 : odef chain (& (SUP.sup sp1))
            z12 with o≡? (& s) (& (SUP.sup sp1))
            ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc )
@@ -281,7 +292,7 @@
                ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
                ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
            z14 :  f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc))
-           z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 {!!} ))) (me z12 )
+           z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 z11 ))) (me z12 )
            ... | tri< a ¬b ¬c = ⊥-elim z16 where
                z16 : ⊥
                z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
@@ -290,7 +301,7 @@
            ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b )
            ... | tri> ¬a ¬b c = ⊥-elim z17 where
                z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) <  SUP.sup sp1)
-               z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 {!!} ) )
+               z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 z11 ) )
                z17 : ⊥
                z17 with z15
                ... | case1 eq = ¬b eq
@@ -303,11 +314,12 @@
           sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc 
           c = & (SUP.sup sp1)
      premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 :  ZChain A sa f mf supO y ) 
-        → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a)
-        → odef A b → a o< x → Prev< A (ZChain.chain zc0) (incl (ZChain.chain⊆A zc0) (subst (odef (ZChain.chain zc0)) (sym &iso) ca)) f
-          ∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b)
+        → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y
+        →  Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0))
+             (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0))
+             (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b)
        → * a < * b → odef (ZChain.chain zc0) b
-     premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<x P a<b = ZChain.is-max zc0 ca ab {!!} {!!} a<b
+     premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab a<y P a<b
      -- Union of ZFChain
      UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal) 
             → ( (y : Ordinal) → y o< B → ZChain A sa f mf supO y ) → HOD
@@ -324,12 +336,22 @@
           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 = {!!} ; f-immediate = {!!}
-             ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} }
+          zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
+             ; f-next = zc20 (ZChain.f-next zc0) ; f-immediate =  ZChain.f-immediate zc0
+             ; ¬chain∋x>z =  λ {a} x<oa → ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa )
+             ; chain∋x  = ZChain.chain∋x zc0 ; is-max = λ za ba a<x → zc20 (λ za a<x → ZChain.is-max zc0 za ba a<x ) za a<x } where
+              zc20 : {P : Ordinal →  Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a)
+                 → {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) →  P a
+              zc20 {P} prev {a} za a<x with trio< a px
+              ... | tri< a₁ ¬b ¬c = prev za a₁ 
+              ... | tri≈ ¬a b ¬c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (subst (λ k → k o< osuc a) b <-osuc ) za )
+              ... | tri> ¬a ¬b c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (ordtrans c <-osuc ) za )
      ... | 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) 
+          Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x)
+          Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax))))
           --   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 
@@ -347,18 +369,21 @@
                     zc5 : HOD
                     zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f x) } ; odmax = & A ; <odmax = {!!} }
                     ⊆-zc5 : zc5 ⊆ A 
-                    ⊆-zc5 = record { incl = λ lt → {!!} }
+                    ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where
+                        zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f x) ) → odef A z
+                        zc15 {z} (case1 zz) = subst (λ k → odef A k ) &iso ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain  k ) (sym &iso) zz ) )
+                        zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso ax ) )
                     IPO = ⊆-IsPartialOrderSet  ⊆-zc5 PO
                     zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x
                     zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P)))
                     fx=zc :  odef (ZChain.chain zc0) x → Tri  (* (f x) < * x ) (* (f x) ≡ * x) (* x < * (f x) )
                     fx=zc  c with mf x (subst (λ k → odef A k) &iso  ax )
-                    ... | ⟪ case1 x=fx , afx ⟫ = tri≈ {!!} zc13 {!!} where
+                    ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax (Afx ax) (case1 (sym zc13))) zc13 (z01 (Afx ax) ax (case1 zc13)) where
                         zc13 : * (f x) ≡ * x
                         zc13 = subst (λ k → k ≡ * x ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx ))
-                    ... | ⟪ case2 x<fx , afx ⟫ = tri> {!!} {!!} zc14 where
+                    ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax (Afx ax) (case2 zc14)) (λ lt → z01 (Afx ax) ax (case1 lt) zc14) zc14 where
                         zc14 : * x < * (f x)
-                        zc14 = subst₂ (λ j k → j < k ) {!!} (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
+                        zc14 = subst (λ k → * x < k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
                     cmp : Trichotomous _ _ 
                     cmp record { elm = a ; is-elm = aa } record { elm = b ; is-elm = ab } with aa | ab
                     ... | case1 x | case1 x₁ = IsStrictTotalOrder.compare (ZChain.f-total zc0) (me x) (me x₁)
@@ -368,19 +393,13 @@
                     ... | case2 n = {!!}
                     ... | case1 fb with IsStrictTotalOrder.compare (ZChain.f-total zc0) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay y))) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb)))
                     ... | tri< a₁ ¬b ¬c = {!!}
-                    ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx=zc (subst (λ k → odef chain k) {!!} c )) where
-                         zc11 : & a ≡ f x
-                         zc11 = fx 
+                    ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where
                          zc10 : * x ≡ b
                          zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax y ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) 
-                         zc12 : Tri (a < b) (a ≡ b) (b < a)
-                         zc12 with mf x (subst (λ k → odef A k) &iso  ax )
-                         ... | ⟪ case1 x=fx , afx ⟫ = tri≈ {!!} zc13 {!!} where
-                             zc13 : a ≡ b
-                             zc13 = subst₂ (λ j k → j ≡ k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym zc11))) zc10 (sym ( x=fx ))
-                         ... | ⟪ case2 x<fx , afx ⟫ = tri> {!!} {!!} zc14 where
-                             zc14 : b < a
-                             zc14 = subst₂ (λ j k → j < k ) zc10 (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym zc11))) x<fx
+                         zc11 : * (f x) ≡ a
+                         zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx))
+                         zc12 : odef chain x
+                         zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10)))  c 
                     ... | tri> ¬a ¬b c₁ = {!!}
                     zc6 : IsTotalOrderSet zc5
                     zc6 =  record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z}
@@ -390,11 +409,15 @@
           ... | case2 not = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!}
                      ; f-immediate = {!!} ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} }  -- no extention
      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 =  {!!}
-              ; f-immediate = {!!} ; chain∋x  = {!!}  ; is-max = {!!} } where
+     ... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
+              ; f-next = {!!}
+              ; f-immediate = {!!} ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} } where
           zc0 = prev (& A) a
      ... | tri≈ ¬a b ¬c = {!!}
-     ... | tri> ¬a ¬b c = {!!}
+     ... | tri> ¬a ¬b c =  record { chain = uzc ; chain⊆A = record { incl = λ {x} lt → proj1 lt } ; f-total = {!!} ; f-next =  {!!}
+              ; f-immediate = {!!} ; chain∋x  = {!!}  ; is-max = {!!} } where
+         uzc : HOD
+         uzc = UZFChain f mf x prev
      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