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changeset 1059:bd2a258f309c

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 10 Dec 2022 17:41:17 +0900
parents 12ce8d0224d2
children a09f5e728f92
files src/zorn.agda
diffstat 1 files changed, 72 insertions(+), 133 deletions(-) [+]
line wrap: on
line diff
--- a/src/zorn.agda	Sat Dec 10 09:56:32 2022 +0900
+++ b/src/zorn.agda	Sat Dec 10 17:41:17 2022 +0900
@@ -338,10 +338,9 @@
       asupf :  {x : Ordinal } → odef A (supf x)
       supf-mono : {x y : Ordinal } → x o≤  y → supf x o≤ supf y
       supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z
+      zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x
 
       is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x)
-      sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
-           → IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b
 
    chain : HOD
    chain = UnionCF A f ay supf z
@@ -430,38 +429,6 @@
    f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ =
       subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso  (fcn-cmp y f mf fca fcb )
 
-   supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z  → supf (supf b) ≡ supf b
-   supf-idem {b} b≤z sfb≤x = z52 where
-       z54 :  {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b)
-       z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc
-       z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x)  fc where
-               u<b : u o< b
-               u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x )
-       z52 : supf (supf b) ≡ supf b
-       z52 = sup=u asupf sfb≤x  record { ax = asupf  ; x≤sup = z54  } 
-
-   x≤supfx : {x : Ordinal } → x o≤ z → supf x o≤ z → x o≤ supf x 
-   x≤supfx {x} x≤z sx≤z with x<y∨y≤x (supf x) x
-   ... | case2 le = le
-   ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where 
-         z46 : odef (UnionCF A f ay supf x) (f (supf x))
-         z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf  z47 )) ⟫ where
-             z47 : supf (supf x) ≡ supf x
-             z47 = supf-idem x≤z  sx≤z
-         z45 : f (supf x) ≤ supf x
-         z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46 
-
-   sup=u0 : {b : Ordinal} → (ab : odef A b) → b o≤ z
-       → IsSUP A (UnionCF A f ay supf b) b  → supf b ≡ b
-   sup=u0 {b} ab b≤z is-sup with trio< (supf b) b
-   ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
-         z47 : b o≤ supf b
-         z47 = x≤supfx b≤z ?
-   ... | tri≈ ¬a b ¬c = b
-   ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) where
-         z48 : supf b o≤ b
-         z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
-
    supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b 
    supf-mono< {a} {b} b≤z sa<sb  with order {a} {b} b≤z sa<sb (init asupf refl)
    ... | case2 lt = lt
@@ -492,6 +459,40 @@
                  z  ≤⟨ z≤sa ⟩
                  supf a ∎ )) where open o≤-Reasoning O
 
+   sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
+       → IsSUP A (UnionCF A f ay supf b) b  → supf b ≡ b
+   sup=u {b} ab b≤z is-sup = z50 where
+           z48 : supf b o≤ b
+           z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
+           z50 : supf b ≡ b
+           z50 with trio< (supf b) b
+           ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
+                 z47 : b o≤ supf b
+                 z47 = zo≤sz b≤z
+           ... | tri≈ ¬a b ¬c = b
+           ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) 
+
+   supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z  → supf (supf b) ≡ supf b
+   supf-idem {b} b≤z sfb≤x = z52 where
+       z54 :  {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b)
+       z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc
+       z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x)  fc where
+               u<b : u o< b
+               u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x )
+       z52 : supf (supf b) ≡ supf b
+       z52 = sup=u asupf sfb≤x  record { ax = asupf  ; x≤sup = z54  } 
+
+   x≤supfx : {x : Ordinal } → x o≤ z → supf x o≤ z → x o≤ supf x 
+   x≤supfx {x} x≤z sx≤z with x<y∨y≤x (supf x) x
+   ... | case2 le = le
+   ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where 
+         z46 : odef (UnionCF A f ay supf x) (f (supf x))
+         z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf  z47 )) ⟫ where
+             z47 : supf (supf x) ≡ supf x
+             z47 = supf-idem x≤z  sx≤z
+         z45 : f (supf x) ≤ supf x
+         z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46 
+
 record ZChain1 ( A : HOD )    ( f : Ordinal → Ordinal )  (mf< : <-monotonic-f A f)
         {y : Ordinal} (ay : odef A y)  (zc : ZChain A f mf< ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where
    supf = ZChain.supf zc
@@ -879,8 +880,8 @@
              m14 {z} uz = MinSUP.x≤sup sup1 (case1 uz)
 
           zc41 : ZChain A f mf< ay x
-          zc41 =  record { supf = supf1 ; sup=u = sup=u ; asupf = asupf1 ; supf-mono = supf1-mono ; order = order
-              ; supfmax = supfmax ; is-minsup = is-minsup ;  cfcs = cfcs  }  where
+          zc41 =  record { supf = supf1 ; asupf = asupf1 ; supf-mono = supf1-mono ; order = order
+              ; zo≤sz = zo≤sz ; supfmax = supfmax ; is-minsup = is-minsup ;  cfcs = cfcs  }  where
 
                  supf1 : Ordinal → Ordinal
                  supf1 z with trio< z px
@@ -1117,90 +1118,6 @@
                                 u≤px : u o≤ px
                                 u≤px = ordtrans u<x z≤px
 
-                 sup=u : {b : Ordinal} (ab : odef A b) →
-                    b o≤ x → IsSUP A (UnionCF A f ay supf1 b) b  → supf1 b ≡ b
-                 sup=u {b} ab b≤x is-sup with trio< b px
-                 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a)  zc40 where
-                     zc42 : {w : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) b) w → w ≤ b
-                     zc42 {w} ⟪ ua , ch-init fc ⟫ = IsSUP.x≤sup is-sup  ⟪ ua , ch-init fc ⟫ 
-                     zc42 {w} ⟪ ua ,  ch-is-sup u u<x su=u fc  ⟫ = IsSUP.x≤sup is-sup ⟪ ua ,  
-                         ch-is-sup u u<x (trans (sf1=sf0 zc44) su=u) (fcpu fc zc44)  ⟫ where
-                           zc44 : u o≤ px
-                           zc44 = ordtrans u<x (o<→≤ a)
-                     zc40 : IsSUP A (UnionCF A f ay supf0 b) b
-                     zc40 = record { ax = ab ; x≤sup = zc42 } 
-                 ... | tri≈ ¬a b=px ¬c = ZChain.sup=u zc ab (o≤-refl0 b=px)  record { ax = ab ; x≤sup = zc42 } where
-                     zc42 : {w : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) b) w → w ≤ b
-                     zc42 {w} ⟪ ua , ch-init fc ⟫ = IsSUP.x≤sup is-sup  ⟪ ua , ch-init fc ⟫ 
-                     zc42 {w} ⟪ ua ,  ch-is-sup u u<x su=u fc  ⟫ = IsSUP.x≤sup is-sup ⟪ ua ,  
-                         ch-is-sup u u<x (trans (sf1=sf0 zc44) su=u) (fcpu fc zc44)  ⟫ where
-                           zc44 : u o≤ px
-                           zc44 = o<→≤ ( subst (λ k → u o< k ) b=px u<x )
-                 ... | tri> ¬a ¬b px<b = trans zc36 x=b where
-                     x=b : x ≡ b
-                     x=b with osuc-≡< b≤x
-                     ... | case1 eq = sym (eq)
-                     ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ )
-                     zc31 : sp1 o≤ b
-                     zc31 = MinSUP.minsup sup1 ab zc32 where
-                         zc32 : {w : Ordinal } → odef pchainpx w → (w ≡ b) ∨ (w << b)
-                         zc32 {w} (case1 ⟪ ua , ch-init fc ⟫ ) = IsSUP.x≤sup is-sup ⟪ ua , ch-init fc ⟫
-                         zc32 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫ ) = IsSUP.x≤sup is-sup ⟪ ua , ch-is-sup u z01 z02 z03 ⟫ where
-                             z01 : u o< b 
-                             z01 = ordtrans u<x (subst (λ k → px o< k ) x=b px<x )
-                             z02 : supf1 u ≡ u
-                             z02 = trans (sf1=sf0 (o<→≤ u<x)) su=u
-                             z03 : FClosure A f (supf1 u) w
-                             z03 = fcpu fc (o<→≤ u<x)
-                         zc32 {w} (case2 fc) = IsSUP.x≤sup is-sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) sa<x su=u fc1 ⟫ where
-                             su=u : supf1 (supf0 px) ≡ supf0 px
-                             su=u = trans (sf1=sf0 (zc-b<x _ (proj2 fc))) ( ZChain.supf-idem zc o≤-refl (zc-b<x _ (proj2 fc)) )
-                             fc1 : FClosure A f (supf1 (supf0 px)) w
-                             fc1 = subst (λ k → FClosure A f k w ) (sym su=u) (proj1 fc)
-                             sa<x : supf0 px o< b
-                             sa<x = subst (λ k → supf0 px o< k ) x=b ( proj2 fc)
-                     zc36 : sp1 ≡ x   
-                     zc36 with osuc-≡< zc31
-                     ... | case1 eq = trans eq (sym x=b)
-                     ... | case2 sp1<b = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where
-                        --   sp1 o< x → ⊥
-                        --   supf0 px o≤ sp1 o< x → supf0 px o≤ px
-                        --        px o≤ supf0 px → px ≡ spuf0 px o≤ spuf1 x o< x
-                        --        px ≡ supf1 x
-                        sp1<x : sp1 o<  x
-                        sp1<x = subst (λ k → sp1 o< k ) (sym x=b) sp1<b
-                        zc38 : supf0 px o≤ px
-                        zc38 = begin
-                           supf0 px  ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩
-                           supf1 px  ≤⟨ supf1-mono (o<→≤ px<x) ⟩
-                           supf1 x  ≡⟨ sf1=sp1 px<x ⟩
-                           sp1 ≤⟨ zc-b<x _ sp1<x ⟩
-                           px ∎ where open o≤-Reasoning O
-                        zc37 : supf0 px ≡ px
-                        zc37 with trio< (supf0 px) px
-                        ... | tri< a ¬b ¬c = ⊥-elim ( o≤> (ZChain.x≤supfx zc o≤-refl zc38) a )
-                        ... | tri≈ ¬a b ¬c = b
-                        ... | tri> ¬a ¬b c = ⊥-elim ( o≤> zc38 c )
-                        zc44 : sp1 o≤ supf0 px
-                        zc44 = begin 
-                           sp1 ≤⟨ zc-b<x _ sp1<x ⟩
-                           px ≡⟨ sym zc37 ⟩
-                           supf0 px ∎ where open o≤-Reasoning O
-                        zc45 : supf0 px o≤ sp1
-                        zc45 = begin
-                           supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩
-                           supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩
-                           supf1 x ≡⟨ sf1=sp1 px<x  ⟩
-                           sp1 ∎ where open o≤-Reasoning O
-                        zc39 : supf0 px ≡ sp1
-                        zc39 with trio< (supf0 px) sp1
-                        ... | tri< a ¬b ¬c = ⊥-elim ( o≤> zc44 a ) ---   sp1 o< x ≡ osuc px ≡ osuc (supf0 px)
-                        ... | tri≈ ¬a b ¬c = b
-                        ... | tri> ¬a ¬b c = ⊥-elim ( o≤> zc45 c ) ---   supf0 px o≤ supf1 x ≡ sp1
-                        zc40 : f (supf0 px) ≤ supf0 px
-                        zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39) 
-                           ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫  ))
-
                  supfeq1 : {a b : Ordinal } → a o≤ x →  b o≤ x → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b → supf1 a ≡ supf1 b 
                  supfeq1 {a} {b} a≤z b≤z ua=ub with trio< (supf1 a) (supf1 b)
                  ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
@@ -1226,6 +1143,37 @@
                              x  ≤⟨ z≤sa ⟩
                              supf1 a ∎ )) where open o≤-Reasoning O
 
+
+                 zo≤sz : {z : Ordinal} →  z o≤ x → z o≤ supf1 z
+                 zo≤sz {z} z≤x with osuc-≡< z≤x
+                 ... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x ))
+                 ... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x))) --   px o≤ supf1 px o≤ supf1 x ≡ sp1 → x o≤ sp1
+                 ... | case2 lt = begin 
+                     x ≡⟨ sym (Oprev.oprev=x op) ⟩ 
+                     osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl)  ⟩ 
+                     osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩ 
+                     osuc (supf1 px) ≤⟨ osucc lt ⟩ 
+                     supf1 x ∎ where open o≤-Reasoning O
+                 ... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl )
+                 ... | case2 lt = begin
+                     x ≡⟨ sym (Oprev.oprev=x op) ⟩  
+                     osuc px ≤⟨ osucc lt ⟩ 
+                     supf0 px ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩ 
+                     supf1 px ≤⟨ supf1-mono (o<→≤ px<x)  ⟩ 
+                     supf1 x ∎ where open o≤-Reasoning O
+                 ... | case1 px=spx =  ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where
+                     zc37 : supf0 px ≡ px
+                     zc37 = sym px=spx
+                     zc39 : supf0 px ≡ sp1
+                     zc39 = begin
+                       supf0 px ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩
+                       supf1 px ≡⟨ spx=sx ⟩
+                       supf1 x ≡⟨ sf1=sp1 px<x ⟩
+                       sp1 ∎ where open ≡-Reasoning
+                     zc40 :  f (supf0 px) ≤ supf0 px                        
+                     zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39) 
+                           ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫  ))
+
                  order : {a b : Ordinal} {w : Ordinal} →
                     b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b
                  order {a} {b} {w} b≤x sa<sb fc = z20 where
@@ -1285,15 +1233,15 @@
 
      ... | no lim with trio< x o∅
      ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )
-     ... | tri≈ ¬a b ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; sup=u = ? ; asupf = MinSUP.as (ysup f mf ay) 
+     ... | tri≈ ¬a b ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) 
               ; supf-mono = λ _ → o≤-refl ; order = λ _ s<s → ⊥-elim ( o<¬≡ refl s<s )
-              ; supfmax = λ _ → refl ; is-minsup = ? ; cfcs = ?    } where
+              ; zo≤sz = ? ; supfmax = λ _ → refl ; is-minsup = ? ; cfcs = ?    } where
               mf : ≤-monotonic-f A f
               mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
                  mf00 : * x < * (f x)
                  mf00 = proj1 ( mf< x ax )
-     ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf-mono ; order = ?
-              ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs    }  where
+     ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = ? ; supf-mono = supf-mono ; order = ?
+              ; zo≤sz = λ _ → ?   ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs    }  where
 
           -- mf : ≤-monotonic-f A f
           -- mf x ax = ? -- ⟪ case2 ( proj1 (mf< x ax)) , proj2 (mf< x ax ) ⟫  will result memory exhaust
@@ -1474,15 +1422,6 @@
                ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
                ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u ? ? ?  ⟫
 
-          sup=u : {b : Ordinal} (ab : odef A b) →
-                b o≤ x → IsMinSUP A (UnionCF A f ay supf1 b) b  → supf1 b ≡ b
-          sup=u {b} ab b≤x is-sup with osuc-≡< b≤x
-          ... | case1 b=x = ? where
-                 zc31 : spu o≤ b
-                 zc31 = MinSUP.minsup usup ab zc32 where
-                     zc32 : {w : Ordinal } → odef pchainx w → (w ≡ b) ∨ (w << b)
-                     zc32 = ?
-          ... | case2 b<x = trans (sf1=sf ?) (ZChain.sup=u (pzc (ob<x lim b<x)) ab ? ? )
      ---
      --- the maximum chain  has fix point of any ≤-monotonic function
      ---