Mercurial > hg > Members > kono > Proof > ZF-in-agda

changeset 472:66a7d30d125a

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 29 Mar 2022 11:47:24 +0900
parents 2b048496cb21
children d61f4a89c99e
files src/ODC.agda
diffstat 1 files changed, 29 insertions(+), 21 deletions(-) [+]
line wrap: on
line diff
--- a/src/ODC.agda	Mon Mar 28 15:03:50 2022 +0900
+++ b/src/ODC.agda	Tue Mar 29 11:47:24 2022 +0900
@@ -134,7 +134,7 @@
 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
    field
       maximal : HOD
-      A∋maximal : HOD
+      A∋maximal : A ∋ maximal
       ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x
 
 record ZChain ( A : HOD ) (y : Ordinal)  (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
@@ -144,7 +144,7 @@
       total : TotalOrderSet B _<_
       fb : {x : HOD } → A ∋ x  → HOD
       B∋fb : (x : HOD ) → (ax : A ∋ x)  → B ∋ fb ax
-      ¬x≤sup : (sup : HOD) → (as : A ∋ sup ) → & sup o< y → sup < fb as
+      ¬x≤sup : (sup : HOD) → (as : A ∋ sup ) → & sup o< osuc y → sup < fb as
 
 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
     → o∅ o< & A 
@@ -153,15 +153,21 @@
     → ( ( B : HOD) → (B⊆A : B ⊆ A) → TotalOrderSet B _<_ → SUP A B  _<_  )
     → Maximal A _<_ 
 Zorn-lemma {A} {_<_} 0<A TR PO supP = zorn00 where
+     someA : HOD
+     someA = minimal 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 → (m : Ordinal) → odef A x ∧ odef A m ∧ (¬ (* x < * m))} ; odmax = & A ; <odmax = {!!} }
+     HasMaximal = record { od = record { def = λ x → (m : Ordinal) → odef A x ∧ odef A m ∧ (¬ (* x < * m))} ; odmax = & A ; <odmax = z07 } where
+         z07 :  {y : Ordinal} → ((m : Ordinal) → odef A y ∧ 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 (& someA)) )))
+     Gtx : { x : HOD} → A ∋ x → HOD
+     Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) ∧ ( (& x) o< y )  } ; odmax = & A ; <odmax = {!!} } 
      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 (PO (me  A∋b) (me A∋a)) b<a ⟪ a<b , (λ b=a → proj1 (proj2 (PO (me  A∋b) (me A∋a)) b=a ) b<a ) ⟫
      ZChain→¬SUP :  (z : ZChain A (& A) _<_ ) →  ¬ (SUP A (ZChain.B z) _<_ )
      ZChain→¬SUP z sp = ⊥-elim (z02 (ZChain.fb z (SUP.A∋maximal sp)) (ZChain.B∋fb z  _ (SUP.A∋maximal sp)) (ZChain.¬x≤sup z _  (SUP.A∋maximal sp) z03 )) where
-         z03 : & (SUP.sup sp) o< & A
-         z03 = c<→o< (SUP.A∋maximal sp)
+         z03 : & (SUP.sup sp) o< osuc (& A)
+         z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
          z02 :  (x : HOD) → ZChain.B z ∋ x → SUP.sup sp < x → ⊥
          z02 x xe s<x = ( z01 (incl (ZChain.B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x≤sup sp xe) s<x )
      ind :  HasMaximal =h= od∅
@@ -173,23 +179,19 @@
           px = Oprev.oprev op
           zc1 : ZChain A px _<_
           zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
-          z05 : SUP A (ZChain.B zc1) _<_ 
-          z05 = supP (ZChain.B zc1) (ZChain.B⊆A zc1) (ZChain.total zc1)
-          z06 :  (sup : HOD) (as : A ∋ sup) → & sup o< x → HOD
-          z06 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 = ⊥-elim (¬a s<x)
-          ... | tri< a ¬b ¬c with osuc-≡< (subst (λ k → (& sup) o< k ) (sym (Oprev.oprev=x op)) a )
-          ... | case2 lt = ZChain.fb zc1 as
-          ... | case1 eq = {!!}
-          z04 :  (sup : HOD) (as : A ∋ sup) → & sup o< x → sup < ZChain.fb zc1 as
+          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 = ⊥-elim (¬a s<x)
-          ... | tri< a ¬b ¬c with osuc-≡< (subst (λ k → (& sup) o< k ) (sym (Oprev.oprev=x op)) a )
-          ... | case2 lt = ZChain.¬x≤sup zc1 _ as lt
-          ... | case1 eq = ?
-     ... | yes Ax = {!!}
+          ... | 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 = {!!} where
+          px = Oprev.oprev op
+          zc1 : ZChain A px _<_
+          zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
+          z06 : SUP A (* x , * x) _<_ 
+          z06 = supP (* x , * x)  {!!} {!!}
      -- ... | no ¬Ax = record { B = B (prev B) ; B⊆A = {!!} ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} }
      ind nomx x prev | no ¬ox with trio< (& A) x
      ... | tri< a ¬b ¬c = {!!}
@@ -197,7 +199,13 @@
      ... | tri> ¬a ¬b c = {!!}
      zorn00 : Maximal A _<_
      zorn00 with is-o∅ ( & HasMaximal )
-     ... | no not = record { maximal = minimal HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = {!!}; ¬maximal<x  = {!!} }
+     ... | no not = record { maximal = minimal HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x  = zorn02 } where
+         zorn03 :  odef HasMaximal ( & ( minimal HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ) )
+         zorn03 =  x∋minimal  HasMaximal  (λ eq → not (=od∅→≡o∅ eq))
+         zorn01 :  A ∋ minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq))
+         zorn01 = proj1 ( zorn03 (& someA) )
+         zorn02 : {x : HOD} → A ∋ x → ¬ (minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
+         zorn02 {x} ax m<x = proj2 (proj2 (zorn03 (& x))) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
      ... | yes ¬Maximal = ⊥-elim ( ZChain→¬SUP (z (& A) (≡o∅→=od∅ ¬Maximal)) ( supP B (ZChain.B⊆A (z (& A) (≡o∅→=od∅ ¬Maximal))) (ZChain.total (z (& A) (≡o∅→=od∅ ¬Maximal))) )) where
          z : (x : Ordinal) → HasMaximal =h= od∅  → ZChain A x _<_ 
          z x nomx = TransFinite (ind nomx) x