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

changeset 473:d61f4a89c99e

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 30 Mar 2022 23:44:49 +0900
parents 66a7d30d125a
children 100ceb0fbada
files src/ODC.agda
diffstat 1 files changed, 26 insertions(+), 10 deletions(-) [+]
line wrap: on
line diff
--- a/src/ODC.agda	Tue Mar 29 11:47:24 2022 +0900
+++ b/src/ODC.agda	Wed Mar 30 23:44:49 2022 +0900
@@ -155,27 +155,32 @@
 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 )) 
+     isSomeA : A ∋ someA
+     isSomeA =  x∋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 = z07 } where
+     HasMaximal = record { od = record { def = λ x → (m : Ordinal) →  odef A m → odef A x ∧ (¬ (* x < * m))} ; odmax = & A ; <odmax = {!!} } 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 = {!!} } 
+     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 ) ⟫
+     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 (λ m am  →  P m am ))
      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< 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 )
+         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∅
          → (x : Ordinal) → ((y : Ordinal) → y o< x →  ZChain A y _<_ )
          →  ZChain A x _<_
      ind nomx x prev with Oprev-p x
      ... | yes op with ∋-p A (* x)
-     ... | no ¬Ax = record  { B = ZChain.B zc1 ; B⊆A =  ZChain.B⊆A  zc1 ; total = ZChain.total zc1 ; fb = ZChain.fb zc1 ; B∋fb = ZChain.B∋fb zc1 ; ¬x≤sup = z04 } where
+     ... | no ¬Ax = record  { B = ZChain.B zc1 ; B⊆A =  ZChain.B⊆A  zc1
+               ; total = ZChain.total zc1 ; fb = ZChain.fb zc1 ; B∋fb = ZChain.B∋fb zc1 ; ¬x≤sup = z04 } where
           px = Oprev.oprev op
           zc1 : ZChain A px _<_
           zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
@@ -186,13 +191,24 @@
           ... | 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
+     ... | yes ax = z06 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 = {!!} }
+          z06 : ZChain A x _<_
+          z06 with is-o∅ (& (Gtx ax))
+          ... | yes nogt = ⊥-elim (no-maximum nomx x z06-is-maximal ) where 
+              z06-is-maximal :  (m : Ordinal ) → odef A m  → odef A x ∧ (¬ ( * x < * m ))
+              z06-is-maximal m am  = ⟪ subst (λ k → odef A k) &iso ax , {!!} ⟫ -- ⟪ subst (λ k → odef A k) &iso ax , z07 m am ⟫ where
+                  -- λ  x<m → proj1 (PO (me ax) (me (subst (λ k → odef A k) (sym &iso) am))) x<m  {!!} ) ⟫ where
+                  -- m<x : {!!}
+                  -- m<x = {!!}
+                  -- z07 : ¬ ( * m < * x )
+                  -- z07 =  {!!} -- proj1 ((eq← (≡o∅→=od∅ nogt)) {m} {!!} {!!})
+                  -- proj1 (PO (me ax) (me (subst (λ k → odef A k) (sym &iso) am))) x<m ⟪ {!!} , {!!} ⟫ ) ⟫
+          ... | no not = record { B = ZChain.B (prev px (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc)) , * x
+              ; B⊆A = {!!} ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} }
+          -- minimal (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))
      ind nomx x prev | no ¬ox with trio< (& A) x
      ... | tri< a ¬b ¬c = {!!}
      ... | tri≈ ¬a b ¬c = {!!}
@@ -203,9 +219,9 @@
          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) )
+         zorn01 =  proj1 (zorn03 (& someA) isSomeA ) 
          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 )
+         zorn02 {x} ax m<x = proj2 (zorn03 (& x) ax) (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