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

changeset 475:30da20261771

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 01 Apr 2022 20:15:08 +0900
parents 100ceb0fbada
children 3fc164626468
files src/ODC.agda
diffstat 1 files changed, 25 insertions(+), 22 deletions(-) [+]
line wrap: on
line diff
--- a/src/ODC.agda	Thu Mar 31 10:02:23 2022 +0900
+++ b/src/ODC.agda	Fri Apr 01 20:15:08 2022 +0900
@@ -158,15 +158,13 @@
      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 → odef A x → (m : Ordinal) →  odef A m → ¬ (* x < * m)} ; odmax = & A ; <odmax = {!!} } 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)) )))
-     no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x → ((m : Ordinal) →  odef A m → ¬ (* x < * m) ) →  ⊥
-     no-maximum nomx x ax P = ¬x<0 (eq→ nomx {x} (λ ax m am → P m am )) 
+     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} (λ m am → P m am )) 
      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 = {!!} } 
-     no-gtx : { x : HOD} → (ax : A ∋ x ) → Gtx ax =h= od∅ →  (( y : Ordinal) → odef A y → (x < (* y)) ∧ ( (& x) o< y ))  → ⊥
-     no-gtx {x} ax nogt P = ¬x<0 (eq→ nogt (λ am → P (& x) am ))
+     Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* 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 ) ⟫
@@ -181,8 +179,7 @@
          →  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) 
@@ -199,19 +196,25 @@
           zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
           z06 : ZChain A x _<_
           z06 with is-o∅ (& (Gtx ax))
-          ... | yes nogt = ⊥-elim (no-maximum nomx x (subst (λ k → odef A k) &iso ax) z06-is-maximal ) where 
-              z06-is-maximal :  (m : Ordinal ) → odef A m  → ¬ ( * x < * m )
-              z06-is-maximal m am  = z07  where 
-                  z07 : ¬ ( * x < * m )
-                  z07 x<m =  proj1 (PO (me ax) (me (subst (λ k → odef A k) (sym &iso) am))) x<m ⟪ {!!} , {!!} ⟫ 
-              z08 : ( ( y : Ordinal) → odef A y → (* x < (* y)) ∧ ( (& (* x)) o< y ))  → ⊥
-              z08 p = no-gtx ax (≡o∅→=od∅ nogt) (λ y ay → p y ay ) 
-          ... | 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))
+          ... | yes nogt = ⊥-elim (no-maximum nomx x x-is-maximal ) where
+              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) 
+          ... | no not = record { B = Bx
+              ; B⊆A = {!!} ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} } where
+                 B = ZChain.B (prev px (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc)) 
+                 Bx : HOD
+                 Bx = record { od = record { def = λ y → (x ≡ y) ∨ odef B x } ; odmax = {!!} ; <odmax = {!!}  } 
+                 m = 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 = {!!}
+     ... | tri< a ¬b ¬c = {!!} where
+          zc1 : ZChain A (& A) _<_
+          zc1 = prev (& A) a 
+     ... | tri≈ ¬a b ¬c = record { B = ?
+              ; B⊆A = ? ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} } where
+          zc1 : ZChain A {!!} _<_
+          zc1 = prev {!!} {!!}
      ... | tri> ¬a ¬b c = {!!}
      zorn00 : Maximal A _<_
      zorn00 with is-o∅ ( & HasMaximal )
@@ -219,9 +222,9 @@
          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) isSomeA ) 
+         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 (zorn03 (& x) ax) (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