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

changeset 340:639fbb6284d8

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 13 Jul 2020 09:26:34 +0900
parents feb0fcc430a9
children 27d2933c4bd7
files OD.agda Ordinals.agda
diffstat 2 files changed, 22 insertions(+), 1 deletions(-) [+]
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line diff
--- a/OD.agda	Sun Jul 12 19:55:37 2020 +0900
+++ b/OD.agda	Mon Jul 13 09:26:34 2020 +0900
@@ -387,10 +387,28 @@
             lemma3 = FExists _ (λ {z} t not → not (od→ord (ord→od y , ord→od y)) record { proj1 = case2 refl ; proj2 = lemma4 }) (λ not → not y (infinite-d y)) where
                 lemma4 : def (od (ord→od (od→ord (ord→od y , ord→od y)))) y 
                 lemma4 = subst₂ ( λ j k → def (od j) k ) (sym oiso) diso (case1 refl)
+            lemma5 : y o< odmax (u y)
+            lemma5 = <odmax (u y) lemma3
+            lemma6 : y o< odmax (ord→od y , (ord→od y , ord→od y))
+            lemma6 = <odmax (ord→od y , (ord→od y , ord→od y)) (subst ( λ k → def (od (ord→od y , (ord→od y , ord→od y))) k ) diso (case1 refl))
+            lemma8 : od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y))
+            lemma8 = ho<
+            lemma81 : od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y))
+            lemma81 = nexto=n (subst (λ k →  od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
+            lemma7 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (odmax (ord→od y , (ord→od y , ord→od y)))
+            lemma91 : od→ord (ord→od y) o< od→ord (ord→od y , ord→od y)
+            lemma91 = c<→o< (case1 refl) 
+            lemma92 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next y
+            lemma92 = {!!}
+            lemma9 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y))
+            lemma9 = next< {!!} lemma92
+            lemma7 = ho<
+            lemma71 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y))
+            lemma71 = next< lemma81 lemma9
             lemma1 : od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y))))
             lemma1 = ho<
             lemma2 : od→ord (u y) o< next o∅
-            lemma2 = {!!}
+            lemma2 = next< lemma0 (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1))
         
 
     _=h=_ : (x y : HOD) → Set n
--- a/Ordinals.agda	Sun Jul 12 19:55:37 2020 +0900
+++ b/Ordinals.agda	Mon Jul 13 09:26:34 2020 +0900
@@ -228,6 +228,9 @@
         next< {x} {y} {z} x<nz y<nx | tri> ¬a ¬b c = ⊥-elim (proj2 (proj2 next-limit) (next z) x<nz (ordtrans c y<nx )
            (λ w nz=ow → o<¬≡ (sym nz=ow) (proj1 (proj2 next-limit) _ (subst (λ k → w o< k ) (sym nz=ow) <-osuc ))))
 
+        nexto=n : {x y : Ordinal} → x o< next (osuc y)  → x o< next y 
+        nexto=n {x} {y} x<noy = next< (proj1 (proj2 next-limit) _ (proj1 next-limit)) x<noy
+
         record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
           field
             os→ : (x : Ordinal) → x o< maxordinal → Ordinal