# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1594647637 -32400
# Node ID cfecd05a4061efdae61b6342ac545afe9fee3f12
# Parent  06f10815d0b31ac74fe975117b0d5f6946ab4c30
...

diff -r 06f10815d0b3 -r cfecd05a4061 Ordinals.agda
--- a/Ordinals.agda	Mon Jul 13 19:19:02 2020 +0900
+++ b/Ordinals.agda	Mon Jul 13 22:40:37 2020 +0900
@@ -244,18 +244,22 @@
         nexto≡ {x} | tri> ¬a ¬b c = ⊥-elim ((proj2 (proj2 next-limit)) _ (ordtrans <-osuc (proj1 next-limit)) c
            (λ z eq → o<¬≡ (sym eq) ((proj1 (proj2 next-limit)) _ (osuc< (sym eq)))))
 
+        record prev-choiced  ( x : Ordinal ) : Set (suc n) where
+         field
+            prev : Ordinal
+            is-prev : osuc prev ≡ x 
         not-limit-p :  ( x : Ordinal  ) → Dec ( ¬ ((y : Ordinal) → ¬ (x ≡ osuc y) ))
-        not-limit-p x = TransFinite {λ x → Dec ( ¬ ((y : Ordinal) → ¬ (x ≡ osuc y)))} ind x where
-             ind : (x : Ordinal) → ((y : Ordinal) → y o< x → Dec (¬ ((y₁ : Ordinal) → ¬ y ≡ osuc y₁))) → Dec (¬ ((y : Ordinal) → ¬ x ≡ osuc  y))
-             ind x prev with trio< o∅ x
-             ind x prev | tri< a ¬b ¬c = ?
-             ind x prev | tri≈ ¬a refl ¬c = no (λ not → not lemma) where
-                lemma : (y : Ordinal) → o∅  ≡ osuc y → ⊥
-                lemma y refl with trio< o∅ y
-                lemma y refl | tri< a ¬b ¬c = o<> a <-osuc
-                lemma y refl | tri≈ ¬a b ¬c = o<¬≡ (sym b) <-osuc
-                lemma y refl | tri> ¬a ¬b c = ¬x<0 c
-             ind x prev | tri> ¬a ¬b c = ⊥-elim (  ¬x<0 c )
+        not-limit-p x = {!!} where
+            ψ : ( ox : Ordinal ) → Set (suc n)
+            ψ ox = (( y : Ordinal ) → y o< ox  → ( ¬ (osuc y ≡ x) )) ∨ prev-choiced x
+            ind : (ox : Ordinal) → ((y : Ordinal) → y o< ox → ψ y) → ψ ox 
+            ind ox prev with trio< (osuc ox) x
+            ind ox prev | tri≈ ¬a b ¬c = case2 (record { prev = ox ; is-prev = b })
+            ind ox prev | tri< a ¬b ¬c = case1 (λ y y<ox oy=x → o<> (subst (λ k → k o< osuc ox) oy=x (osucc y<ox )) a ) 
+            --     osuc y = x < osuc ox, y < ox
+            ind ox prev | tri> ¬a ¬b c = {!!}
+            find : (y : Ordinal) → ψ y
+            find ox = TransFinite1 {ψ} ind ox
 
         record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
           field