# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1563480728 -32400
# Node ID acbcbd98d5882409ffa85759ab0f15277750029b
# Parent  b25a4eca06a60a3a964e340fd1cce38fb71808c2
trans finite on ε-induction

diff -r b25a4eca06a6 -r acbcbd98d588 HOD.agda
--- a/HOD.agda	Fri Jul 19 03:27:58 2019 +0900
+++ b/HOD.agda	Fri Jul 19 05:12:08 2019 +0900
@@ -474,16 +474,16 @@
               → (x : OD {suc n} ) → ψ x
          ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x))  <-osuc) where
              ε-induction-ord : ( ox : Ordinal {suc n} ) {oy : Ordinal {suc n} } → oy o< ox  → ψ (ord→od oy)
-             ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case1 ())
-             ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case2 ())
-             ε-induction-ord record { lv = lx ; ord = (OSuc lx ox) } {oy} y<x = 
-                 ind {ord→od oy} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord (record { lv = lx ; ord = ox} ) (lemma y lt ))) where
-                     lemma :  (y : OD) → ord→od oy ∋ y → od→ord y o< record { lv = lx ; ord = ox }
-                     lemma y lt with osuc-≡< y<x
-                     lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
-                     lemma y lt | case2 lt1 = ordtrans  (o<-subst (c<→o< lt) refl diso) lt1
-             ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {record { lv = ly ; ord = OSuc ly oy }} (case1 (s≤s x)) =
-               ind {ord→od record { lv = ly ; ord = OSuc ly oy }} (
-                 λ {y} lt →  subst ( λ k → ψ k ) oiso ( ε-induction-ord record { lv = lx ; ord = (Φ lx) } {od→ord y} {!!})) where
-             ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {record { lv = ly ; ord = Φ ly }} (case1 (s≤s x)) = {!!}
+             ε-induction-ord ox = TransFinite {suc n} {suc n ⊔ m} {λ z → {oy : Ordinal {suc n} } → oy o< z  → ψ (ord→od oy) } lemma1 lemma2 ox where
+                lemma1 : (lx : Nat) {oy : Ordinal} → oy o< record { lv = lx ; ord = Φ lx } → ψ (ord→od oy)
+                lemma1 Zero {oy} (case1 ())
+                lemma1 Zero {oy} (case2 ())
+                lemma1 (Suc lx) {record { lv = Zero ; ord = Φ 0 }} (case1 (s≤s z≤n)) = {!!}
+                lemma1 (Suc lx) {record { lv = Zero ; ord = OSuc 0 oy }} (case1 (s≤s z≤n)) = {!!}
+                lemma1 (Suc (Suc lx)) {record { lv = Suc ly ; ord = Φ (Suc ly) }} (case1 (s≤s (s≤s x))) = {!!}
+                lemma1 (Suc (Suc lx)) {record { lv = Suc ly ; ord = OSuc (Suc ly) oy }} (case1 (s≤s (s≤s x))) = {!!}
+                lemma2 : (lx : Nat) (x₁ : OrdinalD lx) →
+                    ({oy : Ordinal} → oy o< record { lv = lx ; ord = x₁ } → ψ (ord→od oy)) →
+                    {oy : Ordinal} → oy o< record { lv = lx ; ord = OSuc lx x₁ } → ψ (ord→od oy)
+                lemma2 lx x1 p lt = ind ( λ {y} lty → subst (λ k → ψ k) oiso (p {!!} ))