--- a/ordinal.agda	Sat Jul 20 08:04:20 2019 +0900
+++ b/ordinal.agda	Thu Aug 01 12:24:26 2019 +0900
@@ -13,13 +13,11 @@
    OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
 
 record Ordinal {n : Level} : Set n where
+   constructor ordinal  
    field
      lv : Nat
      ord : OrdinalD {n} lv
 
---
---    Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ...  < ℵ (Suc lv)
---
 data _d<_ {n : Level} :   {lx ly : Nat} → OrdinalD {n} lx  →  OrdinalD {n} ly  → Set n where
    Φ<  : {lx : Nat} → {x : OrdinalD {n} lx}  →  Φ lx d< OSuc lx x
    s<  : {lx : Nat} → {x y : OrdinalD {n} lx}  →  x d< y  → OSuc  lx x d< OSuc  lx y
@@ -114,10 +112,22 @@
 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
 ¬a≤a  (s≤s lt) = ¬a≤a  lt
 
+a<sa : {la : Nat} → la < Suc la 
+a<sa {Zero} = s≤s z≤n
+a<sa {Suc la} = s≤s a<sa 
+
 =→¬< : {x : Nat  } → ¬ ( x < x )
 =→¬< {Zero} ()
 =→¬< {Suc x} (s≤s lt) = =→¬< lt
 
+<-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) )
+<-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl
+<-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n)
+<-∨ {Suc x} {Zero} (s≤s ())
+<-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt
+<-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq)
+<-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
+
 case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥
 case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2
 ... | refl = nat-≡< refl lt1
@@ -126,7 +136,7 @@
 case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2
 ... | refl = nat-≡< refl lt1
 
-o<¬≡ : {n : Level } { ox oy : Ordinal {n}} → ox ≡ oy  → ox o< oy  → ⊥
+o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy  → ox o< oy  → ⊥
 o<¬≡ {_} {ox} {ox} refl (case1 lt) =  =→¬< lt
 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt
 
@@ -224,6 +234,12 @@
    lemma1 (case1 x) = ¬a x
    lemma1 (case2 x) = ≡→¬d< x
 
+xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa  → ox o< ob ) → oa o< osuc ob
+xo<ab {n}  {oa} {ob} a→b with trio< oa ob
+xo<ab {n}  {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
+xo<ab {n}  {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
+xo<ab {n}  {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c )  )
+
 maxα : {n : Level} →  Ordinal {suc n} →  Ordinal  → Ordinal
 maxα x y with trio< x y
 maxα x y | tri< a ¬b ¬c = y
@@ -294,8 +310,6 @@
 proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt
 ... | refl = case2 (s< lt)
 
--- omax≡ (omax x x ) (osuc x) (omxx x)
-
 OrdTrans : {n : Level} → Transitive {suc n} _o≤_
 OrdTrans (case1 refl) (case1 refl) = case1 refl
 OrdTrans (case1 refl) (case2 lt2) = case2 lt2
@@ -313,15 +327,31 @@
      }
  }
 
-TransFinite : {n m : Level} → { ψ : Ordinal {n} → Set m }
-  → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
+TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m }
+  → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx)  → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
   → ( ∀ (lx : Nat ) → (x : OrdinalD lx )  → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
   →  ∀ (x : Ordinal)  → ψ x
-TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv
-TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = 
-      caseOSuc lx ox (TransFinite caseΦ caseOSuc  record { lv = lx ; ord = ox })
+TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where
+  TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx)  → ψ x ) )
+  TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where
+      lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x
+      lemma x (case1 ())
+      lemma x (case2 ())
+      lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x
+      lemma1 x (case1 ())
+      lemma1 x (case2 ())
+  TransFinite1 (Suc lx) (Φ (Suc lx)) = record { proj1 = caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) ; proj2 = (λ x → lemma (lv x) (ord x)) } where
+      lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy  o< ordinal lx (Φ lx) → ψ (ordinal ly oy)
+      lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt
+      lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy  o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy)
+      lemma lx1 ox1 (case1 lt) with <-∨ lt
+      lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) )
+      lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa)) 
+      lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0  lx (Φ lx) (case1 (s≤s lt1))
+      lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa ))) 
+  TransFinite1 lx (OSuc lx ox)  = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )}
 
--- we cannot prove this in intutonistic logic 
+-- we cannot prove this in intutionistic logic 
 --  (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p )  → p
 -- postulate 
 --  TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) 
@@ -337,4 +367,3 @@
   → ¬ p
 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) 
 
-