# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1567015384 -32400
# Node ID 8a58e2cd1f55393e495c662f3ee7fea9be3e512e
# Parent  9e0125b06e767ac43d96701b77110b42984507a9
give up product uniquness

diff -r 9e0125b06e76 -r 8a58e2cd1f55 cardinal.agda
--- a/cardinal.agda	Thu Aug 29 01:04:52 2019 +0900
+++ b/cardinal.agda	Thu Aug 29 03:03:04 2019 +0900
@@ -119,43 +119,15 @@
         od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
     ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
         od→ord < x , y >
-    ∎ )
-
-
-lemma44 : {ox oy : Ordinal } → ord-pair (od→ord < ord→od ox , ord→od oy >)
-lemma44 {ox} {oy} = pair ox oy
-
-lemma55 : {ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
-lemma55 {ox} {oy} = pair ox oy
-
-lemma66 : {ox oy : Ordinal } →  pair ( pi1 ( pair ox oy )) ( pi2 ( pair ox oy )) ≡ pair ox oy
-lemma66  = refl 
-
-lemma77 : {ox oy : Ordinal } → ZFProduct ∋ < ord→od (pi1 ( pair ox oy ))  , ord→od (pi2 ( pair ox oy ))  >  ≡ ZFProduct ∋ < ord→od ox , ord→od oy > 
-lemma77  = refl
+    ∎ ) 
 
 
-p-iso :  { x  : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
-p-iso {x} p = {!!} where
-
-pair-iso : {op ox oy : Ordinal} (x : ord-pair (od→ord < ord→od ox , ord→od oy >) )  → pi1 x ≡ ox → pi2 x ≡ oy → x ≡ pair ox oy
-pair-iso (pair ox oy) = {!!}
+p-iso1 :  { ox oy  : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >  
+p-iso1 {ox} {oy} = pair ox oy
 
-p-iso3 : { ox oy  : Ordinal } →  (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → p ≡ pair ox oy
-p-iso3 {ox} {oy} p with p-iso p
-... | eq with prod-eq ( ord→== (cong (λ k → od→ord k) eq ) )
-... | record { proj1 = eq1 ; proj2 = eq2 } = lemma eq1 eq2 where
-    lemma : ord→od (pi1 p) ≡ ord→od ox → ord→od (pi2 p) ≡ ord→od oy → p ≡ pair ox oy
-    lemma eq1 eq2 with od≡→≡ eq1 | od≡→≡ eq2
-    ... | eq1' | eq2' = pair-iso {od→ord  < ord→od ox , ord→od oy >} {ox} {oy} p eq1' eq2'
+postulate
+    p-iso :  { x  : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
 
-p-iso2 :  { ox oy  : Ordinal } → p-cons (ord→od ox) (ord→od oy) ≡ pair ox oy
-p-iso2  {ox} {oy} = p-iso3 (p-cons (ord→od ox) (ord→od oy))
-
-p-iso1 :  { ox oy  : Ordinal  } → (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → < ord→od (π1 p) , ord→od (π2 p) > ≡  < ord→od ox , ord→od oy >
-p-iso1 {x} {y} p with p-cons (ord→od (π1 p)) (ord→od (π2 p))
-... | t with p-iso3 p | p-iso3 t
-... | refl | refl  = refl
     
 ∋-p : (A x : OD ) → Dec ( A ∋ x ) 
 ∋-p A x with p∨¬p ( A ∋ x )