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

changeset 11:2df90eb0896c

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 13 May 2019 20:51:45 +0900
parents 8022e14fce74
children b531d2b417ad e11e95d5ddee
files zf.agda
diffstat 1 files changed, 59 insertions(+), 28 deletions(-) [+]
line wrap: on
line diff
--- a/zf.agda	Mon May 13 18:25:38 2019 +0900
+++ b/zf.agda	Mon May 13 20:51:45 2019 +0900
@@ -117,55 +117,86 @@
   minimul =   IsZF.minimul  ( ZF.isZF zf )
   regularity =   IsZF.regularity  ( ZF.isZF zf )
 
-  russel : Select ( λ x →  x ∋ x  ) ≈ ∅
-  russel with Select ( λ x →  x ∋ x  ) 
-  ... | s = {!!}
+--  russel : Select ( λ x →  x ∋ x  ) ≈ ∅
+--  russel with Select ( λ x →  x ∋ x  ) 
+--  ... | s = {!!}
 
 module constructible-set  where
 
-  data Nat : Set zero where
-    Zero : Nat
-    Suc : Nat → Nat
-  
-  prev : Nat → Nat
-  prev (Suc n) = n
-  prev Zero = Zero
+  open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ) 
   
   open import  Relation.Binary.PropositionalEquality
   
-  data Transtive {n : Level }  : ( lv : Nat) → Set n where
-     Φ : {lv : Nat} → Transtive {n} lv
-     T-suc : {lv : Nat} → Transtive {n} lv → Transtive lv
-     ℵ_ :  (lv : Nat) → Transtive (Suc lv)
+  data Ordinal {n : Level }  :  Set n where
+     Φ : {lv : Nat} → Ordinal {n} lv
+     T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv
+     ℵ_ :  (lv : Nat) → Ordinal (Suc lv)
+
+  data _o<_ {n : Level } :  Ordinal {n}  →  Ordinal {n}  → Set n where
+     l< : {lx ly : Nat }  → {x : Ordinal {n} lx } →  {y : Ordinal {n} ly } → lx < ly → x o< y
+     Φ<  : {lx : Nat} → {x : Ordinal {n} lx}  →  Φ {n} {lx} o< T-suc {n} {lx} x
+     s<  : {lx : Nat} → {x : Ordinal {n} lx}  →  x o< T-suc {n} {lx} x
+     ℵΦ< : {lx : Nat} → {x : Ordinal {n} lx } →  Φ {n} {lx} o< (ℵ lx) 
+     ℵ<  : {lx : Nat} → {x : Ordinal {n} lx } →  T-suc {n} {lx} x o< (ℵ lx) 
+
+  _o≈_ : {n : Level } {lv : Nat } → Rel ( Ordinal {n} lv ) n
+  _o≈_  = {!!}
+
+  triO : {n : Level } → {lx ly : Nat}   → Trichotomous  _o≈_ ( _o<_ {n} {lx} {ly} )
+  triO {n} {lv} Φ y = {!!}
+  triO {n} {lv} (T-suc x) y = {!!}
+  triO {n} {.(Suc lv)} (ℵ lv) y = {!!}
+
+
+  max = Data.Nat._⊔_
   
-  data Constructible {n : Level } {lv : Nat} ( α : Transtive {n} lv )  :  Set (suc n) where
-     fsub : ( ψ : Transtive {n} lv → Set n ) → Constructible  α
-     xself : Transtive {n} lv → Constructible  α
+  maxα : {n : Level } → { lx ly : Nat } → Ordinal {n} lx  →  Ordinal {n} ly  → Ordinal {n} (max lx ly)
+  maxα x y with x o< y
+  ... | t = {!!}
+
+  -- X' = { x ∈ X |  ψ  x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
+
+  data Constructible {n : Level } {lv : Nat} ( α : Ordinal {n} lv )  :  Set (suc n) where
+     fsub : ( ψ : Ordinal {n} lv → Set n ) → Constructible  α
+     xself : Ordinal {n} lv → Constructible  α
   
   record ConstructibleSet {n : Level } : Set (suc n) where
    field
       level : Nat
-      α : Transtive {n} level 
+      α : Ordinal {n} level 
       constructible : Constructible α
   
   open ConstructibleSet
   
-  data _c∋_ {n : Level } {lv lv' : Nat} {α : Transtive {n} lv } {α' : Transtive {n} lv' } :
+  data _c∋_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
         Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
-     xself-fsub  : (ta : Transtive {n} lv ) ( ψ : Transtive {n} lv' → Set n ) → (xself ta ) t∋ ( fsub ψ)  
-     xself-xself :  (ta : Transtive {n} lv ) (tx : Transtive {n} lv' )  → (xself ta ) t∋ ( xself tx) 
-     fsub-fsub :  ( ψ : Transtive {n} lv → Set n ) ( ψ₁ : Transtive {n} lv' → Set n ) →(  fsub ψ )  t∋ ( fsub ψ₁) 
-     fsub-xself  :   ( ψ : Transtive {n} lv → Set n ) (tx : Transtive {n} lv' ) → (fsub ψ )  t∋ ( xself tx) 
+     c> : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' }
+        (ta : Constructible {n} {lv} α ) ( tx : Constructible {n} {lv'} α' ) → α' o< α →  ta c∋ tx
+     xself-fsub  : {lv : Nat} {α : Ordinal {n} lv } 
+         (ta : Ordinal {n} lv ) ( ψ : Ordinal {n} lv → Set n ) → _c∋_ {n} {_} {_} {α} {α} (xself ta ) ( fsub ψ)  
+     fsub-fsub : {lv lv' : Nat} {α : Ordinal {n} lv } 
+          ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) →
+         ( ∀ ( x :  Ordinal {n} lv ) → ψ x →  ψ₁ x ) →  _c∋_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) 
 
-  _∋_  : {n m : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set m 
-  _∋_ = {!!}
+  _∋_  : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n 
+  a ∋ x  = constructible a c∋ constructible x
+
+  data _c≈_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
+        Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
+      crefl :  {lv : Nat} {α : Ordinal {n} lv } → _c≈_ {n} {_} {_} {α} {α} (xself α ) (xself α )
+      feq :  {lv : Nat} {α : Ordinal {n} lv }
+          → ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) 
+          → (∀ ( x :  Ordinal {n} lv ) → ψ x  ⇔ ψ₁ x ) → _c≈_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
+
+  _≈_  : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n 
+  a ≈ x  = constructible a c≈ constructible x
       
   
-  Transtive→ZF : {n m : Level } → ZF {suc n} {m}
-  Transtive→ZF {n} {m} = record {
+  ConstructibleSet→ZF : {n : Level } → ZF {suc n} {n}
+  ConstructibleSet→ZF {n}  = record {
        ZFSet = ConstructibleSet 
      ; _∋_ = _∋_
-     ; _≈_ = {!!}
+     ; _≈_ = _≈_ 
      ; ∅  = record { level = Zero ; α = Φ ; constructible = xself Φ }
      ; _×_ = {!!}
      ; Union = {!!}