Members/kono/Proof/ZF-in-agda: OD.agda annotate

annotate OD.agda @ 291:ef93c56ad311

...

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Fri, 12 Jun 2020 19:21:14 +0900
parents	359402cc6c3d 5de8905a5a2b
children	be6670af87fa e70980bd80c7

rev	line source
16 ac362cc8b10f ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 15 diff changeset	1 open import Level
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	2 open import Ordinals
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	3 module OD {n : Level } (O : Ordinals {n} ) where
3 e7990ff544bf reocrd ZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: diff changeset	4
14 e11e95d5ddee separete constructible set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 11 diff changeset	5 open import zf
23 7293a151d949 Sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 22 diff changeset	6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
14 e11e95d5ddee separete constructible set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 11 diff changeset	7 open import Relation.Binary.PropositionalEquality
e11e95d5ddee separete constructible set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 11 diff changeset	8 open import Data.Nat.Properties
6 d9b704508281 isEquiv and isZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 4 diff changeset	9 open import Data.Empty
d9b704508281 isEquiv and isZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 4 diff changeset	10 open import Relation.Nullary
d9b704508281 isEquiv and isZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 4 diff changeset	11 open import Relation.Binary
d9b704508281 isEquiv and isZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 4 diff changeset	12 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 4 diff changeset	13
213 22d435172d1a separate logic and nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 210 diff changeset	14 open import logic
22d435172d1a separate logic and nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 210 diff changeset	15 open import nat
22d435172d1a separate logic and nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 210 diff changeset	16
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	17 open inOrdinal O
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	18
27 bade0a35fdd9 OD, HOD, TC Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 26 diff changeset	19 -- Ordinal Definable Set
11 2df90eb0896c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 10 diff changeset	20
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	21 record OD : Set (suc n ) where
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	22 field
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	23 def : (x : Ordinal ) → Set n
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	24
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	25 open OD
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	26
120 ac214eab1c3c inifinite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 119 diff changeset	27 open _∧_
213 22d435172d1a separate logic and nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 210 diff changeset	28 open _∨_
22d435172d1a separate logic and nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 210 diff changeset	29 open Bool
44 fcac01485f32 od→lv : {n : Level} → OD {n} → Nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	30
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	31 record _==_ ( a b : OD ) : Set n where
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	32 field
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	33 eq→ : ∀ { x : Ordinal } → def a x → def b x
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	34 eq← : ∀ { x : Ordinal } → def b x → def a x
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	35
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	36 id : {A : Set n} → A → A
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	37 id x = x
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	38
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	39 ==-refl : { x : OD } → x == x
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	40 ==-refl {x} = record { eq→ = id ; eq← = id }
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	41
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	42 open _==_
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	43
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	44 ==-trans : { x y z : OD } → x == y → y == z → x == z
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	45 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) }
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	46
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	47 ==-sym : { x y : OD } → x == y → y == x
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	49
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	50
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
120 ac214eab1c3c inifinite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 119 diff changeset	54
277 d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	55 -- next assumptions are our axiom
290 359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	56 -- In classical Set Theory, HOD is used, as a subset of OD,
277 d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	57 -- HOD = { x \| TC x ⊆ OD }
290 359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	58 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	59 -- This is not possible because we don't have V yet.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	60 -- We simply assume V=OD here.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	61 --
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	62 -- We also assumes ODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	63 -- ODs have an ovbious maximum, but Ordinals are not. This means, od→ord is not an on-to mapping.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	64 --
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	65 -- ==→o≡ is necessary to prove axiom of extensionality.
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	66 --
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	67 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
359402cc6c3d definition of filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	68 -- we need explict assumption on sup.
277 d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	69
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	70 record ODAxiom : Set (suc n) where
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	71 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	72 field
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	73 od→ord : OD → Ordinal
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	74 ord→od : Ordinal → OD
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	75 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	76 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	77 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	78 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	79 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	80 sup-o : ( OD → Ordinal ) → Ordinal
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	81 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	82 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	83 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	84 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	85
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	86 postulate odAxiom : ODAxiom
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	87 open ODAxiom odAxiom
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	88
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	89 data One : Set n where
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	90 OneObj : One
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	91
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	92 -- Ordinals in OD , the maximum
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	93 Ords : OD
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	94 Ords = record { def = λ x → One }
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	95
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	96 maxod : {x : OD} → od→ord x o< od→ord Ords
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	97 maxod {x} = c<→o< OneObj
d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	98
179 aa89d1b8ce96 fix comments Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 176 diff changeset	99 -- Ordinal in OD ( and ZFSet ) Transitive Set
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	100 Ord : ( a : Ordinal ) → OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	101 Ord a = record { def = λ y → y o< a }
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	102
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	103 od∅ : OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	104 od∅ = Ord o∅
40 9439ff020cbd ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	105
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	106
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	107 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x)
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	108 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	109 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	110 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt))
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	111 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	112 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt )
123 0c2cbf37e002 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 122 diff changeset	113
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	114 _∋_ : ( a x : OD ) → Set n
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	115 _∋_ a x = def a ( od→ord x )
95 f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	116
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	117 _c<_ : ( x a : OD ) → Set n
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	118 x c< a = a ∋ x
103 c8b79d303867 starting over HOD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 100 diff changeset	119
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	120 cseq : {n : Level} → OD → OD
140 312e27aa3cb5 remove otrans again. start over Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 139 diff changeset	121 cseq x = record { def = λ y → def x (osuc y) } where
113 5f97ebaeb89b ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 112 diff changeset	122
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	123 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
95 f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	124 def-subst df refl refl = df
f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	125
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	126 sup-od : ( OD → OD ) → OD
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	127 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )
95 f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	128
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	129 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	130 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )}
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	131 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	132 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x))
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	133 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) )
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	134
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	135 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
187 ac872f6b8692 add Todo Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 186 diff changeset	136 otrans x<a y<x = ordtrans y<x x<a
123 0c2cbf37e002 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 122 diff changeset	137
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	138 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	139 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso
44 fcac01485f32 od→lv : {n : Level} → OD {n} → Nat Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	140
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	141
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	142 -- avoiding lv != Zero error
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	143 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	144 orefl refl = refl
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	145
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	146 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	147 ==-iso {x} {y} eq = record {
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	148 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	149 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	150 where
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	151 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	152 lemma {x} {z} d = def-subst d oiso refl
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	153
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	154 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y)
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	155 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso)
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	156
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	157 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	158 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	159 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	160 lemma ox ox refl = ==-refl
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	161
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	162 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	163 o≡→== {x} {.x} refl = ==-refl
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	164
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	165 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	166 o∅≡od∅ = ==→o≡ lemma where
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	167 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	168 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	169 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	170 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	171 lemma : ord→od o∅ == od∅
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	172 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	173
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	174 ord-od∅ : od→ord (od∅ ) ≡ o∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	175 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
80 461690d60d07 remove ∅-base-def Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 79 diff changeset	176
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	177 ∅0 : record { def = λ x → Lift n ⊥ } == od∅
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	178 eq→ ∅0 {w} (lift ())
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	179 eq← ∅0 {w} lt = lift (¬x<0 lt)
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	180
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	181 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ )
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	182 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	183 ∅< {x} {y} d eq \| lift ()
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	184
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	185 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	186 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	187
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	188 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x
76 8e8f54e7a030 extensionality done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 74 diff changeset	189 def-iso refl t = t
8e8f54e7a030 extensionality done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 74 diff changeset	190
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	191 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	192 is-o∅ x with trio< x o∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	193 is-o∅ x \| tri< a ¬b ¬c = no ¬b
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	194 is-o∅ x \| tri≈ ¬a b ¬c = yes b
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	195 is-o∅ x \| tri> ¬a ¬b c = no ¬b
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	196
254 2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	197 _,_ : OD → OD → OD
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	198 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y))
188 1f2c8b094908 axiom of choice → p ∨ ¬ p Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 187 diff changeset	199
79 c07c548b2ac1 add some lemma Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 78 diff changeset	200 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	201 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
59 d13d1351a1fa lemma = cong₂ (λ x not → minimul x not ) oiso { }6 Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 58 diff changeset	202
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	203 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	204 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	205
96 f239ffc27fd0 Power Set and L Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 95 diff changeset	206 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
97 f2b579106770 power set using sup on Def Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 96 diff changeset	207
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	208 ZFSubset : (A x : OD ) → OD
191 9eb6a8691f02 choice function cannot jump between ordinal level Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 190 diff changeset	209 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set
97 f2b579106770 power set using sup on Def Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 96 diff changeset	210
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	211 Def : (A : OD ) → OD
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	212 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) )
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	213
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	214 -- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	215 -- _⊆_ A B {x} = A ∋ x → B ∋ x
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	216
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	217 record _⊆_ ( A B : OD ) : Set (suc n) where
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	218 field
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	219 incl : { x : OD } → A ∋ x → B ∋ x
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	220
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	221 open _⊆_
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	222
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	223 infixr 220 _⊆_
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	224
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	225 subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	226 subset-lemma {A} {x} = record {
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	227 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	228 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	229 }
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	230
261 d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	231 open import Data.Unit
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	232
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	233 ε-induction : { ψ : OD → Set (suc n)}
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	234 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	235 → (x : OD ) → ψ x
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	236 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	237 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	238 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	239 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	240 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy
d9d178d1457c ε-induction from TransFinite induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 260 diff changeset	241
262 53744836020b CH trying ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 261 diff changeset	242 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
53744836020b CH trying ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 261 diff changeset	243 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )
53744836020b CH trying ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 261 diff changeset	244
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	245 OD→ZF : ZF
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	246 OD→ZF = record {
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	247 ZFSet = OD
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	248 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	249 ; _≈_ = _==_
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	250 ; ∅ = od∅
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	251 ; _,_ = _,_
f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	252 ; Union = Union
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	253 ; Power = Power
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	254 ; Select = Select
f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	255 ; Replace = Replace
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	256 ; infinite = infinite
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	257 ; isZF = isZF
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	258 } where
287 5de8905a5a2b ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	259 ZFSet = OD -- is less than Ords because of maxod
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	260 Select : (X : OD ) → ((x : OD ) → Set n ) → OD
156 3e7475fb28db differeent Union approach Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 155 diff changeset	261 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	262 Replace : OD → (OD → OD ) → OD
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	263 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x }
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	264 _∩_ : ( A B : ZFSet ) → ZFSet
145 f0fa9a755513 all done but ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 144 diff changeset	265 A ∩ B = record { def = λ x → def A x ∧ def B x }
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	266 Union : OD → OD
156 3e7475fb28db differeent Union approach Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 155 diff changeset	267 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	268 _∈_ : ( A B : ZFSet ) → Set n
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	269 A ∈ B = B ∋ A
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	270 Power : OD → OD
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	271 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
277 d9d3654baee1 seperate choice from LEM Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 276 diff changeset	272 -- ｛_｝ : ZFSet → ZFSet
287 5de8905a5a2b ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 277 diff changeset	273 -- ｛ x ｝ = ( x , x ) -- it works but we don't use
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	274
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	275 data infinite-d : ( x : Ordinal ) → Set n where
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	276 iφ : infinite-d o∅
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	277 isuc : {x : Ordinal } → infinite-d x →
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	278 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	279
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	280 infinite : OD
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	281 infinite = record { def = λ x → infinite-d x }
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	282
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	283 infixr 200 _∈_
96 f239ffc27fd0 Power Set and L Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 95 diff changeset	284 -- infixr 230 _∩_ _∪_
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	285 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	286 isZF = record {
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	287 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
247 d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	288 ; pair→ = pair→
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	289 ; pair← = pair←
72 f39f1a90d154 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 71 diff changeset	290 ; union→ = union→
f39f1a90d154 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 71 diff changeset	291 ; union← = union←
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	292 ; empty = empty
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	293 ; power→ = power→
76 8e8f54e7a030 extensionality done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 74 diff changeset	294 ; power← = power←
186 914cc522c53a fix extensionality Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 185 diff changeset	295 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
274 29a85a427ed2 ε-induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 272 diff changeset	296 ; ε-induction = ε-induction
78 9a7a64b2388c infinite and replacement begin Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 77 diff changeset	297 ; infinity∅ = infinity∅
160 ebac6fa116fa ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 159 diff changeset	298 ; infinity = infinity
116 47541e86c6ac axiom of selection Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 115 diff changeset	299 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
135 b60b6e8a57b0 otrans in repl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 134 diff changeset	300 ; replacement← = replacement←
b60b6e8a57b0 otrans in repl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 134 diff changeset	301 ; replacement→ = replacement→
274 29a85a427ed2 ε-induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 272 diff changeset	302 -- ; choice-func = choice-func
29a85a427ed2 ε-induction Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 272 diff changeset	303 -- ; choice = choice
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	304 } where
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	305
247 d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	306 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	307 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x ))
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	308 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y ))
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	309
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	310 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	311 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x))
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	312 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y))
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	313
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	314 empty : (x : OD ) → ¬ (od∅ ∋ x)
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	315 empty x = ¬x<0
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	316
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	317 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	318 o<→c< lt = record { incl = λ z → ordtrans z lt }
155 53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	319
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	320 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
155 53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	321 ⊆→o< {x} {y} lt with trio< x y
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	322 ⊆→o< {x} {y} lt \| tri< a ¬b ¬c = ordtrans a <-osuc
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	323 ⊆→o< {x} {y} lt \| tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	324 ⊆→o< {x} {y} lt \| tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
155 53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	325 ... \| ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
151 b5a337fb7a6d recovering... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 150 diff changeset	326
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	327 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
157 afc030b7c8d0 explict logical definition of Union failed Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 156 diff changeset	328 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
afc030b7c8d0 explict logical definition of Union failed Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 156 diff changeset	329 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
159 3675bd617ac8 infinite continue... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 158 diff changeset	330 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	331 union← X z UX∋z = FExists _ lemma UX∋z where
165 d16b8bf29f4f minor fix Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 162 diff changeset	332 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
d16b8bf29f4f minor fix Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 162 diff changeset	333 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	334
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	335 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	336 ψiso {ψ} t refl = t
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	337 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	338 selection {ψ} {X} {y} = record {
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	339 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	340 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	341 }
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	342 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	343 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	344 lemma : def (in-codomain X ψ) (od→ord (ψ x))
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	345 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	346 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	347 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	348 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	349 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	350 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	351 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y))
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	352 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq )
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	353 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	354 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 ))
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	355
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	356 ---
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	357 --- Power Set
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	358 ---
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	359 --- First consider ordinals in OD
100 a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	360 ---
a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	361 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A
a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	362 --
a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	363 --
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	364 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	365 ∩-≡ {a} {b} inc = record {
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	366 eq→ = λ {x} x<a → record { proj2 = x<a ;
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	367 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	368 eq← = λ {x} x<a∩b → proj2 x<a∩b }
100 a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	369 --
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	370 -- Transitive Set case
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	371 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	372 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	373 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
100 a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	374 --
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	375 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	376 ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t}
127 870fe02c7a07 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 126 diff changeset	377 lemma refl (lemma1 lemma-eq )where
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	378 lemma-eq : ZFSubset (Ord a) t == t
97 f2b579106770 power set using sup on Def Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 96 diff changeset	379 eq→ lemma-eq {z} w = proj2 w
f2b579106770 power set using sup on Def Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 96 diff changeset	380 eq← lemma-eq {z} w = record { proj2 = w ;
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	381 proj1 = def-subst {_} {_} {(Ord a)} {z}
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	382 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	383 lemma1 : {a : Ordinal } { t : OD }
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	384 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	385 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	386 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x))
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	387 lemma = sup-o<
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	388
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	389 --
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	390 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	391 -- then replace of all elements of the Power set by A ∩ y
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	392 --
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	393 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
166 ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	394
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	395 -- we have oly double negation form because of the replacement axiom
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	396 --
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	397 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	398 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	399 a = od→ord A
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	400 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y)))
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	401 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
166 ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	402 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	403 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	404 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	405 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
166 ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	406 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x))
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	407 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	408
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	409 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	410 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	411 a = od→ord A
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	412 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	413 lemma0 {x} t∋x = c<→o< (t→A t∋x)
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	414 lemma3 : Def (Ord a) ∋ t
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	415 lemma3 = ord-power← a t lemma0
152 996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	416 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	417 lemma4 = let open ≡-Reasoning in begin
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	418 A ∩ ord→od (od→ord t)
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	419 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	420 A ∩ t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	421 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	422 t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	423 ∎
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	424 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x))
6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	425 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x)))
6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	426 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} )
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	427 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
151 b5a337fb7a6d recovering... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 150 diff changeset	428 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
b5a337fb7a6d recovering... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 150 diff changeset	429 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
b5a337fb7a6d recovering... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 150 diff changeset	430 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A )))
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	431
271 2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	432 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	433 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	434 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	435 lemma lt y<x with osuc-≡< lt
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	436 lemma lt y<x \| case1 refl = c<→o< y<x
2169d948159b fix incl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 262 diff changeset	437 lemma lt y<x \| case2 x<a = ordtrans (c<→o< y<x) x<a
262 53744836020b CH trying ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 261 diff changeset	438
276 6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	439 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
6f10c47e4e7a separate choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 274 diff changeset	440 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	441
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	442 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	443 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	444 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
186 914cc522c53a fix extensionality Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 185 diff changeset	445
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	446 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
186 914cc522c53a fix extensionality Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 185 diff changeset	447 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
914cc522c53a fix extensionality Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 185 diff changeset	448 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	449
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	450 infinity∅ : infinite ∋ od∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	451 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	452 lemma : o∅ ≡ od→ord od∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	453 lemma = let open ≡-Reasoning in begin
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	454 o∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	455 ≡⟨ sym diso ⟩
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	456 od→ord ( ord→od o∅ )
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	457 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	458 od→ord od∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	459 ∎
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	460 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	461 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	462 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	463 ≡ od→ord (Union (x , (x , x)))
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	464 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	465
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	466
226 176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	467 Union = ZF.Union OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	468 Power = ZF.Power OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	469 Select = ZF.Select OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	470 Replace = ZF.Replace OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	471 isZF = ZF.isZF OD→ZF

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

annotate OD.agda @ 291:ef93c56ad311