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

annotate OD.agda @ 260:8b85949bde00

sup with limit give up

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Thu, 05 Sep 2019 10:58:06 +0900
parents	f714fe999102
children	d9d178d1457c

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
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	39 eq-refl : { x : OD } → x == x
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	40 eq-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
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	44 eq-sym : { x y : OD } → x == y → y == x
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	45 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	46
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	47 eq-trans : { x y z : OD } → x == y → y == z → x == z
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	48 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	49
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	50 ⇔→== : { 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	51 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	52 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	53
179 aa89d1b8ce96 fix comments Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 176 diff changeset	54 -- 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	55 Ord : ( a : Ordinal ) → OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	56 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	57
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	58 od∅ : OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	59 od∅ = Ord o∅
40 9439ff020cbd ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	60
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	61 -- next assumptions are our axiom
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	62 -- it defines a subset of OD, which is called HOD, usually defined as
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	63 -- HOD = { x \| TC x ⊆ OD }
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	64 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	65
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	66 postulate
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	67 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	68 od→ord : OD → Ordinal
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	69 ord→od : Ordinal → OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	70 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	71 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	72 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	73 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	74 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal is allowed as OD
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	75 -- o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
159 3675bd617ac8 infinite continue... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 158 diff changeset	76 -- ord→od x ≡ Ord x results the same
100 a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	77 -- supermum as Replacement Axiom
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	78 sup-o : ( Ordinal → Ordinal ) → Ordinal
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	79 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
111 1daa1d24348c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 109 diff changeset	80 -- contra-position of mimimulity of supermum required in Power Set Axiom
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	81 -- sup-x : {n : Level } → ( Ordinal → Ordinal ) → Ordinal
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	82 -- sup-lb : {n : Level } → { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	83 -- mimimul and x∋minimal is an Axiom of choice
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	84 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
117 a4c97390d312 minimum assuption Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 116 diff changeset	85 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x )
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	86 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	87 -- minimality (may proved by ε-induction )
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	88 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	89
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	90 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	91 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	92 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	93 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	94 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	95 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	96
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	97 _∋_ : ( a x : OD ) → Set n
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	98 _∋_ a x = def a ( od→ord x )
95 f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	99
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	100 _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	101 x c< a = a ∋ x
103 c8b79d303867 starting over HOD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 100 diff changeset	102
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	103 cseq : {n : Level} → OD → OD
140 312e27aa3cb5 remove otrans again. start over Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 139 diff changeset	104 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	105
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	106 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	107 def-subst df refl refl = df
f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	108
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	109 sup-od : ( OD → OD ) → OD
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	110 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
95 f3da2c87cee0 clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 94 diff changeset	111
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	112 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	113 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od 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	114 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	115 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	116 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	117
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	118 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	119 otrans x<a y<x = ordtrans y<x x<a
123 0c2cbf37e002 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 122 diff changeset	120
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	121 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	122 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	123
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	124
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	125 -- avoiding lv != Zero error
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	126 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	127 orefl refl = refl
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	128
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	129 ==-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	130 ==-iso {x} {y} eq = record {
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	131 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	132 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	133 where
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	134 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	135 lemma {x} {z} d = def-subst d oiso refl
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	136
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	137 =-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	138 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso)
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	139
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	140 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	141 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	142 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	143 lemma ox ox refl = eq-refl
83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	144
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	145 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	146 o≡→== {x} {.x} refl = eq-refl
51 83b13f1f4f42 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 50 diff changeset	147
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	148 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	149 o∅≡od∅ = ==→o≡ lemma where
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	150 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	151 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	152 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	153 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
150 ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	154 lemma : ord→od o∅ == od∅
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	155 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
ebcbfd9d9c8e fix some Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 148 diff changeset	156
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	157 ord-od∅ : od→ord (od∅ ) ≡ o∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	158 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	159
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	160 ∅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	161 eq→ ∅0 {w} (lift ())
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	162 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	163
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	164 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ )
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	165 ∅< {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	166 ∅< {x} {y} d eq \| lift ()
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	167
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	168 ∅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	169 ∅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	170
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	171 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	172 def-iso refl t = t
8e8f54e7a030 extensionality done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 74 diff changeset	173
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	174 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	175 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	176 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	177 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	178 is-o∅ x \| tri> ¬a ¬b c = no ¬b
57 419688a279e0 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	179
254 2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	180 _,_ : OD → OD → OD
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	181 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y))
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	182 <_,_> : (x y : OD) → OD
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	183 < x , y > = (x , x ) , (x , y )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	184
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	185 exg-pair : { x y : OD } → (x , y ) == ( y , x )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	186 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	187 left : {z : Ordinal} → def (x , y) z → def (y , x) z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	188 left (case1 t) = case2 t
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	189 left (case2 t) = case1 t
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	190 right : {z : Ordinal} → def (y , x) z → def (x , y) z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	191 right (case1 t) = case2 t
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	192 right (case2 t) = case1 t
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	193
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	194 ==-trans : { x y z : OD } → x == y → y == z → x == z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	195 ==-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) }
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	196
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	197 ==-sym : { x y : OD } → x == y → y == x
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	198 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	199
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	200 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	201 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	202
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	203 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	204 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	205
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	206 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	207 eq-prod refl refl = refl
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	208
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	209 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	210 prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	211 lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	212 lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	213 lemma0 {x} {y} eq \| tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	214 lemma0 {x} {y} eq \| tri< a ¬b ¬c \| case1 s = ⊥-elim ( o<¬≡ (sym s) a )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	215 lemma0 {x} {y} eq \| tri< a ¬b ¬c \| case2 s = ⊥-elim ( o<¬≡ (sym s) a )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	216 lemma0 {x} {y} eq \| tri≈ ¬a b ¬c = ord≡→≡ b
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	217 lemma0 {x} {y} eq \| tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	218 lemma0 {x} {y} eq \| tri> ¬a ¬b c \| case1 s = ⊥-elim ( o<¬≡ s c )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	219 lemma0 {x} {y} eq \| tri> ¬a ¬b c \| case2 s = ⊥-elim ( o<¬≡ s c )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	220 lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	221 lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	222 lemma3 : ( x , x ) == ( y , z )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	223 lemma3 = ==-trans eq exg-pair
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	224 lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	225 lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	226 lemma1 {x} {y} eq \| case1 s = ord≡→≡ (sym s)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	227 lemma1 {x} {y} eq \| case2 s = ord≡→≡ (sym s)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	228 lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	229 lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	230 lemma4 {x} {y} {z} eq \| case1 s with ord≡→≡ s -- x ≡ z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	231 ... \| refl with lemma2 (==-sym eq )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	232 ... \| refl = refl
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	233 lemma4 {x} {y} {z} eq \| case2 s = ord≡→≡ (sym s) -- y ≡ z
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	234 lemmax : x ≡ x'
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	235 lemmax with eq→ eq {od→ord (x , x)} (case1 refl)
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	236 lemmax \| case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x')
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	237 lemmax \| case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	238 ... \| refl = lemma1 (ord→== s )
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	239 lemmay : y ≡ y'
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	240 lemmay with lemmax
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	241 ... \| refl with lemma4 eq -- with (x,y)≡(x,y')
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	242 ... \| eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
2ea2a19f9cd6 ordered pair clean up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 247 diff changeset	243
257 53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	244 data ord-pair : (p : Ordinal) → Set n where
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	245 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	246
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	247 ZFProduct : OD
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	248 ZFProduct = record { def = λ x → ord-pair x }
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	249
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	250 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	251 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	252 -- eq-pair refl refl = HE.refl
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	253
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	254 pi1 : { p : Ordinal } → ord-pair p → Ordinal
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	255 pi1 ( pair x y) = x
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	256
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	257 π1 : { p : OD } → ZFProduct ∋ p → OD
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	258 π1 lt = ord→od (pi1 lt )
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	259
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	260 pi2 : { p : Ordinal } → ord-pair p → Ordinal
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	261 pi2 ( pair x y ) = y
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	262
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	263 π2 : { p : OD } → ZFProduct ∋ p → OD
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	264 π2 lt = ord→od (pi2 lt )
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	265
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	266 op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	267 op-cons {ox} {oy} = pair ox oy
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	268
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	269 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y >
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	270 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	271 let open ≡-Reasoning in begin
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	272 od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	273 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	274 od→ord < x , y >
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	275 ∎ )
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	276
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	277 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	278 op-iso (pair ox oy) = refl
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	279
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	280 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	281 p-iso {x} p = ord≡→≡ (op-iso p)
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	282
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	283 p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	284 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	285
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	286 p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	287 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
188 1f2c8b094908 axiom of choice → p ∨ ¬ p Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 187 diff changeset	288
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	289 --
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	290 -- Axiom of choice in intutionistic logic implies the exclude middle
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	291 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	292 --
257 53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	293
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	294 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	295 ppp {p} {a} d = d
53b7acd63481 move product to OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 254 diff changeset	296
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	297 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	298 p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } ))
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	299 p∨¬p p \| yes eq = case2 (¬p eq) where
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	300 ps = record { def = λ x → p }
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	301 lemma : ps == od∅ → p → ⊥
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	302 lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 )
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	303 ¬p : (od→ord ps ≡ o∅) → p → ⊥
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	304 ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq ))
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	305 p∨¬p p \| no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	306 ps = record { def = λ x → p }
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	307 eqo∅ : ps == od∅ → od→ord ps ≡ o∅
188 1f2c8b094908 axiom of choice → p ∨ ¬ p Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 187 diff changeset	308 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq))
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	309 lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq))
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	310 lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq))
188 1f2c8b094908 axiom of choice → p ∨ ¬ p Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 187 diff changeset	311
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	312 decp : ( p : Set n ) → Dec p -- assuming axiom of choice
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	313 decp p with p∨¬p p
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	314 decp p \| case1 x = yes x
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	315 decp p \| case2 x = no x
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	316
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	317 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	318 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice
189 540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	319 ... \| yes p = p
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	320 ... \| no ¬p = ⊥-elim ( notnot ¬p )
540b845ea2de Axiom of choies implies p ∨ ( ¬ p ) Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 188 diff changeset	321
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	322 OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y )
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	323 OrdP x y with trio< x (od→ord y)
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	324 OrdP x y \| tri< a ¬b ¬c = no ¬c
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	325 OrdP x y \| tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	326 OrdP x y \| tri> ¬a ¬b c = yes c
119 6e264c78e420 infinite Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 118 diff changeset	327
79 c07c548b2ac1 add some lemma Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 78 diff changeset	328 -- 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	329 -- 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	330
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	331 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	332 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	333
96 f239ffc27fd0 Power Set and L Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 95 diff changeset	334 -- 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	335
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	336 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	337 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	338
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	339 Def : (A : OD ) → OD
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	340 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	341
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	342 _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	343 _⊆_ A B {x} = A ∋ x → B ∋ x
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	344
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	345 infixr 220 _⊆_
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	346
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	347 subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} )
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	348 subset-lemma {A} {x} {y} = record {
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	349 proj1 = λ z lt → proj1 (z lt)
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	350 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt }
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	351 }
6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	352
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	353 OD→ZF : ZF
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	354 OD→ZF = record {
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	355 ZFSet = OD
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	356 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	357 ; _≈_ = _==_
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	358 ; ∅ = od∅
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	359 ; _,_ = _,_
f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	360 ; Union = Union
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	361 ; Power = Power
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	362 ; Select = Select
f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	363 ; Replace = Replace
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	364 ; infinite = infinite
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	365 ; isZF = isZF
28 f36e40d5d2c3 OD continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 27 diff changeset	366 } where
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	367 ZFSet = OD
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	368 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	369 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	370 Replace : OD → (OD → OD ) → OD
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	371 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	372 _∩_ : ( A B : ZFSet ) → ZFSet
145 f0fa9a755513 all done but ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 144 diff changeset	373 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	374 Union : OD → OD
156 3e7475fb28db differeent Union approach Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 155 diff changeset	375 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	376 _∈_ : ( 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	377 A ∈ B = B ∋ A
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	378 Power : OD → OD
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	379 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
103 c8b79d303867 starting over HOD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 100 diff changeset	380 ｛_｝ : ZFSet → ZFSet
c8b79d303867 starting over HOD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 100 diff changeset	381 ｛ x ｝ = ( x , x )
109 dab56d357fa3 remove o<→c< and add otrans in OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 103 diff changeset	382
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	383 data infinite-d : ( x : Ordinal ) → Set n where
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	384 iφ : infinite-d o∅
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	385 isuc : {x : Ordinal } → infinite-d x →
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	386 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	387
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	388 infinite : OD
161 4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	389 infinite = record { def = λ x → infinite-d x }
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	390
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	391 infixr 200 _∈_
96 f239ffc27fd0 Power Set and L Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 95 diff changeset	392 -- infixr 230 _∩_ _∪_
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	393 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	394 isZF = record {
43 0d9b9db14361 equalitu and internal parametorisity Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	395 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
247 d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	396 ; pair→ = pair→
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	397 ; pair← = pair←
72 f39f1a90d154 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 71 diff changeset	398 ; union→ = union→
f39f1a90d154 ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 71 diff changeset	399 ; union← = union←
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	400 ; empty = empty
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	401 ; power→ = power→
76 8e8f54e7a030 extensionality done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 74 diff changeset	402 ; power← = power←
186 914cc522c53a fix extensionality Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 185 diff changeset	403 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	404 -- ; ε-induction = {!!}
78 9a7a64b2388c infinite and replacement begin Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 77 diff changeset	405 ; infinity∅ = infinity∅
160 ebac6fa116fa ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 159 diff changeset	406 ; infinity = infinity
116 47541e86c6ac axiom of selection Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 115 diff changeset	407 ; 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	408 ; replacement← = replacement←
b60b6e8a57b0 otrans in repl Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 134 diff changeset	409 ; replacement→ = replacement→
183 de3d87b7494f fix zf Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 182 diff changeset	410 ; choice-func = choice-func
de3d87b7494f fix zf Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 182 diff changeset	411 ; choice = choice
29 fce60b99dc55 posturate OD is isomorphic to Ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	412 } where
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	413
247 d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	414 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	415 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	416 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	417
d09437fcfc7c fix pair Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 235 diff changeset	418 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	419 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	420 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	421
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	422 empty : (x : OD ) → ¬ (od∅ ∋ x)
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	423 empty x = ¬x<0
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	424
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	425 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z}
155 53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	426 o<→c< lt lt1 = ordtrans lt1 lt
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	427
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	428 ⊆→o< : {x y : Ordinal } → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y
155 53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	429 ⊆→o< {x} {y} lt with trio< x y
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	430 ⊆→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	431 ⊆→o< {x} {y} lt \| tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	432 ⊆→o< {x} {y} lt \| tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
53371f91ce63 union continue Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 154 diff changeset	433 ... \| ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
151 b5a337fb7a6d recovering... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 150 diff changeset	434
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	435 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	436 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	437 ; 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	438 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	439 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	440 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	441 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	442
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	443 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	444 ψiso {ψ} t refl = t
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	445 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	446 selection {ψ} {X} {y} = record {
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	447 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	448 ; 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	449 }
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	450 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	451 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	452 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	453 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	454 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	455 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	456 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	457 → ¬ ((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	458 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	459 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	460 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	461 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	462 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	463
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	464 ---
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	465 --- Power Set
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	466 ---
3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	467 --- First consider ordinals in OD
100 a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	468 ---
a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	469 --- 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	470 --
a402881cc341 add comment Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 99 diff changeset	471 --
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	472 ∩-≡ : { 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	473 ∩-≡ {a} {b} inc = record {
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	474 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	475 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	476 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	477 --
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	478 -- Transitive Set case
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	479 -- 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	480 -- 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	481 -- 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	482 --
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	483 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	484 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	485 lemma refl (lemma1 lemma-eq )where
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	486 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	487 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	488 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	489 proj1 = def-subst {_} {_} {(Ord a)} {z}
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	490 ( 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	491 lemma1 : {a : Ordinal } { t : OD }
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	492 → (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	493 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 ))
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	494 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	495 lemma = sup-o<
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	496
144 3ba37037faf4 Union again Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 143 diff changeset	497 --
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	498 -- 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	499 -- 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	500 --
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	501 -- 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	502
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	503 -- 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	504 --
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	505 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	506 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	507 a = od→ord A
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	508 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y)))
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	509 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	510 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
ea0e7927637a use double negation Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 165 diff changeset	511 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	512 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	513 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	514 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	515 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	516
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	517 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	518 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	519 a = od→ord A
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	520 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	521 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	522 lemma3 : Def (Ord a) ∋ t
c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	523 lemma3 = ord-power← a t lemma0
152 996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	524 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	525 lemma4 = let open ≡-Reasoning in begin
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	526 A ∩ ord→od (od→ord t)
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	527 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	528 A ∩ t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	529 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	530 t
996a67042f50 power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 151 diff changeset	531 ∎
260 8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	532 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x))
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	533 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x)))
8b85949bde00 sup with limit give up Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 259 diff changeset	534 lemma4 (sup-o< {λ x → od→ord (A ∩ ord→od x)} {od→ord t} )
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	535 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	536 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	537 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	538 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	539
190 6e778b0a7202 add filter Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 189 diff changeset	540 -- assuming axiom of choice
141 21b2654985c4 fix or Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 140 diff changeset	541 regularity : (x : OD) (not : ¬ (x == od∅)) →
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	542 (x ∋ minimal x not) ∧ (Select (minimal x not) (λ x₁ → (minimal x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	543 proj1 (regularity x not ) = x∋minimal x not
117 a4c97390d312 minimum assuption Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 116 diff changeset	544 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	545 lemma1 : {x₁ : Ordinal} → def (Select (minimal x not) (λ x₂ → (minimal x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	546 lemma1 {x₁} s = ⊥-elim ( minimal-1 x not (ord→od x₁) lemma3 ) where
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	547 lemma3 : def (minimal x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	548 lemma3 = record { proj1 = def-subst {_} {_} {minimal x not} {_} (proj1 s) refl (sym diso)
142 c30bc9f5bd0d Power Set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 141 diff changeset	549 ; proj2 = proj2 (proj2 s) }
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	550 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimal x not) (λ x₂ → (minimal x not ∋ x₂) ∧ (x ∋ x₂))) x₁
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	551 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
129 2a5519dcc167 ord power set Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 128 diff changeset	552
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	553 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	554 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	555 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	556
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	557 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	558 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	559 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	560
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	561 infinity∅ : infinite ∋ od∅
2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	562 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	563 lemma : o∅ ≡ od→ord od∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	564 lemma = let open ≡-Reasoning in begin
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	565 o∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	566 ≡⟨ sym diso ⟩
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	567 od→ord ( ord→od o∅ )
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	568 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	569 od→ord od∅
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	570 ∎
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	571 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	572 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	573 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	574 ≡ od→ord (Union (x , (x , x)))
4c704b7a62e4 ininite done Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 160 diff changeset	575 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	576
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	577 -- Axiom of choice ( is equivalent to the existence of minimal in our case )
162 b06f5d2f34b1 Axiom of choice Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 161 diff changeset	578 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	579 choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	580 choice-func X {x} not X∋x = minimal x not
223 2e1f19c949dc sepration of ordinal from OD Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 219 diff changeset	581 choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
258 63df67b6281c ... Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 257 diff changeset	582 choice X {A} X∋A not = x∋minimal A not
78 9a7a64b2388c infinite and replacement begin Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 77 diff changeset	583
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	584 ---
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	585 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	586 ---
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	587 record choiced ( X : OD) : Set (suc n) where
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	588 field
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	589 a-choice : OD
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	590 is-in : X ∋ a-choice
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	591
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	592 choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	593 choice-func' X p∨¬p not = have_to_find where
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	594 ψ : ( ox : Ordinal ) → Set (suc n)
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	595 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	596 lemma-ord : ( ox : Ordinal ) → ψ ox
235 846e0926bb89 fix cardinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 234 diff changeset	597 lemma-ord ox = TransFinite {ψ} induction ox where
234 e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	598 ∋-p : (A x : OD ) → Dec ( A ∋ x )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	599 ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x ))
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	600 ∋-p A x \| case1 (lift t) = yes t
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	601 ∋-p A x \| case2 t = no (λ x → t (lift x ))
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	602 ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) }
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	603 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	604 ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x))
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	605 ∀-imply-or {A} {B} ∀AB \| case1 (lift t) = case1 t
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	606 ∀-imply-or {A} {B} ∀AB \| case2 x = case2 (lemma (λ not → x (lift not ))) where
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	607 lemma : ¬ ((x : Ordinal ) → A x) → B
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	608 lemma not with p∨¬p B
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	609 lemma not \| case1 b = b
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	610 lemma not \| case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b ))
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	611 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	612 induction x prev with ∋-p X ( ord→od x)
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	613 ... \| yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	614 ... \| no ¬p = lemma where
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	615 lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	616 lemma1 y with ∋-p X (ord→od y)
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	617 lemma1 y \| yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	618 lemma1 y \| no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	619 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	620 lemma = ∀-imply-or lemma1
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	621 have_to_find : choiced X
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	622 have_to_find with lemma-ord (od→ord X )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	623 have_to_find \| t = dont-or t ¬¬X∋x where
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	624 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥)
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	625 ¬¬X∋x nn = not record {
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	626 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt)
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	627 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	628 }
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	629
e06b76e5b682 ac from LEM in abstract ordinal Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 228 diff changeset	630
226 176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	631 Union = ZF.Union OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	632 Power = ZF.Power OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	633 Select = ZF.Select OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	634 Replace = ZF.Replace OD→ZF
176ff97547b4 set theortic function definition using sup Shinji KONO <kono@ie.u-ryukyu.ac.jp> parents: 223 diff changeset	635 isZF = ZF.isZF OD→ZF

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

annotate OD.agda @ 260:8b85949bde00