Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 83:e013ccf00567
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 04 Jun 2019 12:28:43 +0900 |
parents | 57814596a986 |
children | 4b2aff372b7c |
rev | line source |
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16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
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2 module ordinal-definable where |
3 | 3 |
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e11e95d5ddee
separete constructible set
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4 open import zf |
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5 open import ordinal |
3 | 6 |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
3 | 8 |
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separete constructible set
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9 open import Relation.Binary.PropositionalEquality |
3 | 10 |
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11 open import Data.Nat.Properties |
6 | 12 open import Data.Empty |
13 open import Relation.Nullary | |
14 | |
15 open import Relation.Binary | |
16 open import Relation.Binary.Core | |
17 | |
27 | 18 -- Ordinal Definable Set |
11 | 19 |
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20 record OD {n : Level} : Set (suc n) where |
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21 field |
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22 def : (x : Ordinal {n} ) → Set n |
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23 |
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24 open OD |
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25 open import Data.Unit |
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26 |
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od→lv : {n : Level} → OD {n} → Nat
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27 open Ordinal |
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od→lv : {n : Level} → OD {n} → Nat
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28 |
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29 postulate |
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od∅' {n} = ord→od (o∅ {n})
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30 od→ord : {n : Level} → OD {n} → Ordinal {n} |
36 | 31 ord→od : {n : Level} → Ordinal {n} → OD {n} |
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32 |
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33 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
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34 _∋_ {n} a x = def a ( od→ord x ) |
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35 |
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36 _c<_ : { n : Level } → ( a x : OD {n} ) → Set n |
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37 x c< a = a ∋ x |
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38 |
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39 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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40 field |
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41 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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42 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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43 |
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44 id : {n : Level} {A : Set n} → A → A |
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45 id x = x |
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46 |
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47 eq-refl : {n : Level} { x : OD {n} } → x == x |
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48 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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49 |
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50 open _==_ |
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51 |
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52 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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53 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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54 |
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55 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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56 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) } |
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57 |
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58 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
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59 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
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60 |
40 | 61 od∅ : {n : Level} → OD {n} |
62 od∅ {n} = record { def = λ _ → Lift n ⊥ } | |
63 | |
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64 postulate |
36 | 65 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y |
66 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y | |
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67 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x |
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68 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
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69 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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70 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ |
46 | 71 |
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72 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) |
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73 ∅1 {n} x (lift ()) |
28 | 74 |
37 | 75 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
81 | 76 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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77 c0 : Nat → Ordinal {n} → Set n |
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78 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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79 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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80 c2 Zero not = refl |
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81 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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82 ... | t with t (case1 ≤-refl ) |
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83 c2 (Suc lx) not | t | () |
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84 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) |
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85 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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86 ... | t with t (case2 Φ< ) |
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87 c3 lx (Φ .lx) d not | t | () |
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88 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 89 ... | t with t (case2 (s< s<refl ) ) |
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90 c3 lx (OSuc .lx x₁) d not | t | () |
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91 |
36 | 92 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x |
93 def-subst df refl refl = df | |
94 | |
69 | 95 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
96 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) | |
36 | 97 ... | t = lemma0 (lemma t) where |
98 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) | |
99 lemma xo<z = o<→c< xo<z | |
100 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x) | |
69 | 101 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso) |
36 | 102 |
41 | 103 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
104 field | |
105 mino : Ordinal {n} | |
106 min<x : mino o< x | |
107 | |
57 | 108 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
109 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
110 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
111 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
37 | 112 |
46 | 113 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } |
114 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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115 |
51 | 116 -- avoiding lv != Zero error |
117 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
118 orefl refl = refl | |
119 | |
120 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
121 ==-iso {n} {x} {y} eq = record { | |
122 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
123 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
124 where | |
125 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
126 lemma {x} {z} d = def-subst d oiso refl | |
127 | |
57 | 128 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
129 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
130 | |
51 | 131 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
132 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
133 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
134 lemma ox ox refl = eq-refl | |
135 | |
136 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
137 o≡→== {n} {x} {.x} refl = eq-refl | |
138 | |
139 =→¬< : {x : Nat } → ¬ ( x < x ) | |
140 =→¬< {Zero} () | |
141 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
142 | |
143 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
144 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
145 | |
146 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
147 c≤-refl x = case1 refl | |
148 | |
54 | 149 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ |
51 | 150 o<¬≡ ox ox refl (case1 lt) = =→¬< lt |
151 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt | |
152 | |
54 | 153 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
52 | 154 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
155 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso ) | |
156 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
157 ... | () | |
158 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
159 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso ) | |
160 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
161 ... | () | |
162 | |
79 | 163 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y |
164 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
165 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
166 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
167 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
168 | |
80 | 169 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) |
170 ¬x<0 {n} {x} (case1 ()) | |
171 ¬x<0 {n} {x} (case2 ()) | |
172 | |
173 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} | |
174 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
175 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
176 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
177 lemma lt with def-subst (o<→c< lt) oiso refl | |
178 lemma lt | lift () | |
179 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso | |
180 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
181 | |
51 | 182 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
52 | 183 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
51 | 184 |
185 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) | |
81 | 186 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
51 | 187 |
188 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
189 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) | |
190 | |
191 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) | |
192 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
193 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a ) | |
194 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) | |
195 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso diso) | |
196 | |
54 | 197 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ |
198 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
199 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
200 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
201 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
202 | |
60 | 203 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
204 ∅< {n} {x} {y} d eq with eq→ eq d | |
205 ∅< {n} {x} {y} d eq | lift () | |
57 | 206 |
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207 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
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208 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x |
51 | 209 |
76 | 210 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
211 def-iso refl t = t | |
212 | |
53 | 213 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
214 is-∋ {n} x y with tri-c< x y | |
215 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
216 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
217 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
218 | |
57 | 219 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
220 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
221 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
222 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
223 | |
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224 open _∧_ |
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225 |
66 | 226 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
227 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
228 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
229 lemma ox ne with is-o∅ ox | |
230 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
231 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
80 | 232 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ |
66 | 233 lemma o∅ ne | yes refl | () |
80 | 234 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (∅5 ¬p)) |
69 | 235 |
79 | 236 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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237 |
54 | 238 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
40 | 239 OD→ZF {n} = record { |
54 | 240 ZFSet = OD {suc n} |
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241 ; _∋_ = _∋_ |
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242 ; _≈_ = _==_ |
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243 ; ∅ = od∅ |
28 | 244 ; _,_ = _,_ |
245 ; Union = Union | |
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246 ; Power = Power |
28 | 247 ; Select = Select |
248 ; Replace = Replace | |
81 | 249 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
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250 ; isZF = isZF |
28 | 251 } where |
54 | 252 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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253 Replace X ψ = sup-od ψ |
54 | 254 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
255 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
256 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
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257 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } |
54 | 258 Union : OD {suc n} → OD {suc n} |
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259 Union U = record { def = λ y → osuc y o< (od→ord U) } |
77 | 260 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
54 | 261 Power : OD {suc n} → OD {suc n} |
77 | 262 Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
54 | 263 ZFSet = OD {suc n} |
264 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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265 A ∈ B = B ∋ A |
54 | 266 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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267 _⊆_ A B {x} = A ∋ x → B ∋ x |
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268 _∩_ : ( A B : ZFSet ) → ZFSet |
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269 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) |
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270 _∪_ : ( A B : ZFSet ) → ZFSet |
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271 A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) |
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272 infixr 200 _∈_ |
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273 infixr 230 _∩_ _∪_ |
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274 infixr 220 _⊆_ |
81 | 275 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) |
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276 isZF = record { |
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277 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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278 ; pair = pair |
72 | 279 ; union-u = union-u |
280 ; union→ = union→ | |
281 ; union← = union← | |
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282 ; empty = empty |
76 | 283 ; power→ = power→ |
284 ; power← = power← | |
285 ; extensionality = extensionality | |
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286 ; minimul = minimul |
51 | 287 ; regularity = regularity |
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288 ; infinity∅ = infinity∅ |
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289 ; infinity = {!!} |
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290 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
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291 ; replacement = {!!} |
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292 } where |
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293 open _∧_ |
41 | 294 open Minimumo |
54 | 295 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
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296 proj1 (pair A B ) = case1 refl |
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297 proj2 (pair A B ) = case2 refl |
54 | 298 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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299 empty x () |
77 | 300 --- Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
76 | 301 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
77 | 302 power→ A t P∋t {x} t∋x = proj1 (P∋t (od→ord x) ) |
303 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
304 power← A t t→A z = record { proj1 = lemma2 ; proj2 = lemma1 } where | |
305 lemma1 : def (ord→od (od→ord t)) z | |
76 | 306 lemma1 = {!!} |
77 | 307 lemma2 : def A z |
76 | 308 lemma2 = {!!} |
70 | 309 union-u : (X z : OD {suc n}) → Union X ∋ z → OD {suc n} |
72 | 310 union-u X z U>z = ord→od ( osuc ( od→ord z ) ) |
311 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → union-u X z U>z ∋ z | |
312 union-lemma-u {X} {z} U>z = lemma <-osuc where | |
313 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
314 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl diso | |
73 | 315 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
72 | 316 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
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317 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
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318 union→ X y u xx | tri< a ¬b ¬c | () |
73 | 319 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
320 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX | |
321 lemma refl lt = lt | |
322 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) | |
72 | 323 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z ) |
73 | 324 union← X z X∋z = record { proj1 = def-subst {suc n} (o<→c< X∋z) oiso refl ; proj2 = union-lemma-u X∋z } |
54 | 325 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
326 ψiso {ψ} t refl = t | |
327 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
328 selection {ψ} {X} {y} = record { | |
329 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
330 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
331 } | |
60 | 332 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
333 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
54 | 334 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
68 | 335 minimul x not = od∅ |
57 | 336 regularity : (x : OD) (not : ¬ (x == od∅)) → |
337 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
66 | 338 proj1 (regularity x not ) = ¬∅=→∅∈ not |
339 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where | |
340 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
341 reg {y} t with proj1 t | |
342 ... | x∈∅ = x∈∅ | |
76 | 343 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
344 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
345 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
83 | 346 pair-or : {x y : OD {suc n} } → (x , x) ∋ y → (y == x ) ∨ (x ∋ y ) |
347 pair-or {x} {y} lt with tri-c< x y | |
348 pair-or {x} {y} lt | tri< a ¬b ¬c = {!!} -- x < y < ( x , x ) | |
349 pair-or {x} {y} lt | tri≈ ¬a b ¬c = case1 (eq-sym b) | |
350 pair-or {x} {y} lt | tri> ¬a ¬b c = case2 c | |
351 pair-osuc : (x : OD) → od→ord (x , x) ≡ osuc (od→ord x) | |
352 pair-osuc x with trio< (od→ord (x , x)) (osuc (od→ord x)) | |
353 pair-osuc x | tri< a ¬b ¬c = ⊥-elim ( osuc-< a (c<→o< (proj1 (pair x x )) )) | |
354 pair-osuc x | tri≈ ¬a b ¬c = b | |
355 pair-osuc x | tri> ¬a ¬b c with pair-or (def-subst (o<→c< c ) oiso refl ) -- (x , x) ∋ ord→od (osuc (od→ord x) ) | |
356 pair-osuc x | tri> ¬a ¬b c | case1 eq = ⊥-elim ( o<→¬== (def-subst {suc n} (o<→c< c ) refl diso ) eq ) | |
357 pair-osuc x | tri> ¬a ¬b c | case2 lt = ⊥-elim (c<> c lt ) where | |
358 lemma2 : (z : Ordinal {suc n} ) → lv (od→ord (ord→od z , ord→od z) ) ≡ lv z | |
359 lemma2 z = {!!} | |
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360 next : (x : OD) → Union (x , (x , x)) == ord→od (osuc (od→ord x)) |
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361 eq→ (next x ) {y} z = {!!} |
83 | 362 eq← (next x ) {y} z = def-subst {suc n} {_} {_} {Union (x , (x , x))} {_} |
363 (union→ (x , (x , x)) (ord→od y) (ord→od (osuc y)) (record { proj1 = lemma1 ; proj2 = lemma2 } )) refl diso where | |
364 lemma1 : (x , (x , x)) ∋ ord→od (osuc y) -- z : def (ord→od (osuc (od→ord x))) y | |
365 lemma1 with tri-c< (x , x) (ord→od (osuc (od→ord x))) | |
366 lemma1 | tri< a ¬b ¬c = {!!} | |
367 lemma1 | tri≈ ¬a b ¬c = {!!} | |
368 lemma1 | tri> ¬a ¬b c = {!!} | |
369 lemma2 : ord→od (osuc y) ∋ ord→od y | |
370 lemma2 = o<→c< <-osuc | |
81 | 371 next' : (x : OD) → ord→od ( od→ord ( Union (x , (x , x)))) == ord→od (osuc (od→ord x)) |
372 next' x = subst ( λ k → k == ord→od (osuc (od→ord x))) (sym oiso) (next x) | |
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373 infinite : OD {suc n} |
81 | 374 infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
375 infinity∅ : ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) ∋ od∅ {suc n} | |
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376 infinity∅ = def-subst {suc n} {_} {od→ord (ord→od o∅)} {infinite} {od→ord od∅} |
80 | 377 (o<→c< ( case1 (s≤s z≤n ))) refl (cong (λ k → od→ord k) o∅≡od∅ ) |
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378 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
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379 infinity x ∞∋x = {!!} |
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380 where |
81 | 381 lemma : (ox : Ordinal {suc n} ) → ox o< record { lv = Suc Zero ; ord = Φ 1 } → osuc ox o< record { lv = Suc Zero ; ord = Φ 1 } |
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382 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) |
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383 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) |
81 | 384 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) |
385 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
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386 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 |
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387 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl |
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388 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
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389 replacement {ψ} X x = {!!} |
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390 |