Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 91:b4742cf4ef97
infinity axiom done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 05 Jun 2019 18:24:32 +0900 |
parents | 38d139b5edac |
children | d382a7902f5e |
rev | line source |
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16 | 1 open import Level |
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2 module ordinal-definable where |
3 | 3 |
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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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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 |
90 | 36 _c<_ : { n : Level } → ( x a : 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 | |
90 | 139 |
51 | 140 =→¬< : {x : Nat } → ¬ ( x < x ) |
141 =→¬< {Zero} () | |
142 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
143 | |
144 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
145 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
146 | |
147 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
148 c≤-refl x = case1 refl | |
149 | |
54 | 150 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ |
51 | 151 o<¬≡ ox ox refl (case1 lt) = =→¬< lt |
152 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt | |
153 | |
54 | 154 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
52 | 155 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
156 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso ) | |
157 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
158 ... | () | |
159 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
160 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso ) | |
161 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
162 ... | () | |
163 | |
79 | 164 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y |
165 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
166 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
167 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
168 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
169 | |
90 | 170 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) |
171 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where | |
172 lemma : ord→od x == record { def = λ z → z o< x } | |
173 eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso diso t where | |
174 t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) | |
175 t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) | |
176 eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl diso | |
177 | |
178 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } | |
179 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) | |
180 | |
91 | 181 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
182 ∋→o< {n} {a} {x} lt = t where | |
183 t : (od→ord x) o< (od→ord a) | |
184 t = c<→o< {suc n} {x} {a} lt | |
185 | |
186 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x | |
187 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso diso t where | |
188 t : def (ord→od (od→ord a)) (od→ord (ord→od (od→ord x))) | |
189 t = o<→c< {suc n} {od→ord x} {od→ord a} lt | |
190 | |
80 | 191 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) |
192 ¬x<0 {n} {x} (case1 ()) | |
193 ¬x<0 {n} {x} (case2 ()) | |
194 | |
195 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} | |
196 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
197 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
198 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
199 lemma lt with def-subst (o<→c< lt) oiso refl | |
200 lemma lt | lift () | |
201 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso | |
202 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
203 | |
51 | 204 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
52 | 205 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
51 | 206 |
207 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) | |
81 | 208 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
51 | 209 |
210 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
211 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) | |
212 | |
213 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) | |
214 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
215 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a ) | |
216 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) | |
217 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) | |
218 | |
54 | 219 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ |
220 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
221 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
222 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
223 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
224 | |
60 | 225 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
226 ∅< {n} {x} {y} d eq with eq→ eq d | |
227 ∅< {n} {x} {y} d eq | lift () | |
57 | 228 |
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229 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
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230 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x |
51 | 231 |
76 | 232 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 |
233 def-iso refl t = t | |
234 | |
53 | 235 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
236 is-∋ {n} x y with tri-c< x y | |
237 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
238 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
239 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
240 | |
57 | 241 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
242 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
243 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
244 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
245 | |
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246 open _∧_ |
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247 |
66 | 248 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
249 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
250 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
251 lemma ox ne with is-o∅ ox | |
252 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
253 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
80 | 254 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ |
66 | 255 lemma o∅ ne | yes refl | () |
80 | 256 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (∅5 ¬p)) |
69 | 257 |
79 | 258 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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259 |
89 | 260 postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
261 | |
262 | |
54 | 263 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
40 | 264 OD→ZF {n} = record { |
54 | 265 ZFSet = OD {suc n} |
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266 ; _∋_ = _∋_ |
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267 ; _≈_ = _==_ |
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268 ; ∅ = od∅ |
28 | 269 ; _,_ = _,_ |
270 ; Union = Union | |
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271 ; Power = Power |
28 | 272 ; Select = Select |
273 ; Replace = Replace | |
81 | 274 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
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275 ; isZF = isZF |
28 | 276 } where |
54 | 277 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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278 Replace X ψ = sup-od ψ |
54 | 279 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
280 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
281 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
84 | 282 x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } |
54 | 283 Union : OD {suc n} → OD {suc n} |
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284 Union U = record { def = λ y → osuc y o< (od→ord U) } |
77 | 285 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
54 | 286 Power : OD {suc n} → OD {suc n} |
77 | 287 Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
54 | 288 ZFSet = OD {suc n} |
289 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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290 A ∈ B = B ∋ A |
54 | 291 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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292 _⊆_ A B {x} = A ∋ x → B ∋ x |
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293 _∩_ : ( A B : ZFSet ) → ZFSet |
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294 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) |
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295 _∪_ : ( A B : ZFSet ) → ZFSet |
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296 A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) |
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297 infixr 200 _∈_ |
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298 infixr 230 _∩_ _∪_ |
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299 infixr 220 _⊆_ |
81 | 300 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) |
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301 isZF = record { |
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302 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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303 ; pair = pair |
72 | 304 ; union-u = union-u |
305 ; union→ = union→ | |
306 ; union← = union← | |
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307 ; empty = empty |
76 | 308 ; power→ = power→ |
309 ; power← = power← | |
310 ; extensionality = extensionality | |
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311 ; minimul = minimul |
51 | 312 ; regularity = regularity |
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313 ; infinity∅ = infinity∅ |
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314 ; infinity = {!!} |
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315 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
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316 ; replacement = {!!} |
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317 } where |
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318 open _∧_ |
41 | 319 open Minimumo |
54 | 320 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
87 | 321 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
322 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
54 | 323 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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324 empty x () |
77 | 325 --- Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
76 | 326 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
77 | 327 power→ A t P∋t {x} t∋x = proj1 (P∋t (od→ord x) ) |
328 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
329 power← A t t→A z = record { proj1 = lemma2 ; proj2 = lemma1 } where | |
330 lemma1 : def (ord→od (od→ord t)) z | |
76 | 331 lemma1 = {!!} |
77 | 332 lemma2 : def A z |
76 | 333 lemma2 = {!!} |
70 | 334 union-u : (X z : OD {suc n}) → Union X ∋ z → OD {suc n} |
72 | 335 union-u X z U>z = ord→od ( osuc ( od→ord z ) ) |
336 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → union-u X z U>z ∋ z | |
337 union-lemma-u {X} {z} U>z = lemma <-osuc where | |
338 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
339 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl diso | |
73 | 340 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
72 | 341 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
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342 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
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343 union→ X y u xx | tri< a ¬b ¬c | () |
73 | 344 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
345 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX | |
346 lemma refl lt = lt | |
347 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 | 348 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z ) |
73 | 349 union← X z X∋z = record { proj1 = def-subst {suc n} (o<→c< X∋z) oiso refl ; proj2 = union-lemma-u X∋z } |
54 | 350 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
351 ψiso {ψ} t refl = t | |
352 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
353 selection {ψ} {X} {y} = record { | |
354 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
355 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
356 } | |
60 | 357 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
358 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
54 | 359 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
68 | 360 minimul x not = od∅ |
57 | 361 regularity : (x : OD) (not : ¬ (x == od∅)) → |
362 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
66 | 363 proj1 (regularity x not ) = ¬∅=→∅∈ not |
364 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where | |
365 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
366 reg {y} t with proj1 t | |
367 ... | x∈∅ = x∈∅ | |
76 | 368 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
369 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
370 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
89 | 371 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
372 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) | |
373 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} | |
374 xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where | |
91 | 375 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) |
376 lemma1 {x} = c<→o< ( proj1 (pair x x ) ) | |
377 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) | |
378 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) | |
89 | 379 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
91 | 380 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
90 | 381 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
382 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where | |
383 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) | |
91 | 384 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) |
385 uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } | |
386 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt | |
387 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
388 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) | |
389 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) | |
390 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
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391 infinite : OD {suc n} |
91 | 392 infinite = ord→od ( omega ) |
393 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} | |
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394 infinity∅ = def-subst {suc n} {_} {od→ord (ord→od o∅)} {infinite} {od→ord od∅} |
80 | 395 (o<→c< ( case1 (s≤s z≤n ))) refl (cong (λ k → od→ord k) o∅≡od∅ ) |
91 | 396 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega |
397 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where | |
398 t : od→ord x o< od→ord (ord→od (omega)) | |
399 t = ∋→o< {n} {infinite} {x} lt | |
400 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) | |
401 infinite∋uxxx x lt = o<∋→ t where | |
402 t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) | |
403 t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt ) | |
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404 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
91 | 405 infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where |
406 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega | |
407 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
408 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
409 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
410 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
411 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
412 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
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413 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
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414 replacement {ψ} X x = {!!} |
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415 |