Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 65:164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
lemma ox = TransFinite {suc n} {λ ox → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} } { }0 { }1 { }2 ox
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 30 May 2019 01:02:47 +0900 |
parents | 87df00599a0e |
children | 92a11dc6425c |
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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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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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 |
40 | 71 ∅-base-def : {n : Level} → def ( ord→od (o∅ {n}) ) ≡ def (od∅ {n}) |
46 | 72 |
60 | 73 congf : {n : Level} {x y : OD {n}} → { f g : OD {n} → OD {n} } → x ≡ y → f ≡ g → f x ≡ g y |
74 congf refl refl = refl | |
75 | |
46 | 76 o∅→od∅ : {n : Level} → ord→od (o∅ {n}) ≡ od∅ {n} |
77 o∅→od∅ {n} = cong ( λ k → record { def = k }) ( ∅-base-def ) | |
78 | |
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79 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) |
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80 ∅1 {n} x (lift ()) |
28 | 81 |
37 | 82 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
83 ∅3 {n} {x} = TransFinite {n} c1 c2 c3 x where | |
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84 c0 : Nat → Ordinal {n} → Set n |
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85 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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86 c1 : ∀ (lx : Nat ) → c0 lx (record { lv = Suc lx ; ord = ℵ lx } ) |
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87 c1 lx not with not ( record { lv = lx ; ord = Φ lx } ) |
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88 ... | t with t (case1 ≤-refl ) |
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89 c1 lx not | t | () |
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90 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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91 c2 Zero not = refl |
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92 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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93 ... | t with t (case1 ≤-refl ) |
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94 c2 (Suc lx) not | t | () |
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95 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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96 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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97 ... | t with t (case2 Φ< ) |
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98 c3 lx (Φ .lx) d not | t | () |
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99 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 100 ... | t with t (case2 (s< s<refl ) ) |
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101 c3 lx (OSuc .lx x₁) d not | t | () |
34 | 102 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } ) |
41 | 103 ... | t with t (case2 (s< ℵΦ< )) |
34 | 104 c3 .(Suc lx) (ℵ lx) d not | t | () |
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105 |
36 | 106 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 |
107 def-subst df refl refl = df | |
108 | |
109 transitive : {n : Level } { x y z : OD {n} } → y ∋ x → z ∋ y → z ∋ x | |
110 transitive {n} {x} {y} {z} x∋y z∋y with ordtrans ( c<→o< {n} {x} {y} x∋y ) ( c<→o< {n} {y} {z} z∋y ) | |
111 ... | t = lemma0 (lemma t) where | |
112 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) | |
113 lemma xo<z = o<→c< xo<z | |
114 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x) | |
115 lemma0 dz = def-subst {n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso) | |
116 | |
41 | 117 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
118 field | |
119 mino : Ordinal {n} | |
120 min<x : mino o< x | |
121 | |
122 ominimal : {n : Level} → (x : Ordinal {n} ) → o∅ o< x → Minimumo {n} x | |
37 | 123 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case1 ()) |
124 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case2 ()) | |
125 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case1 ()) | |
41 | 126 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case2 Φ<) = record { mino = record { lv = Zero ; ord = Φ 0 } ; min<x = case2 Φ< } |
127 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case1 (s≤s x)) = record { mino = record { lv = lv ; ord = Φ lv } ; min<x = case1 (s≤s ≤-refl)} | |
37 | 128 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case2 ()) |
41 | 129 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case1 (s≤s x)) = record { mino = record { lv = (Suc lv) ; ord = ord } ; min<x = case2 s<refl} |
37 | 130 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case2 ()) |
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131 ominimal {n} record { lv = (Suc lx) ; ord = (ℵ .lx) } (case1 (s≤s z≤n)) = record { mino = record { lv = Suc lx ; ord = Φ (Suc lx) } ; min<x = case2 ℵΦ< } |
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132 ominimal {n} record { lv = (Suc lx) ; ord = (ℵ .lx) } (case2 ()) |
37 | 133 |
57 | 134 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
135 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
136 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
137 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
37 | 138 |
39 | 139 ∅8 : {n : Level} → ( x : Ordinal {n} ) → ¬ x o< o∅ {n} |
140 ∅8 {n} x (case1 ()) | |
141 ∅8 {n} x (case2 ()) | |
142 | |
46 | 143 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 } |
144 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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145 |
51 | 146 -- avoiding lv != Zero error |
147 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
148 orefl refl = refl | |
149 | |
150 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
151 ==-iso {n} {x} {y} eq = record { | |
152 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
153 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
154 where | |
155 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
156 lemma {x} {z} d = def-subst d oiso refl | |
157 | |
57 | 158 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
159 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
160 | |
51 | 161 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
162 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
163 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
164 lemma ox ox refl = eq-refl | |
165 | |
166 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
167 o≡→== {n} {x} {.x} refl = eq-refl | |
168 | |
169 ∅7 : {n : Level} → { x : OD {n} } → od→ord x ≡ o∅ {n} → x == od∅ {n} | |
170 ∅7 {n} {x} eq = record { eq→ = e1 (orefl eq) ; eq← = e2 } where | |
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171 e2 : {y : Ordinal {n}} → def od∅ y → def x y |
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172 e2 {y} (lift ()) |
46 | 173 e1 : {ox y : Ordinal {n}} → ox ≡ o∅ {n} → def x y → def od∅ y |
51 | 174 e1 {o∅} {y} refl x>y = lift ( ∅8 y (o<-subst (c<→o< {n} {ord→od y} {x} (def-subst {n} {x} {y} x>y refl (sym diso))) ord-iso eq )) |
175 | |
176 =→¬< : {x : Nat } → ¬ ( x < x ) | |
177 =→¬< {Zero} () | |
178 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
179 | |
180 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
181 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
182 | |
183 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
184 c≤-refl x = case1 refl | |
185 | |
186 o<> : {n : Level } ( ox oy : Ordinal {n}) → ox o< oy → oy o< ox → ⊥ | |
187 o<> ox oy (case1 x<y) (case1 y<x) = >→¬< x<y y<x | |
188 o<> ox oy (case1 x<y) (case2 y<x) with d<→lv y<x | |
189 ... | refl = =→¬< x<y | |
190 o<> ox oy (case2 x<y) (case1 y<x) with d<→lv x<y | |
191 ... | refl = =→¬< y<x | |
192 o<> ox oy (case2 x<y) (case2 y<x) with d<→lv x<y | |
193 ... | refl = trio<> x<y y<x | |
194 | |
54 | 195 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ |
51 | 196 o<¬≡ ox ox refl (case1 lt) = =→¬< lt |
197 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt | |
198 | |
54 | 199 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
52 | 200 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
201 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso ) | |
202 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
203 ... | () | |
204 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
205 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso ) | |
206 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
207 ... | () | |
208 | |
51 | 209 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
52 | 210 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
51 | 211 |
212 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) | |
213 o<→¬c> {n} {x} {y} olt clt = o<> (od→ord x) (od→ord y) olt (c<→o< clt ) where | |
214 | |
215 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
216 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) | |
217 | |
218 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) | |
219 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
220 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a ) | |
221 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) | |
222 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) | |
223 | |
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224 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
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225 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where |
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226 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} |
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227 lemma ox = TransFinite {suc n} {λ ox → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} } {!!} {!!} {!!} ox |
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228 |
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229 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc (suc n)) (suc (suc n )) |
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230 |
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231 -- ==∅→≡ : {n : Level} → { x : OD {suc n} } → ( x == od∅ {suc n} ) → x ≡ od∅ {suc n} |
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232 -- ==∅→≡ {n} {x} eq = cong ( λ k → record { def = k } ) (trans {!!} ∅-base-def ) where |
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233 |
54 | 234 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ |
235 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
236 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
237 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
238 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
239 | |
51 | 240 ∅2 : {n : Level} → { x : OD {n} } → o∅ {n} o< od→ord x → ¬ ( x == od∅ {n} ) |
241 ∅2 {n} {x} lt record { eq→ = eq→ ; eq← = eq← } with ominimal (od→ord x ) lt | |
242 ... | min with eq→ ( def-subst (o<→c< (Minimumo.min<x min)) oiso refl ) | |
243 ... | () | |
244 | |
57 | 245 ∅0 : {n : Level} → { x : Ordinal {n} } → o∅ {n} o< x → ¬ ( ord→od x == od∅ {n} ) |
246 ∅0 {n} {x} lt record { eq→ = eq→ ; eq← = eq← } with ominimal x lt | |
247 ... | min with eq→ (o<→c< (Minimumo.min<x min)) | |
248 ... | () | |
60 | 249 |
250 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) | |
251 ∅< {n} {x} {y} d eq with eq→ eq d | |
252 ∅< {n} {x} {y} d eq | lift () | |
57 | 253 |
51 | 254 |
255 is-od∅ : {n : Level} → ( x : OD {suc n} ) → Dec ( x == od∅ {suc n} ) | |
256 is-od∅ {n} x with trio< {n} (od→ord x) (o∅ {suc n}) | |
257 is-od∅ {n} x | tri≈ ¬a b ¬c = yes ( ∅7 (orefl b) ) | |
258 is-od∅ {n} x | tri< (case1 ()) ¬b ¬c | |
259 is-od∅ {n} x | tri< (case2 ()) ¬b ¬c | |
260 is-od∅ {n} x | tri> ¬a ¬b c = no ( ∅2 c ) | |
46 | 261 |
53 | 262 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
263 is-∋ {n} x y with tri-c< x y | |
264 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
265 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
266 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
267 | |
57 | 268 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
269 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
270 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
271 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
272 | |
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273 open _∧_ |
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274 |
57 | 275 ∅9 : {n : Level} → {x : OD {n} } → ¬ x == od∅ → o∅ o< od→ord x |
276 ∅9 {_} {x} not = ∅5 lemma where | |
38 | 277 lemma : ¬ od→ord x ≡ o∅ |
51 | 278 lemma eq = not ( ∅7 eq ) |
37 | 279 |
60 | 280 ∅10 : {n : Level} → {ox : Ordinal {n}} → {x : OD{n}} → ( x ≡ ord→od ox ) → (not : ¬ (x == od∅)) → o∅ o< ox |
281 ∅10 {n} {ox} {x} refl not = subst (λ k → o∅ o< k) diso (∅9 not) | |
282 | |
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283 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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284 |
54 | 285 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
40 | 286 OD→ZF {n} = record { |
54 | 287 ZFSet = OD {suc n} |
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288 ; _∋_ = _∋_ |
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289 ; _≈_ = _==_ |
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290 ; ∅ = od∅ |
28 | 291 ; _,_ = _,_ |
292 ; Union = Union | |
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293 ; Power = Power |
28 | 294 ; Select = Select |
295 ; Replace = Replace | |
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296 ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } } |
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297 ; isZF = isZF |
28 | 298 } where |
54 | 299 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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300 Replace X ψ = sup-od ψ |
54 | 301 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
302 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
303 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
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304 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } |
54 | 305 Union : OD {suc n} → OD {suc n} |
306 Union x = record { def = λ y → {z : Ordinal {suc n}} → def x z → def (ord→od z) y } | |
307 Power : OD {suc n} → OD {suc n} | |
308 Power x = record { def = λ y → (z : Ordinal {suc n} ) → ( def x y ∧ def (ord→od z) y ) } | |
309 ZFSet = OD {suc n} | |
310 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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311 A ∈ B = B ∋ A |
54 | 312 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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313 _⊆_ A B {x} = A ∋ x → B ∋ x |
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314 _∩_ : ( A B : ZFSet ) → ZFSet |
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315 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) |
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316 _∪_ : ( A B : ZFSet ) → ZFSet |
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317 A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) |
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318 infixr 200 _∈_ |
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319 infixr 230 _∩_ _∪_ |
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320 infixr 220 _⊆_ |
54 | 321 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) |
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322 isZF = record { |
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323 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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324 ; pair = pair |
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325 ; union→ = {!!} |
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326 ; union← = {!!} |
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327 ; empty = empty |
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328 ; power→ = {!!} |
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329 ; power← = {!!} |
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330 ; extensionality = {!!} |
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331 ; minimul = minimul |
51 | 332 ; regularity = regularity |
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333 ; infinity∅ = {!!} |
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334 ; infinity = {!!} |
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335 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
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336 ; replacement = {!!} |
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337 } where |
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338 open _∧_ |
41 | 339 open Minimumo |
54 | 340 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
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341 proj1 (pair A B ) = case1 refl |
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342 proj2 (pair A B ) = case2 refl |
54 | 343 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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344 empty x () |
54 | 345 union→ : (X x y : OD {suc n} ) → (X ∋ x) → (x ∋ y) → (Union X ∋ y) |
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346 union→ X x y X∋x x∋y = {!!} where |
54 | 347 lemma : {z : Ordinal {suc n} } → def X z → z ≡ od→ord y |
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348 lemma {z} X∋z = {!!} |
54 | 349 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
350 ψiso {ψ} t refl = t | |
351 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
352 selection {ψ} {X} {y} = record { | |
353 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
354 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
355 } | |
60 | 356 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
357 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
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358 minord : (x : OD {suc n} ) → ¬ (x == od∅ ) → Minimumo (od→ord x) |
57 | 359 minord x not = ominimal (od→ord x) (∅9 not) |
54 | 360 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
41 | 361 minimul x not = ord→od ( mino (minord x not)) |
62 | 362 omin→cmin : {x : OD {suc n}} → {not : ¬ x == od∅ } → mino (minord x not) o< (od→ord x) → def x (od→ord (ord→od (mino (minord x not)))) |
363 omin→cmin {x} {not} m<x = def-subst {suc n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl | |
54 | 364 minimul<x : (x : OD {suc n} ) → (not : ¬ x == od∅ ) → x ∋ minimul x not |
62 | 365 minimul<x x not = omin→cmin {x} {not} (min<x (minord x not)) |
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366 omin∅→min∅ : (ox : Ordinal {suc n}) → {not : ¬ ( ord→od ox == od∅)} |
64 | 367 → mino (ominimal ox (∅10 refl not)) ≡ o∅ → mino (ominimal (od→ord (ord→od ox)) (∅9 not)) ≡ o∅ |
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368 omin∅→min∅ ox {not} eq with ominimal ox (∅10 refl not) |
64 | 369 omin∅→min∅ record { lv = Zero ; ord = (Φ .0) } eq | record { mino = mino ; min<x = case1 () } |
370 omin∅→min∅ record { lv = Zero ; ord = (Φ .0) } eq | record { mino = mino ; min<x = case2 () } | |
371 omin∅→min∅ record { lv = Zero ; ord = (OSuc .0 ord₁) } refl | record { mino = .o∅ ; min<x = case1 () } | |
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372 omin∅→min∅ record { lv = Zero ; ord = (OSuc .0 (Φ .0)) } {not} refl | record { mino = record { lv = .0 ; ord = .(Φ 0) } ; min<x = case2 Φ< } = {!!} where |
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373 lemma : mino (ominimal (od→ord (ord→od (record { lv = Zero ; ord = OSuc Zero (Φ Zero) }))) (∅5 (λ eq → not (∅7 eq)))) ≡ o∅ |
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374 lemma = {!!} |
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375 omin∅→min∅ record { lv = Zero ; ord = (OSuc .0 (OSuc .0 ord₁)) } {_} refl | record { mino = record { lv = .0 ; ord = .(Φ 0) } ; min<x = case2 Φ< } = {!!} |
64 | 376 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } refl | record { mino = .o∅ ; min<x = case1 (s≤s z≤n) } = {!!} |
377 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } eq | record { mino = mino ; min<x = case2 () } | |
378 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } refl | record { mino = .o∅ ; min<x = case1 (s≤s z≤n) } = {!!} | |
379 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } refl | record { mino = .o∅ ; min<x = case2 () } | |
380 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (ℵ .lv₁) } refl | record { mino = .o∅ ; min<x = case1 (s≤s z≤n) } = {!!} | |
381 omin∅→min∅ record { lv = (Suc lv₁) ; ord = (ℵ .lv₁) } refl | record { mino = .o∅ ; min<x = case2 () } | |
57 | 382 regularity : (x : OD) (not : ¬ (x == od∅)) → |
383 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
61 | 384 regularity x not = regularity-ord (od→ord x) {x} (sym oiso ) not where |
385 regularity-ord : (ox : Ordinal ) → {x : OD} → ( x ≡ ord→od ox ) → (not : ¬ (x == od∅)) | |
60 | 386 → (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
62 | 387 regularity-ord ox {x} refl not with ominimal ox (∅10 refl not) | minimul x not | is-o∅ (mino (minord x not)) | is-o∅ ox |
388 regularity-ord record { lv = Zero ; ord = (Φ .0) } refl not | record { mino = mino ; min<x = case1 () } | r | t | s | |
389 regularity-ord record { lv = Zero ; ord = (Φ .0) } refl not | record { mino = mino ; min<x = case2 () } | r | t | s | |
390 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = min ; min<x = case1 () } | r | t | s | |
391 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = record { lv = Suc lv₁ ; ord = ord } ; min<x = case2 () } | r | t | s | |
392 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = record { lv = Zero ; ord = Φ .0 } ; min<x = case2 Φ< } | r | yes p | s | |
63 | 393 = record { proj1 = {!!} ; proj2 = record { eq→ = {!!} ; eq← = λ () } } |
62 | 394 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = record { lv = Zero ; ord = Φ .0 } ; min<x = case2 Φ< } | r | no ¬p | yes p = {!!} |
395 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = record { lv = Zero ; ord = Φ .0 } ; min<x = case2 Φ< } | r | no ¬p | no ¬p₁ = {!!} | |
396 regularity-ord record { lv = Zero ; ord = (OSuc .0 ord₁) } refl not | record { mino = record { lv = Zero ; ord = OSuc .0 ord₂ } ; min<x = case2 (s< lt) } | r | t | s = {!!} | |
397 regularity-ord record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } refl not | record { mino = min ; min<x = case1 lt } | r | t | s = {!!} | |
398 regularity-ord record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } refl not | record { mino = min ; min<x = case2 () } | r | t | s | |
399 regularity-ord record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } refl not | record { mino = min ; min<x = min<x } | r | t | s = {!!} | |
400 regularity-ord record { lv = (Suc lv₁) ; ord = (ℵ .lv₁) } refl not | record { mino = min ; min<x = min<x } | r | t | s = {!!} |