Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 97:f2b579106770
power set using sup on Def
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 09 Jun 2019 19:41:53 +0900 |
parents | f239ffc27fd0 |
children | 1ff0ddc991ac |
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⊔_ ) |
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8 open import Relation.Binary.PropositionalEquality |
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9 open import Data.Nat.Properties |
6 | 10 open import Data.Empty |
11 open import Relation.Nullary | |
12 open import Relation.Binary | |
13 open import Relation.Binary.Core | |
14 | |
27 | 15 -- Ordinal Definable Set |
11 | 16 |
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17 record OD {n : Level} : Set (suc n) where |
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18 field |
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19 def : (x : Ordinal {n} ) → Set n |
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20 |
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21 open OD |
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22 open import Data.Unit |
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23 |
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od→lv : {n : Level} → OD {n} → Nat
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24 open Ordinal |
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od→lv : {n : Level} → OD {n} → Nat
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25 |
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26 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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27 field |
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28 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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29 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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30 |
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31 id : {n : Level} {A : Set n} → A → A |
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32 id x = x |
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33 |
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34 eq-refl : {n : Level} { x : OD {n} } → x == x |
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35 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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36 |
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37 open _==_ |
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38 |
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39 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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40 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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41 |
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42 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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43 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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44 |
40 | 45 od∅ : {n : Level} → OD {n} |
46 od∅ {n} = record { def = λ _ → Lift n ⊥ } | |
47 | |
96 | 48 -- OD can be iso to a subset of Ordinal |
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49 postulate |
95 | 50 od→ord : {n : Level} → OD {n} → Ordinal {n} |
51 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
52 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y | |
53 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x | |
54 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x | |
55 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
56 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} | |
57 sup-o< : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ | |
58 | |
59 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n | |
60 _∋_ {n} a x = def a ( od→ord x ) | |
61 | |
62 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n | |
63 x c< a = a ∋ x | |
64 | |
65 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) | |
66 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) | |
67 | |
68 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 | |
69 def-subst df refl refl = df | |
70 | |
71 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} | |
72 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) | |
73 | |
74 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) | |
75 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )} | |
76 ( o<→c< ( sup-o< ( λ y → od→ord (ψ (ord→od y ))) {od→ord x } )) refl (cong ( λ k → od→ord (ψ k) ) oiso) | |
46 | 77 |
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78 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) |
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79 ∅1 {n} x (lift ()) |
28 | 80 |
37 | 81 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
81 | 82 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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83 c0 : Nat → Ordinal {n} → Set n |
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84 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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85 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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86 c2 Zero not = refl |
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87 c2 (Suc 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 c2 (Suc lx) not | t | () |
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90 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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91 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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92 ... | t with t (case2 Φ< ) |
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93 c3 lx (Φ .lx) d not | t | () |
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94 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 95 ... | t with t (case2 (s< s<refl ) ) |
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96 c3 lx (OSuc .lx x₁) d not | t | () |
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97 |
69 | 98 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
99 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 | 100 ... | t = lemma0 (lemma t) where |
95 | 101 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x) |
36 | 102 lemma xo<z = o<→c< xo<z |
95 | 103 lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x) |
104 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl | |
36 | 105 |
41 | 106 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
107 field | |
108 mino : Ordinal {n} | |
109 min<x : mino o< x | |
110 | |
57 | 111 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
112 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
113 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
114 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
37 | 115 |
46 | 116 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 } |
117 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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118 |
51 | 119 -- avoiding lv != Zero error |
120 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
121 orefl refl = refl | |
122 | |
123 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
124 ==-iso {n} {x} {y} eq = record { | |
125 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
126 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
127 where | |
128 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
129 lemma {x} {z} d = def-subst d oiso refl | |
130 | |
57 | 131 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
132 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
133 | |
51 | 134 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
135 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
136 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
137 lemma ox ox refl = eq-refl | |
138 | |
139 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
140 o≡→== {n} {x} {.x} refl = eq-refl | |
141 | |
142 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
143 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
144 | |
145 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
146 c≤-refl x = case1 refl | |
147 | |
54 | 148 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
52 | 149 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
95 | 150 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl ) |
52 | 151 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) |
152 ... | () | |
153 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
95 | 154 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl ) |
52 | 155 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) |
156 ... | () | |
157 | |
79 | 158 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y |
159 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
160 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
161 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
162 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
163 | |
90 | 164 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) |
165 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where | |
166 lemma : ord→od x == record { def = λ z → z o< x } | |
95 | 167 eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where |
168 t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) | |
169 t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) | |
170 eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl | |
90 | 171 |
172 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } | |
173 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) | |
174 | |
91 | 175 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
176 ∋→o< {n} {a} {x} lt = t where | |
177 t : (od→ord x) o< (od→ord a) | |
178 t = c<→o< {suc n} {x} {a} lt | |
179 | |
180 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x | |
95 | 181 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where |
182 t : def (ord→od (od→ord a)) (od→ord x) | |
91 | 183 t = o<→c< {suc n} {od→ord x} {od→ord a} lt |
184 | |
80 | 185 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
186 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
187 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
188 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
189 lemma lt with def-subst (o<→c< lt) oiso refl | |
190 lemma lt | lift () | |
191 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso | |
192 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
193 | |
51 | 194 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
52 | 195 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
51 | 196 |
197 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) | |
81 | 198 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
51 | 199 |
200 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
201 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) | |
202 | |
203 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) | |
204 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
95 | 205 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a ) |
51 | 206 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) |
95 | 207 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 refl) |
51 | 208 |
54 | 209 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ |
210 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
211 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
212 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
213 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
214 | |
60 | 215 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
216 ∅< {n} {x} {y} d eq with eq→ eq d | |
217 ∅< {n} {x} {y} d eq | lift () | |
57 | 218 |
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219 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
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220 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x |
51 | 221 |
76 | 222 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 |
223 def-iso refl t = t | |
224 | |
53 | 225 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
226 is-∋ {n} x y with tri-c< x y | |
227 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
228 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
229 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
230 | |
57 | 231 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
232 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
233 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
234 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
235 | |
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236 open _∧_ |
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237 |
66 | 238 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
239 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
240 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
241 lemma ox ne with is-o∅ ox | |
242 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
243 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
80 | 244 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ |
66 | 245 lemma o∅ ne | yes refl | () |
95 | 246 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) ) |
69 | 247 |
79 | 248 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
94 | 249 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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250 |
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251 csuc : {n : Level} → OD {suc n} → OD {suc n} |
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252 csuc x = ord→od ( osuc ( od→ord x )) |
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253 |
96 | 254 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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255 |
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256 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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257 ZFSubset A x = record { def = λ y → def A y ∧ def x y } |
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258 |
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259 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
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260 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
96 | 261 |
262 -- Constructible Set on α | |
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263 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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264 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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265 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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266 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
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267 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } |
89 | 268 |
96 | 269 -- L∋ord : {n : Level} → (x : Ordinal {suc n} ) → L (osuc x) ∋ ord→od x |
270 -- L∋ord {n} record { lv = Zero ; ord = (Φ .0) } = {!!} | |
271 -- L∋ord {n} record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } = {!!} | |
272 -- L∋ord {n} record { lv = lv ; ord = (OSuc .(lv) ord₁) } = {!!} | |
273 | |
274 -- X ⊆ OD → (P X ∩ OD ) ⊆ L (supP X) ∈ OD | |
275 | |
276 ord-of : {n : Level} → (A : OD {suc n} ) → Ordinal {suc n} → Ordinal {suc n} | |
277 ord-of {n} A x with def A x | |
278 ... | t = x | |
279 | |
280 supP : {n : Level} → (A : OD {suc n} ) → Ordinal {suc n} | |
281 supP A = sup-o ( ord-of A ) | |
282 | |
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283 L→P : {n : Level} → (A x : OD {suc n} ) → L {n} (supP A) ∋ ZFSubset A x |
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284 L→P {n} A x with sup-o< (ord-of A) {{!!}} |
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285 ... | lt = {!!} |
89 | 286 |
54 | 287 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
40 | 288 OD→ZF {n} = record { |
54 | 289 ZFSet = OD {suc n} |
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290 ; _∋_ = _∋_ |
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291 ; _≈_ = _==_ |
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292 ; ∅ = od∅ |
28 | 293 ; _,_ = _,_ |
294 ; Union = Union | |
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295 ; Power = Power |
28 | 296 ; Select = Select |
297 ; Replace = Replace | |
81 | 298 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
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299 ; isZF = isZF |
28 | 300 } where |
54 | 301 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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302 Replace X ψ = sup-od ψ |
54 | 303 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
304 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
305 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
84 | 306 x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } |
54 | 307 Union : OD {suc n} → OD {suc n} |
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308 Union U = record { def = λ y → osuc y o< (od→ord U) } |
77 | 309 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
54 | 310 Power : OD {suc n} → OD {suc n} |
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311 Power A = Def A |
54 | 312 ZFSet = OD {suc n} |
313 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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314 A ∈ B = B ∋ A |
54 | 315 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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316 _⊆_ A B {x} = A ∋ x → B ∋ x |
96 | 317 -- _∩_ : ( A B : ZFSet ) → ZFSet |
318 -- A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) | |
319 -- _∪_ : ( A B : ZFSet ) → ZFSet | |
320 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) | |
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321 infixr 200 _∈_ |
96 | 322 -- infixr 230 _∩_ _∪_ |
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323 infixr 220 _⊆_ |
81 | 324 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) |
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325 isZF = record { |
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326 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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327 ; pair = pair |
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328 ; union-u = λ _ z _ → csuc z |
72 | 329 ; union→ = union→ |
330 ; union← = union← | |
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331 ; empty = empty |
76 | 332 ; power→ = power→ |
333 ; power← = power← | |
334 ; extensionality = extensionality | |
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335 ; minimul = minimul |
51 | 336 ; regularity = regularity |
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337 ; infinity∅ = infinity∅ |
93 | 338 ; infinity = λ _ → infinity |
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339 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
93 | 340 ; replacement = replacement |
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341 } where |
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342 open _∧_ |
41 | 343 open Minimumo |
54 | 344 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
87 | 345 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
346 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
54 | 347 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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348 empty x () |
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349 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } |
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350 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
76 | 351 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
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352 power→ A t P∋t {x} t∋x = proj1 lemma-s where |
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353 lemma-t : ZFSubset A {!!} ∋ csuc t |
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354 lemma-t = record { proj1 = {!!} ; proj2 = {!!} } |
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355 lemma-s : ZFSubset A {!!} ∋ x |
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356 lemma-s = {!!} -- transitive {!!} t∋x |
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357 lemma : od→ord (ZFSubset A (ord→od (od→ord (csuc t))) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
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358 lemma = sup-o< {suc n} ( λ x → od→ord ( ZFSubset A (ord→od x ))) |
77 | 359 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
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360 power← A t t→A = {!!} where |
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361 lemma-eq : ZFSubset A t == t |
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362 eq→ lemma-eq {z} w = proj2 w |
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363 eq← lemma-eq {z} w = record { proj2 = w ; |
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364 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
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365 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
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366 lemma = sup-o< {suc n} ( λ x → od→ord ( ZFSubset A (ord→od x ))) |
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367 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z |
72 | 368 union-lemma-u {X} {z} U>z = lemma <-osuc where |
369 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
95 | 370 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl refl |
73 | 371 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
72 | 372 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
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373 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
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374 union→ X y u xx | tri< a ¬b ¬c | () |
73 | 375 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
376 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX | |
377 lemma refl lt = lt | |
378 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 )) | |
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379 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) |
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380 union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym diso) ; proj2 = union-lemma-u X∋z } |
54 | 381 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
382 ψiso {ψ} t refl = t | |
383 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
384 selection {ψ} {X} {y} = record { | |
385 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
386 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
387 } | |
93 | 388 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
389 replacement {ψ} X x = sup-c< ψ {x} | |
60 | 390 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
391 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
54 | 392 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
68 | 393 minimul x not = od∅ |
57 | 394 regularity : (x : OD) (not : ¬ (x == od∅)) → |
395 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
66 | 396 proj1 (regularity x not ) = ¬∅=→∅∈ not |
397 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where | |
398 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
399 reg {y} t with proj1 t | |
400 ... | x∈∅ = x∈∅ | |
76 | 401 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
402 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
403 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
89 | 404 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
405 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) | |
406 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} | |
407 xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where | |
91 | 408 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) |
409 lemma1 {x} = c<→o< ( proj1 (pair x x ) ) | |
410 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) | |
411 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) | |
89 | 412 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
91 | 413 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
90 | 414 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
415 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where | |
416 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) | |
91 | 417 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) |
418 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) } | |
419 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt | |
420 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
421 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) | |
422 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 )) | |
423 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
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424 infinite : OD {suc n} |
91 | 425 infinite = ord→od ( omega ) |
426 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} | |
95 | 427 infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} |
428 (o<→c< ( case1 (s≤s z≤n ))) refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) o∅≡od∅ )) | |
91 | 429 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega |
430 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where | |
431 t : od→ord x o< od→ord (ord→od (omega)) | |
432 t = ∋→o< {n} {infinite} {x} lt | |
433 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) | |
434 infinite∋uxxx x lt = o<∋→ t where | |
435 t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) | |
436 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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437 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
91 | 438 infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where |
439 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega | |
440 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
441 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
442 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
443 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
444 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
445 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
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446 |