Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 70:cd9cf8b09610
Union needs +1 space
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 01 Jun 2019 10:01:38 +0900 |
parents | e584686a1307 |
children | d088eb66564e |
rev | line source |
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34 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
16 | 2 open import Level |
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posturate OD is isomorphic to Ordinal
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3 module ordinal where |
3 | 4 |
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5 open import zf |
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 |
24 | 11 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
12 Φ : (lv : Nat) → OrdinalD lv | |
13 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
17 | 14 ℵ_ : (lv : Nat) → OrdinalD (Suc lv) |
3 | 15 |
24 | 16 record Ordinal {n : Level} : Set n where |
16 | 17 field |
18 lv : Nat | |
24 | 19 ord : OrdinalD {n} lv |
16 | 20 |
34 | 21 data ¬ℵ {n : Level} {lx : Nat } : ( x : OrdinalD {n} lx ) → Set where |
22 ¬ℵΦ : ¬ℵ (Φ lx) | |
23 ¬ℵs : {x : OrdinalD {n} lx } → ¬ℵ x → ¬ℵ (OSuc lx x) | |
24 | |
70 | 25 -- |
26 -- Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ... < ℵ (Suc lv) | |
27 -- | |
24 | 28 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
29 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
30 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
41 | 31 ℵΦ< : {lx : Nat} → Φ (Suc lx) d< (ℵ lx) |
34 | 32 ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → ¬ℵ x → OSuc (Suc lx) x d< (ℵ lx) |
33 ℵs< : {lx : Nat} → (ℵ lx) d< OSuc (Suc lx) (ℵ lx) | |
35 | 34 ℵss< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → (ℵ lx) d< x → (ℵ lx) d< OSuc (Suc lx) x |
17 | 35 |
36 open Ordinal | |
37 | |
27 | 38 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
17 | 39 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
3 | 40 |
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41 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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42 o<-subst df refl refl = df |
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43 |
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44 open import Data.Nat.Properties |
6 | 45 open import Data.Empty |
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problem on Ordinal ( OSuc ℵ )
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46 open import Data.Unit using ( ⊤ ) |
6 | 47 open import Relation.Nullary |
48 | |
49 open import Relation.Binary | |
50 open import Relation.Binary.Core | |
51 | |
24 | 52 o∅ : {n : Level} → Ordinal {n} |
53 o∅ = record { lv = Zero ; ord = Φ Zero } | |
21 | 54 |
34 | 55 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
56 s<refl {n} {lv} {Φ lv} = Φ< | |
57 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
58 s<refl {n} {Suc lv} {ℵ lv} = ℵs< | |
59 | |
39 | 60 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
61 | |
62 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
63 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
64 ordinal-cong refl refl = refl | |
21 | 65 |
46 | 66 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
67 ordinal-lv refl = refl | |
68 | |
69 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
70 ordinal-d refl = refl | |
71 | |
24 | 72 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
73 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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74 |
24 | 75 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ |
34 | 76 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t |
77 trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) | |
78 trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< | |
35 | 79 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () |
80 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(Φ (Suc _))} ℵΦ< () | |
81 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (Φ (Suc _)))} (ℵ< ¬ℵΦ) (ℵss< ()) | |
82 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} (ℵ< (¬ℵs x)) (ℵss< x<y) = trio<> (ℵ< x) x<y | |
83 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (Φ (Suc _)))} {.(ℵ _)} (ℵss< ()) (ℵ< ¬ℵΦ) | |
84 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} {.(ℵ _)} (ℵss< y<x) (ℵ< (¬ℵs x)) = trio<> y<x (ℵ< x) | |
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85 |
24 | 86 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
17 | 87 trio<≡ refl = ≡→¬d< |
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88 |
24 | 89 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
17 | 90 trio>≡ refl = ≡→¬d< |
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91 |
24 | 92 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
93 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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94 |
35 | 95 fin : {n : Level} → {lx : Nat} → {y : OrdinalD {n} (Suc lx) } → y d< (ℵ lx) → ¬ℵ y |
96 fin {_} {_} {Φ (Suc _)} ℵΦ< = ¬ℵΦ | |
97 fin {_} {_} {OSuc (Suc _) _} (ℵ< x) = ¬ℵs x | |
98 | |
24 | 99 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
100 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
101 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
102 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
41 | 103 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< ℵΦ< (λ ()) ( λ lt → trio<> lt ℵΦ<) |
104 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt ℵΦ< ) (λ ()) ℵΦ< | |
35 | 105 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y ) with triOrdd (ℵ lv) y |
106 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri< a ¬b ¬c = tri< (ℵss< a) (λ ()) (trio<> (ℵss< a) ) | |
107 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri≈ ¬a refl ¬c = tri< ℵs< (λ ()) ( λ lt → trio<> lt ℵs< ) | |
108 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt ( ℵ< (fin c)) ) (λ ()) ( ℵ< (fin c) ) | |
24 | 109 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< |
35 | 110 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) with triOrdd x (ℵ lv) |
111 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri< a ¬b ¬c = tri< (ℵ< (fin a ) ) (λ ()) ( λ lt → trio<> lt (ℵ< (fin a ))) | |
112 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri≈ ¬a refl ¬c = tri> (λ lt → trio<> lt ℵs< ) (λ ()) ℵs< | |
113 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri> ¬a ¬b c = tri> (λ lt → trio<> lt (ℵss< c )) (λ ()) ( ℵss< c ) | |
24 | 114 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y |
115 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) ) | |
116 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
117 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c) | |
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118 |
24 | 119 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y |
17 | 120 d<→lv Φ< = refl |
121 d<→lv (s< lt) = refl | |
122 d<→lv ℵΦ< = refl | |
34 | 123 d<→lv (ℵ< _) = refl |
124 d<→lv ℵs< = refl | |
35 | 125 d<→lv (ℵss< _) = refl |
126 | |
127 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x | |
128 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ | |
129 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t) | |
130 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) () | |
131 xsyℵ {_} {_} {ℵ lv₁} {.(OSuc (Suc lv₁) (ℵ lv₁))} ℵs< (¬ℵs t) = t | |
132 xsyℵ (ℵss< ()) (¬ℵs ¬ℵΦ) | |
133 xsyℵ (ℵss< x<y) (¬ℵs t) = xsyℵ x<y t | |
16 | 134 |
24 | 135 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
136 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
41 | 137 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< (ℵ< _) = ℵΦ< |
24 | 138 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) |
34 | 139 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) |
140 orddtrans ℵs< (ℵ< ()) | |
35 | 141 orddtrans {n} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc ly) y} {ℵ _} (s< x<y) (ℵ< t) = ℵ< ( xsyℵ x<y t ) |
142 orddtrans {n} {.(Suc _)} {.(Φ (Suc _))} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} ℵΦ< ℵs< = Φ< | |
41 | 143 orddtrans {n} {.(Suc _)} {OSuc (Suc _) .(Φ (Suc _))} {.(ℵ _)} {OSuc (Suc _) (ℵ k)} (ℵ< ¬ℵΦ) ℵs< = s< ℵΦ< |
35 | 144 orddtrans {n} {.(Suc _)} {OSuc (Suc lv) (OSuc (Suc _) x)} {ℵ lv} {.(OSuc (Suc _) (ℵ _))} (ℵ< (¬ℵs t)) ℵs< = s< ( ℵ< t ) |
145 orddtrans {n} {.(Suc lv)} {ℵ lv} {OSuc .(Suc lv) (ℵ lv)} {OSuc .(Suc lv) .(OSuc (Suc lv) (ℵ lv))} ℵs< (s< ℵs<) = ℵss< ℵs< | |
146 orddtrans ℵΦ< (ℵss< y<z) = Φ< | |
41 | 147 orddtrans (ℵ< {lx} {Φ .(Suc lx)} nxx) (ℵss< {_} {k} y<z) = s< (orddtrans ℵΦ< y<z) |
35 | 148 orddtrans (ℵ< {lx} {OSuc .(Suc lx) xx} (¬ℵs nxx)) (ℵss< y<z) = s< (orddtrans (ℵ< nxx) y<z) |
149 orddtrans (ℵ< {.lv₁} {ℵ lv₁} ()) (ℵss< y<z) | |
150 orddtrans (ℵss< x<y) (s< y<z) = ℵss< ( orddtrans x<y y<z ) | |
151 orddtrans (ℵss< ()) (ℵ< ¬ℵΦ) | |
152 orddtrans (ℵss< ℵs<) (ℵ< (¬ℵs ())) | |
153 orddtrans (ℵss< (ℵss< x<y)) (ℵ< (¬ℵs x)) = orddtrans (ℵss< x<y) ( ℵ< x ) | |
154 orddtrans {n} {Suc lx} {x} {y} {z} ℵs< (s< (ℵss< {lx} {ss} y<z)) = ℵss< ( ℵss< y<z ) | |
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155 |
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156 max : (x y : Nat) → Nat |
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157 max Zero Zero = Zero |
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158 max Zero (Suc x) = (Suc x) |
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159 max (Suc x) Zero = (Suc x) |
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160 max (Suc x) (Suc y) = Suc ( max x y ) |
3 | 161 |
24 | 162 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
17 | 163 maxαd x y with triOrdd x y |
164 maxαd x y | tri< a ¬b ¬c = y | |
165 maxαd x y | tri≈ ¬a b ¬c = x | |
166 maxαd x y | tri> ¬a ¬b c = x | |
6 | 167 |
24 | 168 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
17 | 169 maxα x y with <-cmp (lv x) (lv y) |
170 maxα x y | tri< a ¬b ¬c = x | |
171 maxα x y | tri> ¬a ¬b c = y | |
172 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
7 | 173 |
24 | 174 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
23 | 175 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
176 | |
27 | 177 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
178 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
179 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ | |
180 ... | refl = case1 x₁ | |
181 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ | |
182 ... | refl = case1 x₂ | |
183 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
184 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
185 | |
186 | |
24 | 187 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
23 | 188 trio< a b with <-cmp (lv a) (lv b) |
24 | 189 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
190 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
191 lemma1 (case1 x) = ¬c x | |
192 lemma1 (case2 x) with d<→lv x | |
193 lemma1 (case2 x) | refl = ¬b refl | |
194 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where | |
195 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
196 lemma1 (case1 x) = ¬a x | |
197 lemma1 (case2 x) with d<→lv x | |
198 lemma1 (case2 x) | refl = ¬b refl | |
23 | 199 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
24 | 200 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where |
201 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
202 lemma1 refl = refl | |
203 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
204 lemma2 (case1 x) = ¬a x | |
205 lemma2 (case2 x) = trio<> x a | |
206 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where | |
207 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
208 lemma1 refl = refl | |
209 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
210 lemma2 (case1 x) = ¬a x | |
211 lemma2 (case2 x) = trio<> x c | |
212 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
213 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
214 lemma1 (case1 x) = ¬a x | |
215 lemma1 (case2 x) = ≡→¬d< x | |
23 | 216 |
24 | 217 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
16 | 218 OrdTrans (case1 refl) (case1 refl) = case1 refl |
219 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
220 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
17 | 221 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) |
222 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y | |
223 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) | |
224 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x | |
225 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) | |
226 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y | |
227 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) | |
16 | 228 |
24 | 229 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
230 OrdPreorder {n} = record { Carrier = Ordinal | |
16 | 231 ; _≈_ = _≡_ |
23 | 232 ; _∼_ = _o≤_ |
16 | 233 ; isPreorder = record { |
234 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
235 ; reflexive = case1 | |
24 | 236 ; trans = OrdTrans |
16 | 237 } |
238 } | |
239 | |
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240 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
22 | 241 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) |
24 | 242 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
243 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
22 | 244 → ∀ (x : Ordinal) → ψ x |
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245 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv |
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246 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ |
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247 ( TransFinite caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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248 TransFinite caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ |
22 | 249 |