Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 423:9984cdd88da3
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 01 Aug 2020 18:05:23 +0900 |
parents | 44a484f17f26 |
children | cc7909f86841 |
rev | line source |
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16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
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2 module ordinal where |
3 | 3 |
423 | 4 open import logic |
5 open import nat | |
23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
75 | 7 open import Data.Empty |
423 | 8 open import Relation.Binary.PropositionalEquality |
9 open import Data.Nat.Properties | |
10 open import Relation.Nullary | |
11 open import Relation.Binary.Core | |
12 | |
3 | 13 |
24 | 14 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
15 Φ : (lv : Nat) → OrdinalD lv | |
16 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
3 | 17 |
24 | 18 record Ordinal {n : Level} : Set n where |
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19 constructor ordinal |
16 | 20 field |
21 lv : Nat | |
24 | 22 ord : OrdinalD {n} lv |
16 | 23 |
24 | 24 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
25 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
26 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
17 | 27 |
28 open Ordinal | |
29 | |
27 | 30 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
17 | 31 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
3 | 32 |
75 | 33 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
34 s<refl {n} {lv} {Φ lv} = Φ< | |
35 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
36 | |
37 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
38 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
39 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
40 | |
41 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
42 d<→lv Φ< = refl | |
43 d<→lv (s< lt) = refl | |
44 | |
24 | 45 o∅ : {n : Level} → Ordinal {n} |
46 o∅ = record { lv = Zero ; ord = Φ Zero } | |
21 | 47 |
39 | 48 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
49 | |
50 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
51 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
52 ordinal-cong refl refl = refl | |
21 | 53 |
24 | 54 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
55 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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separete constructible set
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56 |
24 | 57 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
17 | 58 trio<≡ refl = ≡→¬d< |
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59 |
24 | 60 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
17 | 61 trio>≡ refl = ≡→¬d< |
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62 |
24 | 63 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
64 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
65 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
66 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
67 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
68 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) ) | |
69 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
70 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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71 |
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72 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
75 | 73 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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74 |
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75 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
75 | 76 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
77 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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78 |
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79 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥ |
111 | 80 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt |
81 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt | |
94 | 82 |
83 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) | |
84 ¬x<0 {n} {x} (case1 ()) | |
85 ¬x<0 {n} {x} (case2 ()) | |
86 | |
81 | 87 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
88 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
89 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
90 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
91 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
92 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
93 o<> (case2 y<x) (case2 x<y) | |
16 | 94 |
24 | 95 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
96 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
97 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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98 |
75 | 99 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
100 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
101 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
102 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
103 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
104 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
105 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
106 ... | case1 refl = case1 refl | |
107 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
108 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
109 | |
110 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
111 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
112 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
113 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
114 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
81 | 115 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
116 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
117 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
75 | 118 |
23 | 119 |
27 | 120 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
121 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
81 | 122 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
27 | 123 ... | refl = case1 x₁ |
81 | 124 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
27 | 125 ... | refl = case1 x₂ |
126 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
127 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
128 | |
24 | 129 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
23 | 130 trio< a b with <-cmp (lv a) (lv b) |
24 | 131 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
132 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
133 lemma1 (case1 x) = ¬c x | |
81 | 134 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
24 | 135 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
136 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
137 lemma1 (case1 x) = ¬a x | |
81 | 138 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
23 | 139 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
24 | 140 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 |
141 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
142 lemma1 refl = refl | |
143 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
144 lemma2 (case1 x) = ¬a x | |
145 lemma2 (case2 x) = trio<> x a | |
146 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 | |
147 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
148 lemma1 refl = refl | |
149 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
150 lemma2 (case1 x) = ¬a x | |
151 lemma2 (case2 x) = trio<> x c | |
152 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
153 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
154 lemma1 (case1 x) = ¬a x | |
155 lemma1 (case2 x) = ≡→¬d< x | |
23 | 156 |
86 | 157 |
91 | 158 open _∧_ |
159 | |
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160 TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m } |
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161 → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
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162 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ( (y : Ordinal {suc n} ) → y o< ordinal lx (OSuc lx x) → ψ y ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) |
22 | 163 → ∀ (x : Ordinal) → ψ x |
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164 TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where |
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165 TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx ox → ψ x ) ) |
422 | 166 TransFinite1 Zero (Φ 0) = ⟪ caseΦ Zero lemma , lemma1 ⟫ where |
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167 lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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168 lemma x (case1 ()) |
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169 lemma x (case2 ()) |
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170 lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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171 lemma1 x (case1 ()) |
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172 lemma1 x (case2 ()) |
422 | 173 TransFinite1 (Suc lx) (Φ (Suc lx)) = ⟪ caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) , (λ x → lemma (lv x) (ord x)) ⟫ where |
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174 lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy) |
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175 lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt |
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176 lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy) |
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177 lemma lx1 ox1 (case1 lt) with <-∨ lt |
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178 lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) ) |
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179 lemma lx (Φ lx) (case1 lt) | case2 lt1 = lemma0 lx (Φ lx) (case1 lt1) |
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180 lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 lemma2 where |
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181 lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx ox1) → ψ y |
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182 lemma2 y lt1 with osuc-≡< lt1 |
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183 lemma2 y lt1 | case1 refl = lemma lx ox1 (case1 a<sa) |
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184 lemma2 y lt1 | case2 t = proj2 (TransFinite1 lx ox1) y t |
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185 lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 lemma2 where |
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186 lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx1) ∨ (ord y d< OSuc lx1 ox1) → ψ y |
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187 lemma2 y lt2 with osuc-≡< lt2 |
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188 lemma2 y lt2 | case1 refl = lemma lx1 ox1 (ordtrans lt2 (case1 lt)) |
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189 lemma2 y lt2 | case2 (case1 lt3) = proj2 (TransFinite1 lx (Φ lx)) y (case1 (<-trans lt3 lt1 )) |
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190 lemma2 y lt2 | case2 (case2 lt3) with d<→lv lt3 |
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191 ... | refl = proj2 (TransFinite1 lx (Φ lx)) y (case1 lt1) |
422 | 192 TransFinite1 lx (OSuc lx ox) = ⟪ caseOSuc lx ox lemma , lemma ⟫ where |
222
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193 lemma : (y : Ordinal) → y o< ordinal lx (OSuc lx ox) → ψ y |
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194 lemma y lt with osuc-≡< lt |
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195 lemma y lt | case1 refl = proj1 ( TransFinite1 lx ox ) |
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196 lemma y lt | case2 lt1 = proj2 ( TransFinite1 lx ox ) y lt1 |
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197 |
224 | 198 open import Ordinals |
222
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199 |
224 | 200 C-Ordinal : {n : Level} → Ordinals {suc n} |
222
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201 C-Ordinal {n} = record { |
423 | 202 Ordinal = Ordinal {suc n} |
222
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203 ; o∅ = o∅ |
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204 ; osuc = osuc |
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205 ; _o<_ = _o<_ |
329 | 206 ; next = next |
222
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207 ; isOrdinal = record { |
423 | 208 ordtrans = ordtrans |
209 ; trio< = trio< | |
222
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210 ; ¬x<0 = ¬x<0 |
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211 ; <-osuc = <-osuc |
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212 ; osuc-≡< = osuc-≡< |
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213 ; TransFinite = TransFinite1 |
330 | 214 ; TransFinite1 = TransFinite2 |
416 | 215 ; Oprev-p = Oprev-p |
358 | 216 } ; |
217 isNext = record { | |
218 x<nx = x<nx | |
219 ; osuc<nx = λ {x} {y} → osuc<nx {x} {y} | |
220 ; ¬nx<nx = ¬nx<nx | |
222
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221 } |
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222 } where |
329 | 223 next : Ordinal {suc n} → Ordinal {suc n} |
224 next (ordinal lv ord) = ordinal (Suc lv) (Φ (Suc lv)) | |
358 | 225 x<nx : { y : Ordinal } → (y o< next y ) |
226 x<nx = case1 a<sa | |
227 osuc<nx : { x y : Ordinal } → x o< next y → osuc x o< next y | |
228 osuc<nx (case1 lt) = case1 lt | |
229 ¬nx<nx : {x y : Ordinal} → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ (x ≡ osuc z)) | |
230 ¬nx<nx {x} {y} = lemma2 x where | |
339 | 231 lemma2 : (x : Ordinal) → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ x ≡ osuc z) |
232 lemma2 (ordinal Zero (Φ 0)) (case2 ()) (case1 (s≤s z≤n)) not | |
233 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 Φ<) (case1 (s≤s z≤n)) not = not _ refl | |
234 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 (s< x)) (case1 (s≤s z≤n)) not = not _ refl | |
235 lemma2 (ordinal (Suc lx) (OSuc (Suc lx) ox)) y<x (case1 (s≤s (s≤s lt))) not = not _ refl | |
236 lemma2 (ordinal (Suc lx) (Φ (Suc lx))) (case1 x) (case1 (s≤s (s≤s lt))) not = lemma3 x lt where | |
237 lemma3 : {n l : Nat} → (Suc (Suc n) ≤ Suc l) → l ≤ n → ⊥ | |
238 lemma3 (s≤s sn≤l) (s≤s l≤n) = lemma3 sn≤l l≤n | |
416 | 239 open Oprev |
240 Oprev-p : (x : Ordinal) → Dec ( Oprev (Ordinal {suc n}) osuc x ) | |
241 Oprev-p (ordinal lv (Φ lv)) = no (λ not → lemma (oprev not) (oprev=x not) ) where | |
242 lemma : (x : Ordinal) → osuc x ≡ (ordinal lv (Φ lv)) → ⊥ | |
243 lemma x () | |
244 Oprev-p (ordinal lv (OSuc lv ox)) = yes record { oprev = ordinal lv ox ; oprev=x = refl } | |
222
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245 ord1 : Set (suc n) |
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246 ord1 = Ordinal {suc n} |
324 | 247 TransFinite1 : { ψ : ord1 → Set (suc n) } |
222
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248 → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) |
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249 → ∀ (x : ord1) → ψ x |
324 | 250 TransFinite1 {ψ} lt x = TransFinite {n} {suc n} {ψ} caseΦ caseOSuc x where |
222
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251 caseΦ : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → ψ x₁) → |
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252 ψ (record { lv = lx ; ord = Φ lx }) |
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253 caseΦ lx prev = lt (ordinal lx (Φ lx) ) prev |
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254 caseOSuc : (lx : Nat) (x₁ : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x₁) → ψ y) → |
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255 ψ (record { lv = lx ; ord = OSuc lx x₁ }) |
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256 caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev |
330 | 257 TransFinite2 : { ψ : ord1 → Set (suc (suc n)) } |
258 → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) | |
259 → ∀ (x : ord1) → ψ x | |
260 TransFinite2 {ψ} lt x = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc x where | |
261 caseΦ : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → ψ x₁) → | |
262 ψ (record { lv = lx ; ord = Φ lx }) | |
263 caseΦ lx prev = lt (ordinal lx (Φ lx) ) prev | |
264 caseOSuc : (lx : Nat) (x₁ : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x₁) → ψ y) → | |
265 ψ (record { lv = lx ; ord = OSuc lx x₁ }) | |
266 caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev | |
224 | 267 |
330 | 268 |