Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 324:fbabb20f222e
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 04 Jul 2020 18:18:17 +0900 |
| parents | 6f10c47e4e7a |
| children | 1a54dbe1ea4c |
| 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 |
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4 open import zf |
| 3 | 5 |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 75 | 7 open import Data.Empty |
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8 open import Relation.Binary.PropositionalEquality |
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separate logic and nat
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9 open import logic |
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10 open import nat |
| 3 | 11 |
| 24 | 12 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
| 13 Φ : (lv : Nat) → OrdinalD lv | |
| 14 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
| 3 | 15 |
| 24 | 16 record Ordinal {n : Level} : Set n where |
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17 constructor ordinal |
| 16 | 18 field |
| 19 lv : Nat | |
| 24 | 20 ord : OrdinalD {n} lv |
| 16 | 21 |
| 24 | 22 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 23 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 24 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 17 | 25 |
| 26 open Ordinal | |
| 27 | |
| 27 | 28 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 29 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 30 |
| 75 | 31 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 32 s<refl {n} {lv} {Φ lv} = Φ< | |
| 33 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 34 | |
| 35 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
| 36 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
| 37 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
| 38 | |
| 39 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
| 40 d<→lv Φ< = refl | |
| 41 d<→lv (s< lt) = refl | |
| 42 | |
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equalitu and internal parametorisity
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43 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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44 o<-subst df refl refl = df |
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45 |
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46 open import Data.Nat.Properties |
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problem on Ordinal ( OSuc ℵ )
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47 open import Data.Unit using ( ⊤ ) |
| 6 | 48 open import Relation.Nullary |
| 49 | |
| 50 open import Relation.Binary | |
| 51 open import Relation.Binary.Core | |
| 52 | |
| 24 | 53 o∅ : {n : Level} → Ordinal {n} |
| 54 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 55 |
| 39 | 56 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
| 57 | |
| 58 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
| 59 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
| 60 ordinal-cong refl refl = refl | |
| 21 | 61 |
| 24 | 62 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 63 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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64 |
| 24 | 65 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 66 trio<≡ refl = ≡→¬d< |
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67 |
| 24 | 68 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 69 trio>≡ refl = ≡→¬d< |
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70 |
| 24 | 71 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 72 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 73 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 74 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
| 75 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
| 76 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) ) | |
| 77 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 78 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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79 |
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80 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
| 75 | 81 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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82 |
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83 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
| 75 | 84 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
| 85 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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86 |
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87 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥ |
| 111 | 88 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt |
| 89 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt | |
| 94 | 90 |
| 91 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) | |
| 92 ¬x<0 {n} {x} (case1 ()) | |
| 93 ¬x<0 {n} {x} (case2 ()) | |
| 94 | |
| 81 | 95 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
| 96 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
| 97 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
| 98 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 99 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
| 100 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
| 101 o<> (case2 y<x) (case2 x<y) | |
| 16 | 102 |
| 24 | 103 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 104 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 105 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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106 |
| 75 | 107 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
| 108 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
| 109 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
| 110 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
| 111 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
| 112 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
| 113 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
| 114 ... | case1 refl = case1 refl | |
| 115 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
| 116 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
| 117 | |
| 118 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
| 119 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
| 120 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
| 121 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
| 122 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
| 81 | 123 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
| 124 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 125 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
| 75 | 126 |
| 23 | 127 |
| 27 | 128 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 129 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 81 | 130 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
| 27 | 131 ... | refl = case1 x₁ |
| 81 | 132 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
| 27 | 133 ... | refl = case1 x₂ |
| 134 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 135 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 136 | |
| 24 | 137 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 138 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 139 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 140 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 141 lemma1 (case1 x) = ¬c x | |
| 81 | 142 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
| 24 | 143 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
| 144 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 145 lemma1 (case1 x) = ¬a x | |
| 81 | 146 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
| 23 | 147 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 148 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 |
| 149 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 150 lemma1 refl = refl | |
| 151 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 152 lemma2 (case1 x) = ¬a x | |
| 153 lemma2 (case2 x) = trio<> x a | |
| 154 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 | |
| 155 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 156 lemma1 refl = refl | |
| 157 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 158 lemma2 (case1 x) = ¬a x | |
| 159 lemma2 (case2 x) = trio<> x c | |
| 160 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 161 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 162 lemma1 (case1 x) = ¬a x | |
| 163 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 164 |
| 86 | 165 |
| 91 | 166 open _∧_ |
| 167 | |
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168 TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m } |
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169 → ( ∀ (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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170 → ( ∀ (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 | 171 → ∀ (x : Ordinal) → ψ x |
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172 TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where |
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173 TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx ox → ψ x ) ) |
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174 TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where |
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175 lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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176 lemma x (case1 ()) |
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177 lemma x (case2 ()) |
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178 lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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179 lemma1 x (case1 ()) |
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180 lemma1 x (case2 ()) |
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181 TransFinite1 (Suc lx) (Φ (Suc lx)) = record { proj1 = caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) ; proj2 = (λ x → lemma (lv x) (ord x)) } where |
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182 lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy) |
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183 lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt |
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184 lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy) |
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185 lemma lx1 ox1 (case1 lt) with <-∨ lt |
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186 lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) ) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
187 lemma lx (Φ lx) (case1 lt) | case2 lt1 = lemma0 lx (Φ lx) (case1 lt1) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
188 lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 lemma2 where |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
189 lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx ox1) → ψ y |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
190 lemma2 y lt1 with osuc-≡< lt1 |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
191 lemma2 y lt1 | case1 refl = lemma lx ox1 (case1 a<sa) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
192 lemma2 y lt1 | case2 t = proj2 (TransFinite1 lx ox1) y t |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
193 lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 lemma2 where |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
194 lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx1) ∨ (ord y d< OSuc lx1 ox1) → ψ y |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
195 lemma2 y lt2 with osuc-≡< lt2 |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
196 lemma2 y lt2 | case1 refl = lemma lx1 ox1 (ordtrans lt2 (case1 lt)) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
197 lemma2 y lt2 | case2 (case1 lt3) = proj2 (TransFinite1 lx (Φ lx)) y (case1 (<-trans lt3 lt1 )) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
198 lemma2 y lt2 | case2 (case2 lt3) with d<→lv lt3 |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
199 ... | refl = proj2 (TransFinite1 lx (Φ lx)) y (case1 lt1) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
200 TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox lemma ; proj2 = lemma } where |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
201 lemma : (y : Ordinal) → y o< ordinal lx (OSuc lx ox) → ψ y |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
202 lemma y lt with osuc-≡< lt |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
203 lemma y lt | case1 refl = proj1 ( TransFinite1 lx ox ) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
204 lemma y lt | case2 lt1 = proj2 ( TransFinite1 lx ox ) y lt1 |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
205 |
| 224 | 206 open import Ordinals |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
207 |
| 224 | 208 C-Ordinal : {n : Level} → Ordinals {suc n} |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
209 C-Ordinal {n} = record { |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
210 ord = Ordinal {suc n} |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
211 ; o∅ = o∅ |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
212 ; osuc = osuc |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
213 ; _o<_ = _o<_ |
| 324 | 214 ; next = ? |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
215 ; isOrdinal = record { |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
216 Otrans = ordtrans |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
217 ; OTri = trio< |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
218 ; ¬x<0 = ¬x<0 |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
219 ; <-osuc = <-osuc |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
220 ; osuc-≡< = osuc-≡< |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
221 ; TransFinite = TransFinite1 |
| 324 | 222 ; is-limit = ? |
| 223 ; next-limit = ? | |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
224 } |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
225 } where |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
226 ord1 : Set (suc n) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
227 ord1 = Ordinal {suc n} |
| 324 | 228 TransFinite1 : { ψ : ord1 → Set (suc n) } |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
229 → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
230 → ∀ (x : ord1) → ψ x |
| 324 | 231 TransFinite1 {ψ} lt x = TransFinite {n} {suc n} {ψ} caseΦ caseOSuc x where |
|
222
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
232 caseΦ : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → ψ x₁) → |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
233 ψ (record { lv = lx ; ord = Φ lx }) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
234 caseΦ lx prev = lt (ordinal lx (Φ lx) ) prev |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
235 caseOSuc : (lx : Nat) (x₁ : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x₁) → ψ y) → |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
236 ψ (record { lv = lx ; ord = OSuc lx x₁ }) |
|
59771eb07df0
TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
220
diff
changeset
|
237 caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev |
| 224 | 238 |