Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 204:d4802eb159ff
Transfinite induction fixed
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Wed, 31 Jul 2019 15:29:51 +0900 |
| parents | 8edd2a13a7f3 |
| children | 22d435172d1a |
| rev | line source |
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| 34 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 16 | 2 open import Level |
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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⊔_ ) |
| 75 | 8 open import Data.Empty |
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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 | |
| 3 | 14 |
| 24 | 15 record Ordinal {n : Level} : Set n where |
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16 constructor ordinal |
| 16 | 17 field |
| 18 lv : Nat | |
| 24 | 19 ord : OrdinalD {n} lv |
| 16 | 20 |
| 24 | 21 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 22 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 23 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 17 | 24 |
| 25 open Ordinal | |
| 26 | |
| 27 | 27 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 28 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 29 |
| 75 | 30 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 31 s<refl {n} {lv} {Φ lv} = Φ< | |
| 32 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 33 | |
| 34 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
| 35 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
| 36 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
| 37 | |
| 38 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
| 39 d<→lv Φ< = refl | |
| 40 d<→lv (s< lt) = refl | |
| 41 | |
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42 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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43 o<-subst df refl refl = df |
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44 |
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45 open import Data.Nat.Properties |
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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 |
| 39 | 55 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
| 56 | |
| 57 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
| 58 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
| 59 ordinal-cong refl refl = refl | |
| 21 | 60 |
| 46 | 61 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
| 62 ordinal-lv refl = refl | |
| 63 | |
| 64 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
| 65 ordinal-d refl = refl | |
| 66 | |
| 24 | 67 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 68 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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69 |
| 24 | 70 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 71 trio<≡ refl = ≡→¬d< |
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72 |
| 24 | 73 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 74 trio>≡ refl = ≡→¬d< |
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75 |
| 24 | 76 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 77 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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78 |
| 24 | 79 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 80 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 81 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 82 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
| 83 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
| 84 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) ) | |
| 85 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 86 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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87 |
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88 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
| 75 | 89 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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90 |
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91 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
| 75 | 92 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
| 93 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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94 |
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95 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
| 75 | 96 osuc-lveq {n} = refl |
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97 |
| 113 | 98 osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox |
| 99 osucc {n} {ox} {oy} (case1 x) = case1 x | |
| 100 osucc {n} {ox} {oy} (case2 x) with d<→lv x | |
| 101 ... | refl = case2 (s< x) | |
| 102 | |
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103 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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104 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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105 |
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106 nat-<≡ : { x : Nat } → x < x → ⊥ |
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107 nat-<≡ (s≤s lt) = nat-<≡ lt |
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108 |
| 81 | 109 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥ |
| 110 nat-≡< refl lt = nat-<≡ lt | |
| 111 | |
| 75 | 112 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ |
| 113 ¬a≤a (s≤s lt) = ¬a≤a lt | |
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114 |
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115 a<sa : {la : Nat} → la < Suc la |
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116 a<sa {Zero} = s≤s z≤n |
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117 a<sa {Suc la} = s≤s a<sa |
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118 |
| 94 | 119 =→¬< : {x : Nat } → ¬ ( x < x ) |
| 120 =→¬< {Zero} () | |
| 121 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
| 122 | |
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123 <-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) ) |
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124 <-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl |
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125 <-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n) |
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126 <-∨ {Suc x} {Zero} (s≤s ()) |
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127 <-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt |
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128 <-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq) |
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129 <-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1) |
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130 |
| 147 | 131 case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥ |
| 132 case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
| 133 ... | refl = nat-≡< refl lt1 | |
| 134 | |
| 135 case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥ | |
| 136 case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2 | |
| 137 ... | refl = nat-≡< refl lt1 | |
| 138 | |
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139 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥ |
| 111 | 140 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt |
| 141 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt | |
| 94 | 142 |
| 143 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) | |
| 144 ¬x<0 {n} {x} (case1 ()) | |
| 145 ¬x<0 {n} {x} (case2 ()) | |
| 146 | |
| 81 | 147 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
| 148 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
| 149 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
| 150 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 151 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
| 152 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
| 153 o<> (case2 y<x) (case2 x<y) | |
| 16 | 154 |
| 24 | 155 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 156 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 157 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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158 |
| 75 | 159 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
| 160 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
| 161 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
| 162 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
| 163 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
| 164 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
| 165 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
| 166 ... | case1 refl = case1 refl | |
| 167 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
| 168 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
| 169 | |
| 170 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
| 171 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
| 172 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
| 173 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
| 174 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
| 81 | 175 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
| 176 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 177 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
| 75 | 178 |
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179 max : (x y : Nat) → Nat |
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180 max Zero Zero = Zero |
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181 max Zero (Suc x) = (Suc x) |
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182 max (Suc x) Zero = (Suc x) |
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183 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 184 |
| 24 | 185 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 186 maxαd x y with triOrdd x y |
| 187 maxαd x y | tri< a ¬b ¬c = y | |
| 188 maxαd x y | tri≈ ¬a b ¬c = x | |
| 189 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 190 |
| 127 | 191 minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 192 minαd x y with triOrdd x y | |
| 193 minαd x y | tri< a ¬b ¬c = x | |
| 194 minαd x y | tri≈ ¬a b ¬c = y | |
| 195 minαd x y | tri> ¬a ¬b c = x | |
| 196 | |
| 24 | 197 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 198 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 199 | |
| 27 | 200 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 201 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 81 | 202 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
| 27 | 203 ... | refl = case1 x₁ |
| 81 | 204 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
| 27 | 205 ... | refl = case1 x₂ |
| 206 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 207 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 208 | |
| 24 | 209 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 210 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 211 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 212 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 213 lemma1 (case1 x) = ¬c x | |
| 81 | 214 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
| 24 | 215 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
| 216 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 217 lemma1 (case1 x) = ¬a x | |
| 81 | 218 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
| 23 | 219 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 220 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 |
| 221 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 222 lemma1 refl = refl | |
| 223 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 224 lemma2 (case1 x) = ¬a x | |
| 225 lemma2 (case2 x) = trio<> x a | |
| 226 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 | |
| 227 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 228 lemma1 refl = refl | |
| 229 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 230 lemma2 (case1 x) = ¬a x | |
| 231 lemma2 (case2 x) = trio<> x c | |
| 232 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 233 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 234 lemma1 (case1 x) = ¬a x | |
| 235 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 236 |
| 180 | 237 xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob |
| 238 xo<ab {n} {oa} {ob} a→b with trio< oa ob | |
| 239 xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 240 xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc | |
| 241 xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) | |
| 242 | |
| 129 | 243 maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
| 244 maxα x y with trio< x y | |
| 127 | 245 maxα x y | tri< a ¬b ¬c = y |
| 246 maxα x y | tri> ¬a ¬b c = x | |
| 129 | 247 maxα x y | tri≈ ¬a refl ¬c = x |
| 84 | 248 |
| 129 | 249 minα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal |
| 250 minα {n} x y with trio< {n} x y | |
| 127 | 251 minα x y | tri< a ¬b ¬c = x |
| 252 minα x y | tri> ¬a ¬b c = y | |
| 129 | 253 minα x y | tri≈ ¬a refl ¬c = x |
| 254 | |
| 255 min1 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y | |
| 256 min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y | |
| 257 min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x | |
| 258 min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x | |
| 259 min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y | |
| 260 | |
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261 -- |
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262 -- max ( osuc x , osuc y ) |
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263 -- |
| 88 | 264 |
| 84 | 265 omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n} |
| 88 | 266 omax {n} x y with trio< x y |
| 84 | 267 omax {n} x y | tri< a ¬b ¬c = osuc y |
| 268 omax {n} x y | tri> ¬a ¬b c = osuc x | |
| 88 | 269 omax {n} x y | tri≈ ¬a refl ¬c = osuc x |
| 84 | 270 |
| 271 omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y | |
| 88 | 272 omax< {n} x y lt with trio< x y |
| 84 | 273 omax< {n} x y lt | tri< a ¬b ¬c = refl |
| 88 | 274 omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) |
| 275 omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) | |
| 276 | |
| 277 omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y | |
| 278 omax≡ {n} x y eq with trio< x y | |
| 279 omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) | |
| 280 omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl | |
| 281 omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) | |
| 84 | 282 |
| 86 | 283 omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y |
| 88 | 284 omax-x {n} x y with trio< x y |
| 285 omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 86 | 286 omax-x {n} x y | tri> ¬a ¬b c = <-osuc |
| 88 | 287 omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc |
| 86 | 288 |
| 289 omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y | |
| 88 | 290 omax-y {n} x y with trio< x y |
| 86 | 291 omax-y {n} x y | tri< a ¬b ¬c = <-osuc |
| 88 | 292 omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc |
| 293 omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc | |
| 86 | 294 |
| 88 | 295 omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x |
| 296 omxx {n} x with trio< x x | |
| 297 omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) | |
| 298 omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) | |
| 299 omxx {n} x | tri≈ ¬a refl ¬c = refl | |
| 86 | 300 |
| 301 omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x) | |
| 88 | 302 omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) |
| 86 | 303 |
| 91 | 304 open _∧_ |
| 305 | |
| 306 osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y) | |
| 307 proj1 (osuc2 {n} x y) (case1 lt) = case1 lt | |
| 308 proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt | |
| 309 proj2 (osuc2 {n} x y) (case1 lt) = case1 lt | |
| 310 proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt | |
| 311 ... | refl = case2 (s< lt) | |
| 312 | |
| 24 | 313 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 314 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 315 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 316 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 81 | 317 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y) |
| 16 | 318 |
| 24 | 319 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 320 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 321 ; _≈_ = _≡_ |
| 23 | 322 ; _∼_ = _o≤_ |
| 16 | 323 ; isPreorder = record { |
| 324 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 325 ; reflexive = case1 | |
| 24 | 326 ; trans = OrdTrans |
| 16 | 327 } |
| 328 } | |
| 329 | |
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330 TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m } |
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331 → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 24 | 332 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) |
| 22 | 333 → ∀ (x : Ordinal) → ψ x |
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334 TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where |
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335 TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) ) |
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336 TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where |
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337 lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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338 lemma x (case1 ()) |
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339 lemma x (case2 ()) |
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340 lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x |
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341 lemma1 x (case1 ()) |
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342 lemma1 x (case2 ()) |
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343 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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344 lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy) |
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345 lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt |
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346 lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy) |
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347 lemma lx1 ox1 (case1 lt) with <-∨ lt |
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348 lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) ) |
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349 lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa)) |
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350 lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0 lx (Φ lx) (case1 (s≤s lt1)) |
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351 lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa ))) |
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352 TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )} |
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353 |
| 184 | 354 -- we cannot prove this in intutionistic logic |
| 142 | 355 -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p |
| 166 | 356 -- postulate |
| 357 -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
| 358 -- → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
| 359 -- → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → p ) | |
| 360 -- → p | |
| 361 -- | |
| 362 -- Instead we prove | |
| 363 -- | |
| 364 TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) | |
| 165 | 365 → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → ¬ p ) |
| 366 → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
| 367 → ¬ p | |
| 166 | 368 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) |
| 165 | 369 |