Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 237:521290e85527
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Mon, 19 Aug 2019 00:37:35 +0900 |
| parents | 650bdad56729 |
| children | a8c6239176db |
| rev | line source |
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| 16 | 1 open import Level |
| 224 | 2 open import Ordinals |
| 3 module cardinal {n : Level } (O : Ordinals {n}) where | |
| 3 | 4 |
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5 open import zf |
| 219 | 6 open import logic |
| 224 | 7 import OD |
| 23 | 8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 224 | 9 open import Relation.Binary.PropositionalEquality |
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10 open import Data.Nat.Properties |
| 6 | 11 open import Data.Empty |
| 12 open import Relation.Nullary | |
| 13 open import Relation.Binary | |
| 14 open import Relation.Binary.Core | |
| 15 | |
| 224 | 16 open inOrdinal O |
| 17 open OD O | |
| 219 | 18 open OD.OD |
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posturate OD is isomorphic to Ordinal
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19 |
| 120 | 20 open _∧_ |
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21 open _∨_ |
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22 open Bool |
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od→lv : {n : Level} → OD {n} → Nat
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23 |
| 230 | 24 -- we have to work on Ordinal to keep OD Level n |
| 25 -- since we use p∨¬p which works only on Level n | |
| 225 | 26 |
| 233 | 27 <_,_> : (x y : OD) → OD |
| 28 < x , y > = (x , x ) , (x , y ) | |
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29 |
| 237 | 30 Ord< : {x y : Ordinal } → x o< y → od→ord (Ord x) o< od→ord (Ord y) -- x o< y = Ord y ∋ x → ox o< od→ord (Ord y) |
| 31 Ord< {x} {y} lt with trio< (od→ord (Ord x)) (od→ord (Ord y)) | |
| 32 Ord< {x} {y} lt | tri< a ¬b ¬c = a | |
| 33 Ord< {x} {y} lt | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ refl (def-subst {Ord y} {x} {Ord x} {x} lt -- Ord x ∋ x | |
| 34 (subst₂ (λ j k → j ≡ k) oiso oiso (cong (λ k → ord→od k )(sym b)) )refl )) -- Ord x ≡ Ord y | |
| 35 Ord< {x} {y} lt | tri> ¬a ¬b c = ? where | |
| 36 lemma : ¬ ( Ord x ∋ ord→od x ) | |
| 37 lemma lt = ⊥-elim ( o<¬≡ refl ( def-subst {_} {_} {Ord x} {x} lt refl diso )) | |
| 38 lemma1 : od→ord (Ord y) o< od→ord (Ord x) → x o< od→ord (Ord x ) | |
| 39 lemma1 lt1 = ordtrans ( o<-subst (c<→o< lt ) diso refl ) {!!} | |
| 40 | |
| 41 --- Ord (osuc ox) ∋ x | |
| 42 --- ox o< od→ord (Ord (osuc ox)) | |
| 43 --- osuc ox ≡ od→ord (Ord (osuc ox)) | |
| 44 | |
| 45 opair : {x y : OD} → ( < x , y > ∋ x ) ∧ ( < x , y > ∋ y ) | |
| 46 opair {x} {y} = record { proj1 = lemma ; proj2 = {!!} } where | |
| 47 ox = (od→ord x) | |
| 48 oy = (od→ord y) | |
| 49 -- < x , y > ∋ x -- od→ord (Ord (omax ox ox )) | |
| 50 lemma : ox o< (omax (od→ord (x , x)) (od→ord (x , y))) | |
| 51 lemma with trio< ox oy | |
| 52 lemma | tri< a ¬b ¬c with omax< ox oy a | omxx ox | |
| 53 ... | eq | eq1 = {!!} | |
| 54 lemma | tri≈ ¬a b ¬c = {!!} | |
| 55 lemma | tri> ¬a ¬b c = {!!} | |
| 56 | |
| 236 | 57 record SetProduct ( A B : OD ) : Set n where |
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58 field |
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59 π1 : Ordinal |
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60 π2 : Ordinal |
| 233 | 61 A∋π1 : def A π1 |
| 62 B∋π2 : def B π2 | |
| 63 -- opair : x ≡ od→ord (Ord ( omax (omax π1 π1) (omax π1 π2) )) -- < π1 , π2 > | |
| 64 | |
| 65 open SetProduct | |
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66 |
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67 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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68 ∋-p A x with p∨¬p ( A ∋ x ) |
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69 ∋-p A x | case1 t = yes t |
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70 ∋-p A x | case2 t = no t |
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71 |
| 233 | 72 _⊗_ : (A B : OD) → OD |
| 236 | 73 A ⊗ B = record { def = λ x → SetProduct A B } |
| 233 | 74 -- A ⊗ B = record { def = λ x → (y z : Ordinal) → def A y ∧ def B z ∧ ( x ≡ od→ord (< ord→od y , ord→od z >) ) } |
| 75 | |
| 76 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) | |
| 225 | 77 |
| 233 | 78 Func : ( A B : OD ) → OD |
| 79 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 80 | |
| 81 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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82 |
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83 func←od : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → (Ordinal → Ordinal ) |
| 233 | 84 func←od {dom} {cod} {f} lt x = sup-o ( λ y → lemma y ) where |
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85 lemma : Ordinal → Ordinal |
| 235 | 86 lemma y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) |
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87 lemma y | p | no n = o∅ |
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88 lemma y | p | yes f∋y with double-neg-eilm ( p {ord→od y} f∋y ) -- p : {x : OD} → f ∋ x → ¬ ¬ (dom ⊗ cod ∋ x) |
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89 ... | t with decp ( x ≡ π1 t ) |
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90 ... | yes _ = π2 t |
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91 ... | no _ = o∅ |
| 233 | 92 |
| 236 | 93 |
| 233 | 94 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
| 95 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 96 | |
| 236 | 97 record Func←cd { dom cod : OD } {f : Ordinal } (f<F : def (Func dom cod ) f) : Set n where |
| 98 field | |
| 99 func-1 : Ordinal → Ordinal | |
| 100 func→od∈Func-1 : (Func dom (Ord (sup-o (λ x → func-1 x)) )) ∋ func→od func-1 dom | |
| 101 | |
| 102 func←od1 : { dom cod : OD } → {f : Ordinal } → (f<F : def (Func dom cod ) f ) → Func←cd {dom} {cod} {f} f<F | |
| 103 func←od1 {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = {!!} } where | |
| 104 lemma : Ordinal → Ordinal → Ordinal | |
| 105 lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) | |
| 106 lemma x y | p | no n = o∅ | |
| 107 lemma x y | p | yes f∋y with double-neg-eilm ( p {ord→od y} f∋y ) -- p : {x : OD} → f ∋ x → ¬ ¬ (dom ⊗ cod ∋ x) | |
| 108 ... | t with decp ( x ≡ π1 t ) | |
| 109 ... | yes _ = π2 t | |
| 110 ... | no _ = o∅ | |
| 111 | |
| 112 func→od∈Func : (f : Ordinal → Ordinal ) ( dom : OD ) → (Func dom (Ord (sup-o (λ x → f x)) )) ∋ func→od f dom | |
| 113 func→od∈Func f dom = record { proj1 = {!!} ; proj2 = {!!} } | |
| 225 | 114 |
| 227 | 115 -- contra position of sup-o< |
| 116 -- | |
| 117 | |
| 235 | 118 -- postulate |
| 119 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 120 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 121 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 122 |
| 219 | 123 ------------ |
| 124 -- | |
| 125 -- Onto map | |
| 126 -- def X x -> xmap | |
| 127 -- X ---------------------------> Y | |
| 128 -- ymap <- def Y y | |
| 129 -- | |
| 224 | 130 record Onto (X Y : OD ) : Set n where |
| 219 | 131 field |
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132 xmap : Ordinal |
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133 ymap : Ordinal |
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134 xfunc : def (Func X Y) xmap |
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135 yfunc : def (Func Y X) ymap |
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136 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
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137 func←od {X} {Y} {xmap} xfunc ( func←od yfunc y ) ≡ y |
| 230 | 138 |
| 139 open Onto | |
| 140 | |
| 141 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 142 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 143 xmap = xmap1 | |
| 144 ; ymap = zmap | |
| 145 ; xfunc = xfunc1 | |
| 146 ; yfunc = zfunc | |
| 147 ; onto-iso = onto-iso1 | |
| 148 } where | |
| 149 xmap1 : Ordinal | |
| 150 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 151 zmap : Ordinal | |
| 152 zmap = {!!} | |
| 153 xfunc1 : def (Func X Z) xmap1 | |
| 154 xfunc1 = {!!} | |
| 155 zfunc : def (Func Z X) zmap | |
| 156 zfunc = {!!} | |
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157 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func←od xfunc1 ( func←od zfunc z ) ≡ z |
| 230 | 158 onto-iso1 = {!!} |
| 159 | |
| 51 | 160 |
| 224 | 161 record Cardinal (X : OD ) : Set n where |
| 219 | 162 field |
| 224 | 163 cardinal : Ordinal |
| 230 | 164 conto : Onto X (Ord cardinal) |
| 165 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 166 |
| 224 | 167 cardinal : (X : OD ) → Cardinal X |
| 168 cardinal X = record { | |
| 219 | 169 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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170 ; conto = onto |
| 219 | 171 ; cmax = cmax |
| 172 } where | |
| 230 | 173 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 174 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 175 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 176 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 177 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 178 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 179 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 180 lemma1 x prev with trio< x (osuc S) |
| 181 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 182 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 183 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 184 lemma2 : Onto X (Ord x) | |
| 185 lemma2 with prev {!!} {!!} | |
| 186 ... | lift t = t {!!} | |
| 229 | 187 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 188 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 189 onto : Onto X (Ord S) |
| 190 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 191 ... | lift t = t <-osuc | |
| 192 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 193 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 194 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 195 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 196 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 197 lemma | case1 x = refl |
| 198 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 199 |
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200 |
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201 ----- |
| 219 | 202 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 203 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 204 |
| 205 | |
| 206 |