Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 235:846e0926bb89
fix cardinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Thu, 15 Aug 2019 04:51:24 +0900 |
| parents | e06b76e5b682 |
| children | 650bdad56729 |
| 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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| 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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| 233 | 30 record SetProduct ( A B : OD ) (x : Ordinal ) : Set n where |
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31 field |
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32 π1 : Ordinal |
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33 π2 : Ordinal |
| 233 | 34 A∋π1 : def A π1 |
| 35 B∋π2 : def B π2 | |
| 36 -- opair : x ≡ od→ord (Ord ( omax (omax π1 π1) (omax π1 π2) )) -- < π1 , π2 > | |
| 37 | |
| 38 open SetProduct | |
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39 |
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40 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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41 ∋-p A x with p∨¬p ( A ∋ x ) |
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42 ∋-p A x | case1 t = yes t |
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43 ∋-p A x | case2 t = no t |
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44 |
| 233 | 45 _⊗_ : (A B : OD) → OD |
| 46 A ⊗ B = record { def = λ x → SetProduct A B x } | |
| 47 -- A ⊗ B = record { def = λ x → (y z : Ordinal) → def A y ∧ def B z ∧ ( x ≡ od→ord (< ord→od y , ord→od z >) ) } | |
| 48 | |
| 49 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) | |
| 225 | 50 |
| 233 | 51 Func : ( A B : OD ) → OD |
| 52 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 53 | |
| 54 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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55 |
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56 func←od : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → (Ordinal → Ordinal ) |
| 233 | 57 func←od {dom} {cod} {f} lt x = sup-o ( λ y → lemma y ) where |
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58 lemma : Ordinal → Ordinal |
| 235 | 59 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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60 lemma y | p | no n = o∅ |
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61 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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62 ... | t with decp ( x ≡ π1 t ) |
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63 ... | yes _ = π2 t |
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64 ... | no _ = o∅ |
| 233 | 65 |
| 66 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD | |
| 67 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 68 | |
| 225 | 69 |
| 227 | 70 -- contra position of sup-o< |
| 71 -- | |
| 72 | |
| 235 | 73 -- postulate |
| 74 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 75 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 76 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 77 |
| 219 | 78 ------------ |
| 79 -- | |
| 80 -- Onto map | |
| 81 -- def X x -> xmap | |
| 82 -- X ---------------------------> Y | |
| 83 -- ymap <- def Y y | |
| 84 -- | |
| 224 | 85 record Onto (X Y : OD ) : Set n where |
| 219 | 86 field |
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87 xmap : Ordinal |
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88 ymap : Ordinal |
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89 xfunc : def (Func X Y) xmap |
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90 yfunc : def (Func Y X) ymap |
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91 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
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92 func←od {X} {Y} {xmap} xfunc ( func←od yfunc y ) ≡ y |
| 230 | 93 |
| 94 open Onto | |
| 95 | |
| 96 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 97 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 98 xmap = xmap1 | |
| 99 ; ymap = zmap | |
| 100 ; xfunc = xfunc1 | |
| 101 ; yfunc = zfunc | |
| 102 ; onto-iso = onto-iso1 | |
| 103 } where | |
| 104 xmap1 : Ordinal | |
| 105 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 106 zmap : Ordinal | |
| 107 zmap = {!!} | |
| 108 xfunc1 : def (Func X Z) xmap1 | |
| 109 xfunc1 = {!!} | |
| 110 zfunc : def (Func Z X) zmap | |
| 111 zfunc = {!!} | |
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112 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func←od xfunc1 ( func←od zfunc z ) ≡ z |
| 230 | 113 onto-iso1 = {!!} |
| 114 | |
| 51 | 115 |
| 224 | 116 record Cardinal (X : OD ) : Set n where |
| 219 | 117 field |
| 224 | 118 cardinal : Ordinal |
| 230 | 119 conto : Onto X (Ord cardinal) |
| 120 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 121 |
| 224 | 122 cardinal : (X : OD ) → Cardinal X |
| 123 cardinal X = record { | |
| 219 | 124 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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125 ; conto = onto |
| 219 | 126 ; cmax = cmax |
| 127 } where | |
| 230 | 128 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 129 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 130 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 131 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 132 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 133 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 134 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 135 lemma1 x prev with trio< x (osuc S) |
| 136 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 137 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 138 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 139 lemma2 : Onto X (Ord x) | |
| 140 lemma2 with prev {!!} {!!} | |
| 141 ... | lift t = t {!!} | |
| 229 | 142 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 143 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 144 onto : Onto X (Ord S) |
| 145 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 146 ... | lift t = t <-osuc | |
| 147 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 148 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 149 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 150 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 151 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 152 lemma | case1 x = refl |
| 153 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 154 |
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155 |
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156 ----- |
| 219 | 157 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 158 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 159 |
| 160 | |
| 161 |