Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 233:af60c40298a4
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 12 Aug 2019 13:28:59 +0900 |
| parents | 1b1620e2053c |
| children | e06b76e5b682 |
| 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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29 |
| 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 |
| 233 | 40 _⊗_ : (A B : OD) → OD |
| 41 A ⊗ B = record { def = λ x → SetProduct A B x } | |
| 42 -- A ⊗ B = record { def = λ x → (y z : Ordinal) → def A y ∧ def B z ∧ ( x ≡ od→ord (< ord→od y , ord→od z >) ) } | |
| 43 | |
| 44 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) | |
| 225 | 45 |
| 233 | 46 Func : ( A B : OD ) → OD |
| 47 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 48 | |
| 49 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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50 |
| 233 | 51 func←od : { dom cod : OD } → {f : OD } → Func dom cod ∋ f → (Ordinal → Ordinal ) |
| 52 func←od {dom} {cod} {f} lt x = sup-o ( λ y → lemma y ) where | |
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53 lemma : Ordinal → Ordinal |
| 233 | 54 lemma y with IsZF.power→ isZF (dom ⊗ cod) f lt |
| 55 lemma y | p with double-neg-eilm ( p {ord→od y} {!!} ) -- p : {x : OD} → f ∋ x → ¬ ¬ (dom ⊗ cod ∋ x) | |
| 56 ... | t = π2 t | |
| 57 | |
| 58 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD | |
| 59 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 60 | |
| 225 | 61 |
| 227 | 62 -- contra position of sup-o< |
| 63 -- | |
| 64 | |
| 228 | 65 postulate |
| 66 -- contra-position of mimimulity of supermum required in Cardinal | |
| 67 sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 68 sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 69 |
| 219 | 70 ------------ |
| 71 -- | |
| 72 -- Onto map | |
| 73 -- def X x -> xmap | |
| 74 -- X ---------------------------> Y | |
| 75 -- ymap <- def Y y | |
| 76 -- | |
| 224 | 77 record Onto (X Y : OD ) : Set n where |
| 219 | 78 field |
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79 xmap : Ordinal |
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80 ymap : Ordinal |
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81 xfunc : def (Func X Y) xmap |
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82 yfunc : def (Func Y X) ymap |
| 233 | 83 onto-iso : {y : Ordinal } → (lty : def Y y ) → func←od {!!} ( func←od {!!} y ) ≡ y |
| 230 | 84 |
| 85 open Onto | |
| 86 | |
| 87 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 88 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 89 xmap = xmap1 | |
| 90 ; ymap = zmap | |
| 91 ; xfunc = xfunc1 | |
| 92 ; yfunc = zfunc | |
| 93 ; onto-iso = onto-iso1 | |
| 94 } where | |
| 95 xmap1 : Ordinal | |
| 96 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 97 zmap : Ordinal | |
| 98 zmap = {!!} | |
| 99 xfunc1 : def (Func X Z) xmap1 | |
| 100 xfunc1 = {!!} | |
| 101 zfunc : def (Func Z X) zmap | |
| 102 zfunc = {!!} | |
| 233 | 103 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func←od {!!} ( func←od zfunc z ) ≡ z |
| 230 | 104 onto-iso1 = {!!} |
| 105 | |
| 51 | 106 |
| 224 | 107 record Cardinal (X : OD ) : Set n where |
| 219 | 108 field |
| 224 | 109 cardinal : Ordinal |
| 230 | 110 conto : Onto X (Ord cardinal) |
| 111 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 112 |
| 224 | 113 cardinal : (X : OD ) → Cardinal X |
| 114 cardinal X = record { | |
| 219 | 115 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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116 ; conto = onto |
| 219 | 117 ; cmax = cmax |
| 118 } where | |
| 230 | 119 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 120 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 121 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 122 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 123 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 124 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 125 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 126 lemma1 x prev with trio< x (osuc S) |
| 127 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 128 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 129 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 130 lemma2 : Onto X (Ord x) | |
| 131 lemma2 with prev {!!} {!!} | |
| 132 ... | lift t = t {!!} | |
| 229 | 133 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 134 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 135 onto : Onto X (Ord S) |
| 136 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 137 ... | lift t = t <-osuc | |
| 138 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 139 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 140 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 141 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 142 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 143 lemma | case1 x = refl |
| 144 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 145 |
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146 |
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147 ----- |
| 219 | 148 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 149 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 150 |
| 151 | |
| 152 |