Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 242:c10048d69614
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 25 Aug 2019 18:44:41 +0900 |
parents | ccc84f289c98 |
children | f97a2e4df451 |
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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separate logic and nat
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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 |
238 | 30 data ord-pair : (p : Ordinal) → Set n where |
31 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) | |
32 | |
33 ZFProduct : OD | |
34 ZFProduct = record { def = λ x → ord-pair x } | |
35 | |
242 | 36 pi1 : { p : Ordinal } → ord-pair p → Ordinal |
37 pi1 ( pair x y ) = x | |
38 | |
239 | 39 π1 : { p : OD } → ZFProduct ∋ p → Ordinal |
242 | 40 π1 lt = pi1 lt |
41 | |
42 pi2 : { p : Ordinal } → ord-pair p → Ordinal | |
43 pi2 ( pair x y ) = y | |
237 | 44 |
239 | 45 π2 : { p : OD } → ZFProduct ∋ p → Ordinal |
242 | 46 π2 lt = pi2 lt |
237 | 47 |
242 | 48 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > |
49 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( | |
238 | 50 let open ≡-Reasoning in begin |
51 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > | |
52 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ | |
53 od→ord < x , y > | |
54 ∎ ) | |
242 | 55 |
56 π1-iso : { x y : OD } → π1 ( p-cons x y ) ≡ od→ord x | |
57 π1-iso {x} {y} = {!!} where | |
58 lemma : {ox oy : Ordinal} → pi1 ( pair ox oy ) ≡ ox | |
59 lemma = refl | |
60 lemma2 : {ox oy : Ordinal} → | |
61 def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl) ≡ pair ox oy | |
62 lemma2 {ox} {oy} = let open ≡-Reasoning in begin | |
63 def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl) | |
64 ≡⟨ ? ⟩ | |
65 pair ox oy | |
66 ∎ | |
67 lemma1 : {ox oy : Ordinal} → π1 ( p-cons (ord→od ox) (ord→od oy) ) ≡ ox | |
68 lemma1 {ox} {oy} = let open ≡-Reasoning in begin | |
69 π1 ( p-cons (ord→od ox) (ord→od oy) ) | |
70 ≡⟨⟩ | |
71 pi1 (pair (pi1 (def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl))) oy ) | |
72 ≡⟨ cong (λ k → pi1 k) lemma2 ⟩ | |
73 pi1 (pair ox oy ) | |
74 ≡⟨ lemma {ox} {oy} ⟩ | |
75 ox | |
76 ∎ | |
77 | |
78 p-iso : { x : OD } → {p : ZFProduct ∋ x } → < ord→od (π1 p) , ord→od (π2 p) > ≡ x | |
79 p-iso {x} {p} = pi p pc where | |
80 pc : ZFProduct ∋ < ord→od (π1 p) , ord→od (π2 p) > | |
81 pc = p-cons (ord→od (π1 p)) (ord→od (π2 p)) | |
82 pi : { prod prod1 : Ordinal } → ord-pair prod → ord-pair prod1 → {!!} | |
83 pi (pair p1 p2) (pair q1 q2) = {!!} | |
84 | |
238 | 85 |
86 | |
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87 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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88 ∋-p A x with p∨¬p ( A ∋ x ) |
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89 ∋-p A x | case1 t = yes t |
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90 ∋-p A x | case2 t = no t |
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91 |
233 | 92 _⊗_ : (A B : OD) → OD |
239 | 93 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where |
94 checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
95 checkAB (pair x y) = def A x ∧ def B y | |
233 | 96 |
242 | 97 func→od0 : (f : Ordinal → Ordinal ) → OD |
98 func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where | |
99 checkfunc : { p : Ordinal } → def ZFProduct p → Set n | |
100 checkfunc (pair x y) = f x ≡ y | |
101 | |
233 | 102 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
225 | 103 |
233 | 104 Func : ( A B : OD ) → OD |
105 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
106 | |
107 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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108 |
236 | 109 |
233 | 110 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
111 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
112 | |
242 | 113 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where |
236 | 114 field |
115 func-1 : Ordinal → Ordinal | |
242 | 116 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
236 | 117 |
242 | 118 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} |
240 | 119 od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where |
236 | 120 lemma : Ordinal → Ordinal → Ordinal |
121 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) | |
122 lemma x y | p | no n = o∅ | |
240 | 123 lemma x y | p | yes f∋y = lemma2 (proj1 (double-neg-eilm ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) |
124 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
125 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
126 lemma2 (pair x1 y1) | yes p = y1 | |
127 lemma2 (pair x1 y1) | no ¬p = o∅ | |
242 | 128 fod : OD |
129 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) | |
240 | 130 |
131 | |
132 open Func←cd | |
236 | 133 |
227 | 134 -- contra position of sup-o< |
135 -- | |
136 | |
235 | 137 -- postulate |
138 -- -- contra-position of mimimulity of supermum required in Cardinal | |
139 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
140 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
227 | 141 |
219 | 142 ------------ |
143 -- | |
144 -- Onto map | |
145 -- def X x -> xmap | |
146 -- X ---------------------------> Y | |
147 -- ymap <- def Y y | |
148 -- | |
224 | 149 record Onto (X Y : OD ) : Set n where |
219 | 150 field |
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151 xmap : Ordinal |
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152 ymap : Ordinal |
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153 xfunc : def (Func X Y) xmap |
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154 yfunc : def (Func Y X) ymap |
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155 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
240 | 156 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
230 | 157 |
158 open Onto | |
159 | |
160 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
161 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
162 xmap = xmap1 | |
163 ; ymap = zmap | |
164 ; xfunc = xfunc1 | |
165 ; yfunc = zfunc | |
166 ; onto-iso = onto-iso1 | |
167 } where | |
168 xmap1 : Ordinal | |
169 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
170 zmap : Ordinal | |
171 zmap = {!!} | |
172 xfunc1 : def (Func X Z) xmap1 | |
173 xfunc1 = {!!} | |
174 zfunc : def (Func Z X) zmap | |
175 zfunc = {!!} | |
240 | 176 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
230 | 177 onto-iso1 = {!!} |
178 | |
51 | 179 |
224 | 180 record Cardinal (X : OD ) : Set n where |
219 | 181 field |
224 | 182 cardinal : Ordinal |
230 | 183 conto : Onto X (Ord cardinal) |
184 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
151 | 185 |
224 | 186 cardinal : (X : OD ) → Cardinal X |
187 cardinal X = record { | |
219 | 188 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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189 ; conto = onto |
219 | 190 ; cmax = cmax |
191 } where | |
230 | 192 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
193 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
194 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
219 | 195 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
229 | 196 S = sup-o (λ x → proj1 (cardinal-p x)) |
230 | 197 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
198 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
229 | 199 lemma1 x prev with trio< x (osuc S) |
200 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
230 | 201 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
202 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
203 lemma2 : Onto X (Ord x) | |
204 lemma2 with prev {!!} {!!} | |
205 ... | lift t = t {!!} | |
229 | 206 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
207 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
230 | 208 onto : Onto X (Ord S) |
209 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
210 ... | lift t = t <-osuc | |
211 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
229 | 212 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
224 | 213 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
219 | 214 lemma : proj1 (cardinal-p y) ≡ y |
230 | 215 lemma with p∨¬p ( Onto X (Ord y) ) |
219 | 216 lemma | case1 x = refl |
217 lemma | case2 not = ⊥-elim ( not ontoy ) | |
217 | 218 |
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219 |
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220 ----- |
219 | 221 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
222 -- Power ω is larger than ℵ0, so it has no cardinal. | |
218 | 223 |
224 | |
225 |