Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 247:d09437fcfc7c
fix pair
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 26 Aug 2019 12:27:20 +0900 |
parents | 3506f53c7d83 |
children | 9fd85b954477 |
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 |
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 |
243 | 56 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
57 | |
58 eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' | |
59 eq-pair refl refl = HE.refl | |
60 | |
61 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > | |
62 eq-prod refl refl = refl | |
63 | |
247 | 64 -- prod-eq : { x x' y y' : OD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) |
65 -- prod-eq {x} {x'} {y} {y'} eq = {!!} where | |
66 -- lemma : < x , y > ≡ < x , y' > → y ≡ y' | |
67 -- lemma eq1 with trio< (od→ord x) (od→ord y) | |
68 -- lemma eq1 | tri< a ¬b ¬c = {!!} | |
69 -- lemma eq1 | tri≈ ¬a b ¬c = {!!} | |
70 -- lemma eq1 | tri> ¬a ¬b c = {!!} | |
71 | |
245 | 72 postulate |
73 def-eq : { P Q p q : OD } → P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt | |
244 | 74 |
247 | 75 ∈-to-ord : {p : Ordinal } → ( ZFProduct ∋ ord→od p ) → ord-pair p |
76 ∈-to-ord {p} lt = def-subst {ZFProduct} {(od→ord (ord→od p))} {_} {_} lt refl diso | |
77 | |
78 ord-to-∈ : {p : Ordinal } → ord-pair p → ZFProduct ∋ ord→od p | |
79 ord-to-∈ {p} lt = def-subst {_} {_} {ZFProduct} {(od→ord (ord→od p))} lt refl (sym diso) | |
80 | |
81 lemma333 : { A a : OD } → { x : A ∋ a } → def-subst {A} {od→ord a} (def-subst {A} {od→ord a} x refl refl ) refl refl ≡ x | |
82 lemma333 = refl | |
83 | |
84 lemma334 : { A B : OD } → {a b : Ordinal} → { x : A ∋ ord→od a } → { y : B ∋ ord→od b } → (f1 : A ≡ B) → (f2 : a ≡ b) | |
85 → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) refl refl ≅ x | |
86 lemma334 {A} {A} {a} {a} {x} {y} refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y | |
87 ... | HE.refl = HE.refl | |
88 | |
89 lemma335 : { A B C : OD } → {a b c : Ordinal} → { x : A ∋ ord→od a } → { y : C ∋ ord→od c } → (f1 : A ≡ B) → (f2 : a ≡ b) → (g1 : B ≡ C) → (g2 : b ≡ c) | |
90 → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) g1 (cong (λ k → od→ord (ord→od k)) g2 ) | |
91 ≅ def-subst {A} { od→ord (ord→od a)} {C } { od→ord (ord→od c)} x (trans f1 g1) | |
92 (trans (cong (λ k → od→ord (ord→od k)) f2 ) (cong (λ k → od→ord (ord→od k)) g2 )) | |
93 lemma335 {A} {A} {A} {a} {a} {a} {x} {y} refl refl refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y | |
94 ... | HE.refl = HE.refl | |
95 | |
96 ∈-to-ord-oiso : { p : Ordinal } → { x : ord-pair p } → ∈-to-ord (ord-to-∈ x) ≡ x | |
97 ∈-to-ord-oiso {p} {x} = {!!} where | |
98 lemma : def-subst {_} {_} {ZFProduct} {{!!}} (def-subst {_} {_} {ZFProduct} {{!!}} x refl (sym diso)) refl diso ≡ x | |
99 lemma = {!!} | |
100 | |
244 | 101 lemma34 : { p q : Ordinal } → (x : ord-pair p ) → (y : ord-pair q ) → p ≡ q → x ≅ y |
247 | 102 lemma34 {p} {q} x y refl = subst₂ (λ j k → j ≅ k) ∈-to-ord-oiso ∈-to-ord-oiso (HE.cong (λ k → ∈-to-ord k) lemma1 ) where |
245 | 103 lemma : (pt : ZFProduct ∋ ord→od p ) → (qt : ZFProduct ∋ ord→od q ) → p ≡ q → pt ≅ qt |
247 | 104 lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl (cong (λ k → ord→od k) eq) pt qt |
105 lemma1 : (ord-to-∈ x) ≅ (ord-to-∈ y ) | |
106 lemma1 = lemma (ord-to-∈ x) (ord-to-∈ y ) refl | |
244 | 107 |
243 | 108 π1-cong : { p q : OD } → p ≡ q → (pt : ZFProduct ∋ p ) → (qt : ZFProduct ∋ q ) → π1 pt ≅ π1 qt |
245 | 109 π1-cong {p} {q} refl s t = HE.cong (λ k → pi1 k ) (def-eq {ZFProduct} {ZFProduct} refl refl s t ) |
242 | 110 |
246 | 111 π1--iso : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≅ od→ord x |
112 π1--iso {x} {y} p = lemma (od→ord x) (od→ord y) {!!} {!!} refl where | |
244 | 113 lemma1 : ( ox oy op : Ordinal ) → (p : ord-pair op) → op ≡ od→ord ( < ord→od ox , ord→od oy >) → p ≅ pair ox oy |
245 | 114 lemma1 ox oy op (pair x' y') eq = lemma34 {!!} {!!} {!!} |
246 | 115 lemma : ( ox oy op : Ordinal ) → (p : ord-pair op ) → op ≡ od→ord ( < ord→od ox , ord→od oy > ) → pi1 p ≅ ox |
116 lemma ox oy op p eq = {!!} -- HE.cong (λ k → pi1 k ) (lemma1 ox oy op p eq ) | |
244 | 117 |
242 | 118 p-iso : { x : OD } → {p : ZFProduct ∋ x } → < ord→od (π1 p) , ord→od (π2 p) > ≡ x |
243 | 119 p-iso {x} {p} with p-cons (ord→od (π1 p)) (ord→od (π2 p)) |
120 ... | t = {!!} | |
242 | 121 |
238 | 122 |
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123 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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124 ∋-p A x with p∨¬p ( A ∋ x ) |
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125 ∋-p A x | case1 t = yes t |
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126 ∋-p A x | case2 t = no t |
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127 |
233 | 128 _⊗_ : (A B : OD) → OD |
239 | 129 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where |
130 checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
131 checkAB (pair x y) = def A x ∧ def B y | |
233 | 132 |
242 | 133 func→od0 : (f : Ordinal → Ordinal ) → OD |
134 func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where | |
135 checkfunc : { p : Ordinal } → def ZFProduct p → Set n | |
136 checkfunc (pair x y) = f x ≡ y | |
137 | |
233 | 138 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
225 | 139 |
233 | 140 Func : ( A B : OD ) → OD |
141 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
142 | |
143 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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144 |
236 | 145 |
233 | 146 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
147 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
148 | |
242 | 149 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where |
236 | 150 field |
151 func-1 : Ordinal → Ordinal | |
242 | 152 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
236 | 153 |
242 | 154 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} |
240 | 155 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 | 156 lemma : Ordinal → Ordinal → Ordinal |
157 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) | |
158 lemma x y | p | no n = o∅ | |
240 | 159 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) |
160 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
161 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
162 lemma2 (pair x1 y1) | yes p = y1 | |
163 lemma2 (pair x1 y1) | no ¬p = o∅ | |
242 | 164 fod : OD |
165 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) | |
240 | 166 |
167 | |
168 open Func←cd | |
236 | 169 |
227 | 170 -- contra position of sup-o< |
171 -- | |
172 | |
235 | 173 -- postulate |
174 -- -- contra-position of mimimulity of supermum required in Cardinal | |
175 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
176 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
227 | 177 |
219 | 178 ------------ |
179 -- | |
180 -- Onto map | |
181 -- def X x -> xmap | |
182 -- X ---------------------------> Y | |
183 -- ymap <- def Y y | |
184 -- | |
224 | 185 record Onto (X Y : OD ) : Set n where |
219 | 186 field |
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187 xmap : Ordinal |
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188 ymap : Ordinal |
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189 xfunc : def (Func X Y) xmap |
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190 yfunc : def (Func Y X) ymap |
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191 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
240 | 192 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
230 | 193 |
194 open Onto | |
195 | |
196 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
197 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
198 xmap = xmap1 | |
199 ; ymap = zmap | |
200 ; xfunc = xfunc1 | |
201 ; yfunc = zfunc | |
202 ; onto-iso = onto-iso1 | |
203 } where | |
204 xmap1 : Ordinal | |
205 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
206 zmap : Ordinal | |
207 zmap = {!!} | |
208 xfunc1 : def (Func X Z) xmap1 | |
209 xfunc1 = {!!} | |
210 zfunc : def (Func Z X) zmap | |
211 zfunc = {!!} | |
240 | 212 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
230 | 213 onto-iso1 = {!!} |
214 | |
51 | 215 |
224 | 216 record Cardinal (X : OD ) : Set n where |
219 | 217 field |
224 | 218 cardinal : Ordinal |
230 | 219 conto : Onto X (Ord cardinal) |
220 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
151 | 221 |
224 | 222 cardinal : (X : OD ) → Cardinal X |
223 cardinal X = record { | |
219 | 224 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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225 ; conto = onto |
219 | 226 ; cmax = cmax |
227 } where | |
230 | 228 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
229 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
230 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
219 | 231 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
229 | 232 S = sup-o (λ x → proj1 (cardinal-p x)) |
230 | 233 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
234 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
229 | 235 lemma1 x prev with trio< x (osuc S) |
236 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
230 | 237 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
238 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
239 lemma2 : Onto X (Ord x) | |
240 lemma2 with prev {!!} {!!} | |
241 ... | lift t = t {!!} | |
229 | 242 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
243 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
230 | 244 onto : Onto X (Ord S) |
245 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
246 ... | lift t = t <-osuc | |
247 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
229 | 248 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
224 | 249 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
219 | 250 lemma : proj1 (cardinal-p y) ≡ y |
230 | 251 lemma with p∨¬p ( Onto X (Ord y) ) |
219 | 252 lemma | case1 x = refl |
253 lemma | case2 not = ⊥-elim ( not ontoy ) | |
217 | 254 |
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255 |
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256 ----- |
219 | 257 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
258 -- Power ω is larger than ℵ0, so it has no cardinal. | |
218 | 259 |
260 | |
261 |