Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 236:650bdad56729
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Fri, 16 Aug 2019 15:53:29 +0900 |
parents | 846e0926bb89 |
children | 521290e85527 |
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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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 |
236 | 30 record SetProduct ( A B : OD ) : 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 |
236 | 46 A ⊗ B = record { def = λ x → SetProduct A B } |
233 | 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 |
236 | 66 |
233 | 67 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
68 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
69 | |
236 | 70 record Func←cd { dom cod : OD } {f : Ordinal } (f<F : def (Func dom cod ) f) : Set n where |
71 field | |
72 func-1 : Ordinal → Ordinal | |
73 func→od∈Func-1 : (Func dom (Ord (sup-o (λ x → func-1 x)) )) ∋ func→od func-1 dom | |
74 | |
75 func←od1 : { dom cod : OD } → {f : Ordinal } → (f<F : def (Func dom cod ) f ) → Func←cd {dom} {cod} {f} f<F | |
76 func←od1 {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = {!!} } where | |
77 lemma : Ordinal → Ordinal → Ordinal | |
78 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) | |
79 lemma x y | p | no n = o∅ | |
80 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) | |
81 ... | t with decp ( x ≡ π1 t ) | |
82 ... | yes _ = π2 t | |
83 ... | no _ = o∅ | |
84 | |
85 func→od∈Func : (f : Ordinal → Ordinal ) ( dom : OD ) → (Func dom (Ord (sup-o (λ x → f x)) )) ∋ func→od f dom | |
86 func→od∈Func f dom = record { proj1 = {!!} ; proj2 = {!!} } | |
225 | 87 |
227 | 88 -- contra position of sup-o< |
89 -- | |
90 | |
235 | 91 -- postulate |
92 -- -- contra-position of mimimulity of supermum required in Cardinal | |
93 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
94 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
227 | 95 |
219 | 96 ------------ |
97 -- | |
98 -- Onto map | |
99 -- def X x -> xmap | |
100 -- X ---------------------------> Y | |
101 -- ymap <- def Y y | |
102 -- | |
224 | 103 record Onto (X Y : OD ) : Set n where |
219 | 104 field |
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105 xmap : Ordinal |
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106 ymap : Ordinal |
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107 xfunc : def (Func X Y) xmap |
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108 yfunc : def (Func Y X) ymap |
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109 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
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110 func←od {X} {Y} {xmap} xfunc ( func←od yfunc y ) ≡ y |
230 | 111 |
112 open Onto | |
113 | |
114 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
115 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
116 xmap = xmap1 | |
117 ; ymap = zmap | |
118 ; xfunc = xfunc1 | |
119 ; yfunc = zfunc | |
120 ; onto-iso = onto-iso1 | |
121 } where | |
122 xmap1 : Ordinal | |
123 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
124 zmap : Ordinal | |
125 zmap = {!!} | |
126 xfunc1 : def (Func X Z) xmap1 | |
127 xfunc1 = {!!} | |
128 zfunc : def (Func Z X) zmap | |
129 zfunc = {!!} | |
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130 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func←od xfunc1 ( func←od zfunc z ) ≡ z |
230 | 131 onto-iso1 = {!!} |
132 | |
51 | 133 |
224 | 134 record Cardinal (X : OD ) : Set n where |
219 | 135 field |
224 | 136 cardinal : Ordinal |
230 | 137 conto : Onto X (Ord cardinal) |
138 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
151 | 139 |
224 | 140 cardinal : (X : OD ) → Cardinal X |
141 cardinal X = record { | |
219 | 142 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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143 ; conto = onto |
219 | 144 ; cmax = cmax |
145 } where | |
230 | 146 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
147 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
148 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
219 | 149 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
229 | 150 S = sup-o (λ x → proj1 (cardinal-p x)) |
230 | 151 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
152 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
229 | 153 lemma1 x prev with trio< x (osuc S) |
154 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
230 | 155 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
156 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
157 lemma2 : Onto X (Ord x) | |
158 lemma2 with prev {!!} {!!} | |
159 ... | lift t = t {!!} | |
229 | 160 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
161 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
230 | 162 onto : Onto X (Ord S) |
163 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
164 ... | lift t = t <-osuc | |
165 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
229 | 166 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
224 | 167 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
219 | 168 lemma : proj1 (cardinal-p y) ≡ y |
230 | 169 lemma with p∨¬p ( Onto X (Ord y) ) |
219 | 170 lemma | case1 x = refl |
171 lemma | case2 not = ⊥-elim ( not ontoy ) | |
217 | 172 |
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173 |
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174 ----- |
219 | 175 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
176 -- Power ω is larger than ℵ0, so it has no cardinal. | |
218 | 177 |
178 | |
179 |