Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison generic-filter.agda @ 388:19687f3304c9

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 25 Jul 2020 12:54:28 +0900
parents 8b0715e28b33
children cb183674facf
comparison
equal deleted inserted replaced
387:8b0715e28b33 388:19687f3304c9
58 record _f⊆_ (f g : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where 58 record _f⊆_ (f g : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where
59 field 59 field
60 extend : {x : Nat} → (fr : dom f (lift x) ) → dom g (lift x ) 60 extend : {x : Nat} → (fr : dom f (lift x) ) → dom g (lift x )
61 feq : {x : Nat} → {fr : dom f (lift x) } → pmap f (lift x) fr ≡ pmap g (lift x) (extend fr) 61 feq : {x : Nat} → {fr : dom f (lift x) } → pmap f (lift x) fr ≡ pmap g (lift x) (extend fr)
62 62
63 record Gf (f : Nat → Two) (p : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where
64 field
65 gn : Nat
66 f<n : (f ↑ gn) f⊆ p
67
68 record FiniteF (p : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where 63 record FiniteF (p : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where
69 field 64 field
70 f-max : Nat 65 f-max : Nat
71 f-func : Nat → Two 66 f-func : Nat → Two
72 f-p⊆f : p f⊆ (f-func ↑ f-max) 67 f-p⊆f : p f⊆ (f-func ↑ f-max)
82 -- ; dense-f = λ x → record { dom = {!!} ; pmap = {!!} } 77 -- ; dense-f = λ x → record { dom = {!!} ; pmap = {!!} }
83 -- ; dense-d = λ {p} d → {!!} 78 -- ; dense-d = λ {p} d → {!!}
84 -- ; dense-p = λ {p} d → {!!} 79 -- ; dense-p = λ {p} d → {!!}
85 -- } 80 -- }
86 81
87 open Gf
88 open _f⊆_ 82 open _f⊆_
89 open import Data.Nat.Properties 83 open import Data.Nat.Properties
90
91 GF : (Nat → Two ) → F-Filter (PFunc (Lift n Nat) (Lift n Two)) (λ x → Lift (suc n) (One {n}) ) _f⊆_ _f∩_
92 GF f = record {
93 filter = λ p → Gf f p
94 ; f⊆P = lift OneObj
95 ; filter1 = λ {p} {q} _ fp p⊆q → record { gn = gn fp ; f<n = f1 fp p⊆q }
96 ; filter2 = λ {p} {q} fp fq → record { gn = min (gn fp) (gn fq) ; f<n = f2 fp fq }
97 } where
98 f1 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → (f ↑ gn fp) f⊆ q
99 f1 {p} {q} fp p⊆q = record { extend = λ {x} x<g → extend p⊆q (extend (f<n fp ) x<g) ; feq = λ {x} {fr} → lemma {x} {fr} } where
100 lemma : {x : Nat} {fr : Lift n (x ≤ gn fp)} → pmap (f ↑ gn fp) (lift x) fr ≡ pmap q (lift x) (extend p⊆q (extend (f<n fp) fr))
101 lemma {x} {fr} = begin
102 pmap (f ↑ gn fp) (lift x) fr
103 ≡⟨ feq (f<n fp ) ⟩
104 pmap p (lift x) (extend (f<n fp) fr)
105 ≡⟨ feq p⊆q ⟩
106 pmap q (lift x) (extend p⊆q (extend (f<n fp) fr))
107 ∎ where open ≡-Reasoning
108 f2 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf f p ) → (fq : Gf f q ) → (f ↑ (min (gn fp) (gn fq))) f⊆ (p f∩ q)
109 f2 {p} {q} fp fq = record { extend = λ {x} x<g → lemma2 x<g ; feq = λ {x} {fr} → lemma3 fr } where
110 fmin : PFunc (Lift n Nat) (Lift n Two)
111 fmin = f ↑ (min (gn fp) (gn fq))
112 lemma1 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → (fr : dom p (lift x)) (gr : dom q (lift x)) → pmap p (lift x) fr ≡ pmap q (lift x) gr
113 lemma1 {x} x<g fr gr = begin
114 pmap p (lift x) fr
115 ≡⟨ meq p ⟩
116 pmap p (lift x) (extend (f<n fp) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))))
117 ≡⟨ sym (feq (f<n fp)) ⟩
118 pmap (f ↑ (min (gn fp) (gn fq))) (lift x) x<g
119 ≡⟨ feq (f<n fq) ⟩
120 pmap q (lift x) (extend (f<n fq) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))))
121 ≡⟨ meq q ⟩
122 pmap q (lift x) gr
123 ∎ where open ≡-Reasoning
124 lemma2 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → dom (p f∩ q) (lift x)
125 lemma2 x<g = record { proj1 = extend (f<n fp ) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))) ;
126 proj2 = record {proj1 = extend (f<n fq ) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))) ; proj2 = lemma1 x<g }}
127 f∩→⊆ : (p q : PFunc (Lift n Nat) (Lift n Two) ) → (p f∩ q ) f⊆ q
128 f∩→⊆ p q = record {
129 extend = λ {x} pq → proj1 (proj2 pq)
130 ; feq = λ {x} {fr} → (proj2 (proj2 fr)) (proj1 fr) (proj1 (proj2 fr))
131 }
132 lemma3 : {x : Nat} → ( fr : Lift n (x ≤ min (gn fp) (gn fq))) → pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr ≡ pmap (p f∩ q) (lift x) (lemma2 fr)
133 lemma3 {x} fr =
134 pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr
135 ≡⟨ feq (f<n fq) ⟩
136 pmap q (lift x) (extend (f<n fq) ( lift (≤-trans (lower fr) (m⊓n≤n _ _)) ))
137 ≡⟨ sym (feq (f∩→⊆ p q ) {x} {lemma2 fr} ) ⟩
138 pmap (p f∩ q) (lift x) (lemma2 fr)
139 ∎ where open ≡-Reasoning
140 84
141 85
142 ODSuc : (y : HOD) → infinite ∋ y → HOD 86 ODSuc : (y : HOD) → infinite ∋ y → HOD
143 ODSuc y lt = Union (y , (y , y)) 87 ODSuc y lt = Union (y , (y , y))
144 88
198 ω→2f x = {!!} 142 ω→2f x = {!!}
199 143
200 ↑n : (f n : HOD) → ((ω→2 ∋ f ) ∧ (infinite ∋ n)) → HOD 144 ↑n : (f n : HOD) → ((ω→2 ∋ f ) ∧ (infinite ∋ n)) → HOD
201 ↑n f n lt = 3→Hω2 ( ω→2f f (proj1 lt) 3↑ (ω→nat n (proj2 lt) )) 145 ↑n f n lt = 3→Hω2 ( ω→2f f (proj1 lt) 3↑ (ω→nat n (proj2 lt) ))
202 146
203 record Gfo (x : Ordinal) : Set n where
204 field
205 gfunc : Ordinal
206 gmax : Ordinal
207 gcond : (odef ω→2 gfunc) ∧ (odef infinite gmax)
208 gfdef : {!!} -- ( ↑n (ord→od gfunc) (ord→od gmax) (subst₂ ? ? ? gcond) ) ⊆ ord→od x
209 pcond : odef HODω2 x
210
211 open Gfo
212
213 HODGf : HOD
214 HODGf = record { od = record { def = λ x → Gfo x } ; odmax = next o∅ ; <odmax = {!!} }
215
216 G : (Nat → Two) → Filter HODω2
217 G f = record {
218 filter = HODGf
219 ; f⊆PL = {!!}
220 ; filter1 = {!!}
221 ; filter2 = {!!}
222 } where
223 filter0 : HOD
224 filter0 = {!!}
225 f⊆PL1 : filter0 ⊆ Power HODω2
226 f⊆PL1 = {!!}
227 filter11 : { p q : HOD } → q ⊆ HODω2 → filter0 ∋ p → p ⊆ q → filter0 ∋ q
228 filter11 = {!!}
229 filter12 : { p q : HOD } → filter0 ∋ p → filter0 ∋ q → filter0 ∋ (p ∩ q)
230 filter12 = {!!}
231
232 -- the set of finite partial functions from ω to 2 147 -- the set of finite partial functions from ω to 2
233 148
234 Hω2f : Set (suc n) 149 Hω2f : Set (suc n)
235 Hω2f = (Nat → Set n) → Two 150 Hω2f = (Nat → Set n) → Two
236 151
242 field 157 field
243 ctl→ : Nat → Ordinal 158 ctl→ : Nat → Ordinal
244 ctl← : Ordinal → Nat 159 ctl← : Ordinal → Nat
245 ctl-iso→ : { x : Ordinal } → ctl→ (ctl← x ) ≡ x 160 ctl-iso→ : { x : Ordinal } → ctl→ (ctl← x ) ≡ x
246 ctl-iso← : { x : Nat } → ctl← (ctl→ x ) ≡ x 161 ctl-iso← : { x : Nat } → ctl← (ctl→ x ) ≡ x
162
163 record CountableHOD : Set (suc (suc n)) where
164 field
165 phod : HOD
166 ptl→ : Nat → Ordinal
167 ptl→∈P : (i : Nat) → odef phod (ptl→ i)
168 ptl← : (x : Ordinal) → odef phod x → Nat
169 ptl-iso→ : { x : Ordinal } → (lt : odef phod x ) → ptl→ (ptl← x lt ) ≡ x
170 ptl-iso← : { x : Nat } → ptl← (ptl→ x ) (ptl→∈P x) ≡ x
171
247 172
248 open CountableOrdinal 173 open CountableOrdinal
249 174 open CountableHOD
250 PGOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Set n 175
251 PGOD i C p q = ¬ ( odef (ord→od (ctl→ C i)) q ∧ ( (x : Ordinal ) → odef (ord→od p) x → odef (ord→od q) x )) 176 PGHOD : (P : CountableHOD ) (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → HOD
252 177 PGHOD P i C p = record { od = record { def = λ x → odef (ord→od (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (ord→od p) y → odef (ord→od x) y ) }
253 PGHOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → HOD 178 ; odmax = ctl→ C i ; <odmax = λ {y} lt → odefo→o< (proj1 lt)}
254 PGHOD i C p = record { od = record { def = λ x → PGOD i C {!!} {!!} } ; odmax = {!!} ; <odmax = {!!} } 179
255 180 next-p : (P : CountableHOD ) (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → Ordinal
256 ord-compare : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Ordinal 181 next-p P i C p with ODC.decp O ( PGHOD P i C p =h= od∅ )
257 ord-compare i C p q with ODC.p∨¬p O ( (q : Ordinal ) → PGOD i C p q ) 182 next-p P i C p | yes y = p
258 ord-compare i C p q | case1 y = p 183 next-p P i C p | no not = od→ord (ODC.minimal O (PGHOD P i C p ) not)
259 ord-compare i C p q | case2 n = od→ord (ODC.minimal O (PGHOD i C p ) (∅< (subst₂ (λ j k → odef j {!!} ) refl {!!} n)) ) 184
260 185 data PD (P : CountableHOD ) (C : CountableOrdinal) : (x : Ordinal) (i : Nat) → Set n where
261 data PD (P : HOD) (C : CountableOrdinal) : (x : Ordinal) (i : Nat) → Set (suc n) where
262 pd0 : PD P C o∅ 0 186 pd0 : PD P C o∅ 0
263 -- pdq : {q pnx : Ordinal } {n : Nat} → (pn : PD P C pnx n ) → odef (ctl→ C n) q → ord→od p0x ⊆ ord→od q → PD P C q (suc n) 187 pdsuc : {p : Ordinal } {i : Nat} → PD P C p i → PD P C (next-p P i C p) (Suc i)
264 188
265 P-GenericFilter : {P : HOD} → (C : CountableOrdinal) → GenericFilter P 189 record PDN (P : CountableHOD ) (C : CountableOrdinal) (x : Ordinal) : Set n where
190 field
191 px∈ω : odef (phod P) x
192 pdod : PD P C x (ptl← P x px∈ω)
193
194 open PDN
195
196 PDHOD : (P : CountableHOD ) → (C : CountableOrdinal) → HOD
197 PDHOD P C = record { od = record { def = λ x → PDN P C x }
198 ; odmax = odmax (phod P) ; <odmax = λ {y} lt → <odmax (phod P) (px∈ω lt) } where
199
200 --
201 -- p 0 ≡ ∅
202 -- p (suc n) = if ∃ q ∈ ord→od ( ctl→ n ) ∧ p n ⊆ q → q
203 --- else p n
204
205 Gω2r : (x : Ordinal) → (lt : infinite ∋ ord→od x ) → Hω2 (ω→nat (ord→od x) lt ) x
206 Gω2r x = subst (λ k → (lt : odef infinite (od→ord (ord→od x))) → Hω2 (ω→nat (ord→od x) lt ) k ) diso ( ε-induction {ψ} {!!} (ord→od x)) where
207 ψ : HOD → Set n
208 ψ y = (lt : odef infinite (od→ord y) ) → Hω2 (ω→nat y lt ) (od→ord y )
209 ind : {x : HOD} → ({y : HOD} → OD.def (od x) (od→ord y) →
210 (lt : infinite-d (od→ord y)) → Hω2 (ω→nat y lt) (od→ord y)) →
211 (lt : infinite-d (od→ord x)) → Hω2 (ω→nat x lt) (od→ord x)
212 ind {x} prev lt with ω→nat x lt
213 ... | Zero = subst (λ k → Hω2 Zero k) ? hφ
214 ... | Suc t = {!!}
215
216 P-GenericFilter : {P : CountableHOD } → (C : CountableOrdinal) → GenericFilter (phod P)
266 P-GenericFilter {P} C = record { 217 P-GenericFilter {P} C = record {
267 genf = record { filter = {!!} ; f⊆PL = {!!} ; filter1 = {!!} ; filter2 = {!!} } 218 genf = record { filter = PDHOD P C ; f⊆PL = {!!} ; filter1 = {!!} ; filter2 = {!!} }
268 ; generic = λ D → {!!} 219 ; generic = λ D → {!!}
269 } 220 }