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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 23 Dec 2022 12:54:05 +0900
parents 5acf6483a9e3
children 7ce2cc622c92
comparison
equal deleted inserted replaced
1095:08b6aa6870d9 1096:55ab5de1ae02
123 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y 123 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y
124 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt 124 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt
125 125
126 open import Data.Nat.Properties 126 open import Data.Nat.Properties
127 open import nat 127 open import nat
128 open _⊆_
129 128
130 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p))) 129 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p)))
131 p-monotonic1 L p C {n} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) 130 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p))))
132 ... | yes y = refl-⊆ 131 ... | yes y = refl-⊆ {* (find-p L C n (& p))}
133 ... | no not = record { incl = λ {x} lt → proj2 (proj2 fmin∈PGHOD) (& x) lt } where 132 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where
134 fmin : HOD 133 fmin : HOD
135 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) 134 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq))
136 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin 135 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin
137 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) 136 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq))
138 137
139 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : Nat} → n ≤ m → (* (find-p L C m (& p))) ⊆ (* (find-p L C n (& p))) 138 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : Nat} → n ≤ m → (* (find-p L C m (& p))) ⊆ (* (find-p L C n (& p)))
140 p-monotonic L p C {Zero} {Zero} n≤m = refl-⊆ 139 p-monotonic L p C {Zero} {Zero} n≤m = refl-⊆ {* (find-p L C Zero (& p))}
141 p-monotonic L p C {Zero} {Suc m} z≤n = trans-⊆ (p-monotonic1 L p C {m} ) (p-monotonic L p C {Zero} {m} z≤n ) 140 p-monotonic L p C {Zero} {Suc m} z≤n lt = (p-monotonic L p C {Zero} {m} z≤n ) (p-monotonic1 L p C {m} lt )
142 p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m 141 p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m
143 ... | tri< a ¬b ¬c = trans-⊆ (p-monotonic1 L p C {m}) (p-monotonic L p C {Suc n} {m} a) 142 ... | tri< a ¬b ¬c = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt )
144 ... | tri≈ ¬a refl ¬c = refl-⊆ 143 ... | tri≈ ¬a refl ¬c = λ x → x
145 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) 144 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c )
146 145
147 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP ( ctl-M C ) 146 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C )
148 P-GenericFilter P L p0 L⊆PP Lp0 C = record { 147 P-GenericFilter P L p0 L⊆PP Lp0 C = record {
149 genf = record { filter = PDHOD L p0 C ; f⊆L = f⊆PL ; filter1 = λ L∋q PD∋p p⊆q → f1 L∋q PD∋p p⊆q ; filter2 = f2 } 148 genf = record { filter = PDHOD L p0 C ; f⊆L = f⊆PL ; filter1 = λ L∋q PD∋p p⊆q → f1 L∋q PD∋p p⊆q ; filter2 = f2 }
150 ; generic = fdense 149 ; generic = fdense
151 } where 150 } where
152 f⊆PL : PDHOD L p0 C ⊆ L 151 f⊆PL : PDHOD L p0 C ⊆ L
153 f⊆PL = record { incl = λ {x} lt → x∈PP lt } 152 f⊆PL lt = x∈PP lt
154 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q 153 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q
155 f1 {p} {q} L∋q PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where 154 f1 {p} {q} L∋q PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where
156 f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y 155 f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y
157 f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (incl p⊆q (subst₂ (λ j k → odef k j ) (sym &iso) *iso ( pn<gr PD∋p y lt1 ))) 156 f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (p⊆q (subst₂ (λ j k → odef k j ) (sym &iso) *iso ( pn<gr PD∋p y lt1 )))
158 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y 157 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y
159 f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q) 158 f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q)
160 f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p) 159 f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p)
161 ... | tri< a ¬b ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt ) ; x∈PP = L∋pq } where 160 ... | tri< a ¬b ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt ) ; x∈PP = L∋pq } where
162 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y 161 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
163 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y (f5 lt)) ⟫ where 162 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y (f5 lt)) ⟫ where
164 f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y 163 f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y
165 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) &iso ( incl (p-monotonic L p0 C {gr PD∋q} {gr PD∋p} (<to≤ a)) 164 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) &iso ( (p-monotonic L p0 C {gr PD∋q} {gr PD∋p} (<to≤ a))
166 (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) 165 (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) )
167 ... | tri≈ ¬a refl ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f4 y lt) ; x∈PP = L∋pq } where 166 ... | tri≈ ¬a refl ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f4 y lt) ; x∈PP = L∋pq } where
168 f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y 167 f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
169 f4 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ 168 f4 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫
170 ... | tri> ¬a ¬b c = record { gr = gr PD∋q ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt) ; x∈PP = L∋pq } where 169 ... | tri> ¬a ¬b c = record { gr = gr PD∋q ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt) ; x∈PP = L∋pq } where
171 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y 170 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y
172 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y (f5 lt)), subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ where 171 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y (f5 lt)), subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ where
173 f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y 172 f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y
174 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) &iso ( incl (p-monotonic L p0 C {gr PD∋p} {gr PD∋q} (<to≤ c)) 173 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) &iso ( (p-monotonic L p0 C {gr PD∋p} {gr PD∋q} (<to≤ c))
175 (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) 174 (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) )
176 fdense : (D : Dense L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅ 175 fdense : (D : Dense L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅
177 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where 176 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where
178 open Dense 177 open Dense
179 fd09 : (i : Nat ) → odef L (find-p L C i (& p0)) 178 fd09 : (i : Nat ) → odef L (find-p L C i (& p0))
202 fd12 : L ∋ * (find-p L C an (& p0)) 201 fd12 : L ∋ * (find-p L C an (& p0))
203 fd12 = subst (λ k → odef L k) (sym &iso) (fd09 an ) 202 fd12 = subst (λ k → odef L k) (sym &iso) (fd09 an )
204 fd11 : Ordinal 203 fd11 : Ordinal
205 fd11 = & ( dense-f D fd12 ) 204 fd11 = & ( dense-f D fd12 )
206 fd13 : L ∋ ( dense-f D fd12 ) 205 fd13 : L ∋ ( dense-f D fd12 )
207 fd13 = incl (d⊆P D) ( dense-d D fd12 ) 206 fd13 = (d⊆P D) ( dense-d D fd12 )
208 fd14 : (* (ctl→ C an)) ∋ ( dense-f D fd12 ) 207 fd14 : (* (ctl→ C an)) ∋ ( dense-f D fd12 )
209 fd14 = subst (λ k → odef k (& ( dense-f D fd12 ) )) d=an ( dense-d D fd12 ) 208 fd14 = subst (λ k → odef k (& ( dense-f D fd12 ) )) d=an ( dense-d D fd12 )
210 fd15 : (y : Ordinal) → odef (* (& (dense-f D fd12))) y → odef (* (find-p L C an (& p0))) y 209 fd15 : (y : Ordinal) → odef (* (& (dense-f D fd12))) y → odef (* (find-p L C an (& p0))) y
211 fd15 y lt = subst (λ k → odef (* (find-p L C an (& p0))) k ) &iso ( incl (dense-p D fd12 ) fd16 ) where 210 fd15 y lt = subst (λ k → odef (* (find-p L C an (& p0))) k ) &iso ( (dense-p D fd12 ) fd16 ) where
212 fd16 : odef (dense-f D fd12) (& ( * y)) 211 fd16 : odef (dense-f D fd12) (& ( * y))
213 fd16 = subst₂ (λ j k → odef j k ) (*iso) (sym &iso) lt 212 fd16 = subst₂ (λ j k → odef j k ) (*iso) (sym &iso) lt
214 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ 213 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅
215 fd10 = ≡o∅→=od∅ y 214 fd10 = ≡o∅→=od∅ y
216 ... | no not = fd27 where 215 ... | no not = fd27 where
241 240
242 -- 241 --
243 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } 242 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > }
244 -- 243 --
245 244
246 record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter LP (ctl-M C) ) : Set (suc n) where 245 record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (suc n) where
247 field 246 field
248 valx : HOD 247 valx : HOD
249 248
250 record valS (ox oy oG : Ordinal) : Set n where 249 record valS (ox oy oG : Ordinal) : Set n where
251 field 250 field
252 op : Ordinal 251 op : Ordinal
253 p∈G : odef (* oG) op 252 p∈G : odef (* oG) op
254 is-val : odef (* ox) ( & < * oy , * op > ) 253 is-val : odef (* ox) ( & < * oy , * op > )
255 254
256 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} 255 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P}
257 → (G : GenericFilter LP {!!} ) 256 → (G : GenericFilter {L} {P} LP {!!} )
258 → HOD 257 → HOD
259 val x G = TransFinite {λ x → HOD } ind (& x) where 258 val x G = TransFinite {λ x → HOD } ind (& x) where
260 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD 259 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD
261 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } 260 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} }
262 261