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comparison src/generic-filter.agda @ 460:d407cc8499fc

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 19 Mar 2022 12:16:48 +0900
parents 3d84389cc43f
children 0e018784bed3
comparison
equal deleted inserted replaced
459:3d84389cc43f 460:d407cc8499fc
144 ... | tri≈ ¬a refl ¬c = refl-⊆ 144 ... | tri≈ ¬a refl ¬c = refl-⊆
145 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) 145 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c )
146 146
147 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP 147 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP
148 P-GenericFilter P L p0 L⊆PP Lp0 C = record { 148 P-GenericFilter P L p0 L⊆PP Lp0 C = record {
149 genf = record { filter = PDHOD L p0 C ; f⊆L = f⊆PL ; filter1 = {!!} ; filter2 = {!!} } 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 }
150 ; generic = {!!} 150 ; generic = fdense
151 } where 151 } where
152 f⊆PL : PDHOD L p0 C ⊆ L -- Power P
153 f⊆PL = record { incl = λ {x} lt → x∈PP lt }
154 Lq : {q : HOD} → L ∋ q → q ⊆ P
155 Lq {q} lt = ODC.power→⊆ O P q ( incl L⊆PP lt )
156 Pp0 : p0 ∈ Power P
157 Pp0 = incl L⊆PP Lp0
152 PGHOD∈PL : (i : Nat) → (x : Ordinal) → PGHOD i L C x ⊆ Power P 158 PGHOD∈PL : (i : Nat) → (x : Ordinal) → PGHOD i L C x ⊆ Power P
153 PGHOD∈PL i x = record { incl = λ {x} p → {!!} } 159 PGHOD∈PL i x = record { incl = λ {x} p → incl L⊆PP (proj1 p) }
154 Pp0 : p0 ∈ Power P 160 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q
155 Pp0 = {!!} 161 f1 {p} {q} L∋q PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where
156 f⊆PL : PDHOD L p0 C ⊆ L -- Power P 162 f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y
157 f⊆PL = record { incl = λ {x} lt → {!!} } -- x∈PP lt } 163 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 )))
158 f1 : {p q : HOD} → q ⊆ P → PDHOD P p0 C ∋ p → p ⊆ q → PDHOD P p0 C ∋ q
159 f1 {p} {q} q⊆P PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = {!!} } where -- power← _ _ (incl q⊆P) } where
160 f04 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y
161 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 )))
162 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y 164 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y
163 f2 : {p q : HOD} → PDHOD P p0 C ∋ p → PDHOD P p0 C ∋ q → PDHOD P p0 C ∋ (p ∩ q) 165 f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → PDHOD L p0 C ∋ (p ∩ q)
164 f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋q) (gr PD∋p) 166 f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋q) (gr PD∋p)
165 ... | 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 = {!!} } where 167 ... | 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 = {!!} } where
166 -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where 168 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
167 f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
168 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 169 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
169 f5 : odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (find-p P C (gr PD∋q) (& p0))) y 170 f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y
170 f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋q} {gr PD∋p} (<to≤ a)) 171 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))
171 (subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) 172 (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) )
172 ... | 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 = {!!} } where 173 ... | 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 = {!!} } where
173 -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where 174 f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
174 f4 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
175 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) ⟫ 175 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) ⟫
176 ... | 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 = {!!} } where -- 176 ... | 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 = {!!} } where --
177 -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where 177 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y
178 f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (p ∩ q) y 178 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
179 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 179 f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y
180 f5 : odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (* (find-p P C (gr PD∋p) (& p0))) y 180 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))
181 f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋p} {gr PD∋q} (<to≤ c)) 181 (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) )
182 (subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) 182 fdense : (D : Dense L⊆PP ) → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅
183 fdense : (D : Dense L⊆PP ) → ¬ (filter.Dense.dense D ∩ PDHOD P p0 C) ≡ od∅ 183 fdense D eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where
184 fdense D eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD P p0 C} fd01 (≡od∅→=od∅ eq0 )) where
185 open Dense 184 open Dense
186 p0⊆P : L ∋ p0
187 p0⊆P = {!!}
188 fd : HOD 185 fd : HOD
189 fd = dense-f D p0⊆P 186 fd = dense-f D Lp0
190 PP∋D : dense D ⊆ Power P 187 PP∋D : dense D ⊆ Power P
191 PP∋D = {!!} 188 PP∋D = {!!}
192 fd00 : PDHOD P p0 C ∋ p0 189 fd00 : PDHOD P p0 C ∋ p0
193 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = {!!} } 190 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = {!!} }
194 fd02 : dense D ∋ dense-f D {!!} -- p0⊆P 191 fd02 : dense D ∋ dense-f D Lp0
195 fd02 = dense-d D {!!} 192 fd02 = dense-d D Lp0
196 fd04 : dense-f D p0⊆P ⊆ P 193 fd04 : dense-f D Lp0 ⊆ P
197 fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 ) 194 fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 )
198 fd03 : PDHOD P p0 C ∋ dense-f D p0⊆P 195 fd03 : PDHOD L p0 C ∋ dense-f D Lp0
199 fd03 = {!!} 196 fd03 = {!!}
200 -- f1 {p0} {dense-f D p0⊆P} fd04 fd00 ( dense-p D (ODC.power→⊆ O _ _ Pp0 ) ) 197 fd01 : (dense D ∩ PDHOD L p0 C) ∋ fd
201 fd01 : (dense D ∩ PDHOD P p0 C) ∋ fd
202 fd01 = ⟪ fd02 , fd03 ⟫ 198 fd01 = ⟪ fd02 , fd03 ⟫
203 199
204 open GenericFilter 200 open GenericFilter
205 open Filter 201 open Filter
206 202