# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1647659808 -32400
# Node ID d407cc8499fc932805719061b6c0a3d3e00d12d4
# Parent  3d84389cc43ff56ba87eff55f275d60d8cabdff0
...

diff -r 3d84389cc43f -r d407cc8499fc src/generic-filter.agda
--- a/src/generic-filter.agda	Fri Mar 18 23:45:23 2022 +0900
+++ b/src/generic-filter.agda	Sat Mar 19 12:16:48 2022 +0900
@@ -146,59 +146,55 @@
 
 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP
 P-GenericFilter P L p0 L⊆PP Lp0 C = record {
-      genf = record { filter = PDHOD L p0 C ; f⊆L =  f⊆PL ; filter1 = {!!} ; filter2 = {!!}  }
-    ; generic = {!!}
+      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 }
+    ; generic = fdense
    } where
-        PGHOD∈PL :  (i : Nat) → (x : Ordinal) →  PGHOD i L C x ⊆ Power P
-        PGHOD∈PL i x = record { incl = λ {x} p → {!!}  }
-        Pp0 : p0 ∈ Power P
-        Pp0 = {!!}
         f⊆PL :  PDHOD L p0 C ⊆ L -- Power P
-        f⊆PL = record { incl = λ {x} lt →  {!!} } -- x∈PP lt  }
-        f1 : {p q : HOD} → q ⊆ P → PDHOD P p0 C ∋ p → p ⊆ q → PDHOD P p0 C ∋ q
-        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
-           f04 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y
-           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 )))
+        f⊆PL = record { incl = λ {x} lt → x∈PP lt  }
+        Lq : {q : HOD} → L ∋ q → q ⊆ P
+        Lq {q} lt = ODC.power→⊆ O P q ( incl L⊆PP lt )
+        Pp0 : p0 ∈ Power P
+        Pp0 = incl L⊆PP Lp0 
+        PGHOD∈PL :  (i : Nat) → (x : Ordinal) →  PGHOD i L C x ⊆ Power P
+        PGHOD∈PL i x = record { incl = λ {x} p → incl  L⊆PP (proj1 p) }
+        f1 : {p q : HOD} → L ∋  q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q
+        f1 {p} {q} L∋q PD∋p p⊆q =  record { gr = gr PD∋p ;  pn<gr = f04 ; x∈PP = L∋q } where
+           f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y
+           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 )))
                -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y
-        f2 : {p q : HOD} → PDHOD P p0 C ∋ p → PDHOD P p0 C ∋ q → PDHOD P p0 C ∋ (p ∩ q)
+        f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → PDHOD L p0 C ∋ (p ∩ q)
         f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋q) (gr PD∋p)
-        ... | 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
-                -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q)   }  where
-            f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
+        ... | 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
+            f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
             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
-               f5 : odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (find-p P C (gr PD∋q) (& p0))) y
-               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))
-                   (subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) (sym &iso) lt) )
-        ... | 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
-               -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q)   }  where
-            f4 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
+               f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y
+               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))
+                   (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) )
+        ... | 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
+            f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
             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) ⟫ 
         ... | 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 --
-           -- ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q)   } where
-            f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (p ∩ q) y
-            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
-               f5 : odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (* (find-p P C (gr PD∋p) (& p0))) y
-               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))
-                   (subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) (sym &iso) lt) )
-        fdense : (D : Dense L⊆PP ) → ¬ (filter.Dense.dense D ∩ PDHOD P p0 C) ≡ od∅
-        fdense D eq0  = ⊥-elim (  ∅< {Dense.dense D ∩ PDHOD P p0 C} fd01 (≡od∅→=od∅ eq0 )) where
+            f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y
+            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
+               f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y
+               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))
+                   (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) )
+        fdense : (D : Dense L⊆PP ) → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅
+        fdense D eq0  = ⊥-elim (  ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where
            open Dense
-           p0⊆P : L ∋ p0 
-           p0⊆P = {!!}
            fd : HOD
-           fd = dense-f D p0⊆P
+           fd = dense-f D Lp0
            PP∋D : dense D ⊆ Power P
            PP∋D = {!!} 
            fd00 : PDHOD P p0 C ∋ p0
            fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = {!!}  }
-           fd02 : dense D ∋ dense-f D {!!} -- p0⊆P
-           fd02 = dense-d D {!!}
-           fd04 : dense-f D p0⊆P ⊆ P
+           fd02 : dense D ∋ dense-f D Lp0 
+           fd02 = dense-d D Lp0
+           fd04 : dense-f D Lp0 ⊆ P
            fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 )
-           fd03 : PDHOD P p0 C  ∋ dense-f D p0⊆P
+           fd03 : PDHOD L p0 C  ∋ dense-f D Lp0
            fd03 = {!!}
-           -- f1 {p0} {dense-f D p0⊆P} fd04 fd00 ( dense-p D (ODC.power→⊆ O _ _ Pp0 ) )
-           fd01 : (dense D ∩ PDHOD P p0 C) ∋ fd
+           fd01 : (dense D ∩ PDHOD L p0 C) ∋ fd
            fd01 = ⟪ fd02 , fd03 ⟫ 
 
 open GenericFilter