Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff generic-filter.agda @ 427:9b0630f03c4b
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 08 Aug 2020 18:14:14 +0900 |
parents | cc7909f86841 |
children | 94392feeebc5 |
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--- a/generic-filter.agda Thu Aug 06 11:50:35 2020 +0900 +++ b/generic-filter.agda Sat Aug 08 18:14:14 2020 +0900 @@ -185,16 +185,16 @@ -- PGHOD : (i : Nat) → (C : CountableOrdinal) → (P : HOD) → (p : Ordinal) → HOD PGHOD i C P p = record { od = record { def = λ x → - odef P x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } - ; odmax = odmax P ; <odmax = λ {y} lt → <odmax P (proj1 lt) } + odef (Power P) x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } + ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) (proj1 lt) } --- -- p(n+1) = if PGHOD n qn otherwise p(n) -- next-p : (C : CountableOrdinal) (P : HOD ) (i : Nat) → (p : Ordinal) → Ordinal -next-p C P i p with ODC.decp O ( PGHOD i C P p =h= od∅ ) +next-p C P i p with is-o∅ ( & (PGHOD i C P p)) next-p C P i p | yes y = p -next-p C P i p | no not = & (ODC.minimal O (PGHOD i C P p ) not) +next-p C P i p | no not = & (ODC.minimal O (PGHOD i C P p ) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice --- -- ascendant search on p(n) @@ -210,7 +210,7 @@ field gr : Nat pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p C P gr o∅)) y - px∈ω : odef (Power P) x + x∈PP : odef (Power P) x open PDN @@ -219,7 +219,7 @@ -- PDHOD : (C : CountableOrdinal) → (P : HOD ) → HOD PDHOD C P = record { od = record { def = λ x → PDN C P x } - ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) {y} (PDN.px∈ω lt) } where + ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) {y} (PDN.x∈PP lt) } where open PDN @@ -230,17 +230,18 @@ -- p (suc n) = if ∃ q ∈ * ( ctl→ n ) ∧ p n ⊆ q → q (axiom of choice) --- else p n +P∅ : {P : HOD} → odef (Power P) o∅ +P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where + lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) + lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) +x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y +x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt + P-GenericFilter : (C : CountableOrdinal) → (P : HOD ) → GenericFilter P P-GenericFilter C P = record { genf = record { filter = PDHOD C P ; f⊆PL = f⊆PL ; filter1 = {!!} ; filter2 = {!!} } ; generic = λ D → {!!} } where - P∅ : {P : HOD} → odef (Power P) o∅ - P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where - lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) - lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) - x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y - x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt find-p-⊆P : (C : CountableOrdinal) (P : HOD ) (i : Nat) → (x y : Ordinal) → odef (Power P) x → odef (* (find-p C P i x)) y → odef P y find-p-⊆P C P Zero x y Px Py = subst (λ k → odef P k ) &iso ( incl (ODC.power→⊆ O P (* x) (d→∋ (Power P) Px)) (x<y→∋ Py)) @@ -274,7 +275,8 @@ -- -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } -- +-- W (ω , H ( ω , 2 )) = { p ∈ ( Nat → H (ω , 2) ) | { i ∈ Nat → p i ≠ i1 } is finite } +-- -