Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/generic-filter.agda @ 450:b27d92694ed5
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 14 Mar 2022 17:51:16 +0900 |
parents | be685f338fdc |
children | 31f0a5a745a5 |
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--- a/src/generic-filter.agda Sun Mar 13 19:22:12 2022 +0900 +++ b/src/generic-filter.agda Mon Mar 14 17:51:16 2022 +0900 @@ -68,7 +68,6 @@ -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x -- - -- we expect P ∈ * ctl-M ∧ G ⊆ Power P , ¬ G ∈ * ctl-M, open CountableModel @@ -92,7 +91,7 @@ ... | no not = & (ODC.minimal O ( PGHOD i P C (find-p P C i x)) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice --- --- G = { r ∈ Power P | ∃ n → p(n) ⊆ q } +-- G = { r ∈ Power P | ∃ n → p(n) ⊆ r } -- record PDN (P p : HOD ) (C : CountableModel P) (x : Ordinal) : Set n where field @@ -168,7 +167,6 @@ 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∋p} {gr PD∋q} (<to≤ a)) (subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) --- subst (λ k → odef k y) *iso (pn<gr PD∋q y (subst (λ k → odef _ k ) &iso (incl (p-monotonic _ _ C a ) (subst (λ k → odef _ k) &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 = 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 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) ⟫ @@ -181,39 +179,68 @@ fdense : (D : Dense P ) → ¬ (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 open Dense + p0⊆P : p0 ⊆ P + p0⊆P = ODC.power→⊆ O _ _ Pp0 fd : HOD - fd = dense-f D p0 + fd = dense-f D p0⊆P PP∋D : dense D ⊆ Power P PP∋D = d⊆P D fd00 : PDHOD P p0 C ∋ p0 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = Pp0 } - fd02 : dense D ∋ dense-f D p0 - fd02 = dense-d D (ODC.power→⊆ O _ _ Pp0 ) - fd04 : dense-f D p0 ⊆ P + fd02 : dense D ∋ dense-f D p0⊆P + fd02 = dense-d D p0⊆P + fd04 : dense-f D p0⊆P ⊆ P fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 ) - fd03 : PDHOD P p0 C ∋ dense-f D p0 - fd03 = f1 {p0} {dense-f D p0} fd04 fd00 ( dense-p D (ODC.power→⊆ O _ _ Pp0 ) ) + fd03 : PDHOD P p0 C ∋ dense-f D p0⊆P + 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 = ⟪ fd02 , fd03 ⟫ - - - open GenericFilter open Filter -record Incompatible (P : HOD ) : Set (suc (suc n)) where +record Incompatible (P p : HOD ) (PP∋p : p ⊆ P ) : Set (suc (suc n)) where field - q : {p : HOD } → Power P ∋ p → HOD - r : {p : HOD } → Power P ∋ p → HOD - incompatible : { p : HOD } → (P∋p : Power P ∋ p) → Power P ∋ q P∋p → Power P ∋ r P∋p - → ( p ⊆ q P∋p) ∧ ( p ⊆ r P∋p) - → ∀ ( s : HOD ) → Power P ∋ s → ¬ (( q P∋p ⊆ s ) ∧ ( r P∋p ⊆ s )) + q r : HOD + PP∋q : q ⊆ P + PP∋r : r ⊆ P + p⊆q : p ⊆ q + p⊆r : p ⊆ r + incompatible : ∀ ( s : HOD ) → s ⊆ P → (¬ ( q ⊆ s )) ∨ (¬ ( r ⊆ s )) lemma725 : (P p : HOD ) (C : CountableModel P) - → * (ctl-M C) ∋ Power P - → Incompatible P → ¬ ( * (ctl-M C) ∋ filter ( genf ( P-GenericFilter P p {!!} C ))) -lemma725 = {!!} + → (pp0 : Power P ∋ p ) + → * (ctl-M C) ∋ (Power P ∩ * (ctl-M C)) -- M is a Model of ZF + → * (ctl-M C) ∋ ( (Power P ∩ * (ctl-M C)) \ filter ( genf ( P-GenericFilter P p pp0 C)) ) -- M ∋ G and M is a Model of ZF + → ((p : HOD) → (PP∋p : p ⊆ P ) → Incompatible P p PP∋p ) + → ¬ ( * (ctl-M C) ∋ filter ( genf ( P-GenericFilter P p pp0 C ))) +lemma725 P p C PP∋p M∋PM M∋D I M∋G = D∩G≠∅ D∩G=∅ where + G = filter ( genf ( P-GenericFilter P p PP∋p C )) + M = * (ctl-M C) + D : HOD + D = Power P \ G + p⊆P : p ⊆ P + p⊆P = ODC.power→⊆ O _ _ PP∋p + df : {x : HOD} → x ⊆ P → HOD + df {x} PP∋x with Incompatible.incompatible (I p p⊆P) x PP∋x + ... | case1 q = Incompatible.q (I p p⊆P) + ... | case2 r = Incompatible.r (I p p⊆P) + df-d : {x : HOD} → (lt : x ⊆ P) → D ∋ df lt + df-d = {!!} + df-p : {x : HOD} → (lt : x ⊆ P) → x ⊆ df lt + df-p = {!!} + D-Dense : Dense P + D-Dense = record { + dense = D + ; d⊆P = record { incl = λ {x} lt → proj1 lt } + ; dense-f = df + ; dense-d = df-d + ; dense-p = df-p + } + D∩G=∅ : ( D ∩ G ) =h= od∅ + D∩G=∅ = {!!} + D∩G≠∅ : ¬ (( D ∩ G ) =h= od∅ ) + D∩G≠∅ eq = generic (P-GenericFilter P p PP∋p C) D-Dense ( ==→o≡ eq ) open import PFOD O @@ -222,11 +249,11 @@ -- ω→2 : HOD -- ω→2 = Power infinite -lemma725-1 : Incompatible HODω2 +lemma725-1 : (p : HOD) → (PP∋p : p ⊆ HODω2 ) → Incompatible HODω2 p PP∋p lemma725-1 = {!!} lemma726 : (C : CountableModel HODω2) - → Union ( Replace HODω2 (λ p → filter ( genf ( P-GenericFilter HODω2 p {!!} C )))) =h= ω→2 + → Union ( Replace' (Power HODω2) (λ p lt → filter ( genf ( P-GenericFilter HODω2 p lt C )))) =h= ω→2 -- HODω2 ∋ p lemma726 = {!!} --