Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 461:0e018784bed3
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 20 Mar 2022 11:41:48 +0900 |
| parents | d407cc8499fc |
| children | 667c54e6fa1f |
| files | src/filter.agda src/generic-filter.agda |
| diffstat | 2 files changed, 68 insertions(+), 104 deletions(-) [+] |
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--- a/src/filter.agda Sat Mar 19 12:16:48 2022 +0900 +++ b/src/filter.agda Sun Mar 20 11:41:48 2022 +0900 @@ -43,19 +43,19 @@ filter : HOD f⊆L : filter ⊆ L filter1 : { p q : HOD } → L ∋ q → filter ∋ p → p ⊆ q → filter ∋ q - filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) + filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → L ∋ (p ∩ q) → filter ∋ (p ∩ q) open Filter record prime-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) - prime : {p q : HOD } → L ∋ p → L ∋ q → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) + prime : {p q : HOD } → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) record ultra-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) - ultra : {p : HOD } → L ∋ p → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) + ultra : {p : HOD } → L ∋ p → L ∋ ( P ＼ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) open _⊆_ @@ -81,13 +81,16 @@ -- ultra filter is prime -- -filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) → (F : Filter LP) → ultra-filter F → prime-filter F -filter-lemma1 {P} {L} LP F u = record { +filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) + → ({p : HOD} → L ∋ p → L ∋ (P ＼ p)) + → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) + → (F : Filter LP) → ultra-filter F → prime-filter F +filter-lemma1 {P} {L} LP NG IN F u = record { proper = ultra-filter.proper u ; prime = lemma3 } where - lemma3 : {p q : HOD} → L ∋ p → L ∋ q → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) - lemma3 {p} {q} Lp Lq lt with ultra-filter.ultra u Lp + lemma3 : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) + lemma3 {p} {q} Lp Lq _ lt with ultra-filter.ultra u Lp (NG Lp) ... | case1 p∈P = case1 p∈P ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P ＼ p)} Lq lemma7 lemma8) where lemma5 : ((p ∪ q ) ∩ (P ＼ p)) =h= (q ∩ (P ＼ p)) @@ -98,8 +101,10 @@ lemma4 x lt with proj1 lt lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) lemma4 x lt | case2 qx = qx + lemma9 : L ∋ ((p ∪ q ) ∩ (P ＼ p)) + lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (IN Lq (NG Lp)) lemma6 : filter F ∋ ((p ∪ q ) ∩ (P ＼ p)) - lemma6 = filter2 F lt ¬p∈P + lemma6 = filter2 F lt ¬p∈P lemma9 lemma7 : filter F ∋ (q ∩ (P ＼ p)) lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 lemma8 : (q ∩ (P ＼ p)) ⊆ q @@ -110,11 +115,12 @@ -- if Filter contains L, prime filter is ultra -- -filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) - → (F : Filter LP) → filter F ∋ P → prime-filter F → ultra-filter F +filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) + → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) + → (F : Filter LP) → filter F ∋ P → prime-filter F → ultra-filter F filter-lemma2 {P} {L} LP Lm F f∋P prime = record { proper = prime-filter.proper prime - ; ultra = λ {p} L∋p → prime-filter.prime prime L∋p (Lm L∋p) (lemma p (p⊆L L∋p )) + ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (Lm L∋p) (lemma10 L∋p (incl (f⊆L F) f∋P) ) (lemma p (p⊆L L∋p )) } where open _==_ p⊆L : {p : HOD} → L ∋ p → p ⊆ P @@ -127,11 +133,13 @@ eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P ＼ p)) lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P + lemma10 : {p : HOD} → L ∋ p → L ∋ P → L ∋ (p ∪ (P ＼ p)) + lemma10 {p} L∋p lt = subst (λ k → L ∋ k ) (==→o≡ (p+1-p=1 {p} (p⊆L L∋p))) lt ----- -- -- if there is a filter , there is a ultra filter under the axiom of choise --- +-- Zorn Lemma -- filter→ultra : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → (F : Filter LP) → ultra-filter F -- filter→ultra {P} {L} LP Lm F = {!!} @@ -160,8 +168,8 @@ prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) -record GenericFilter {L P : HOD} (LP : L ⊆ Power P) : Set (suc n) where +record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where field genf : Filter LP - generic : (D : Dense LP ) → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) + generic : (D : Dense LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ )
--- a/src/generic-filter.agda Sat Mar 19 12:16:48 2022 +0900 +++ b/src/generic-filter.agda Sun Mar 20 11:41:48 2022 +0900 @@ -56,11 +56,11 @@ record CountableModel : Set (suc (suc n)) where field - ctl-M : Ordinal + ctl-M : HOD ctl→ : Nat → Ordinal - ctl<M : (x : Nat) → odef (* ctl-M) (ctl→ x) - ctl← : (x : Ordinal )→ odef (* ctl-M ) x → Nat - ctl-iso→ : { x : Ordinal } → (lt : odef (* ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x + ctl<M : (x : Nat) → odef (ctl-M) (ctl→ x) + ctl← : (x : Ordinal )→ odef (ctl-M ) x → Nat + ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x -- -- almmost universe @@ -144,12 +144,12 @@ ... | tri≈ ¬a refl ¬c = refl-⊆ ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) -P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP +P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP ( ctl-M C ) P-GenericFilter P L p0 L⊆PP Lp0 C = record { 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 - f⊆PL : PDHOD L p0 C ⊆ L -- Power P + f⊆PL : PDHOD L p0 C ⊆ L 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 ) @@ -162,118 +162,74 @@ 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 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 + f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q) + f2 {p} {q} PD∋p PD∋q L∋pq 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 = L∋pq } 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 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 + ... | 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 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 -- + ... | 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 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 + fdense : (D : Dense L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅ + fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where open Dense - fd : HOD - 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 = {!!} } + PP∋D = trans-⊆ (d⊆P D) L⊆PP + fd00 : PDHOD L p0 C ∋ p0 + fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = Lp0 } 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 L p0 C ∋ dense-f D Lp0 - fd03 = {!!} - fd01 : (dense D ∩ PDHOD L p0 C) ∋ fd - fd01 = ⟪ fd02 , fd03 ⟫ + an : Nat + an = ctl← C (& (dense D)) MD + pn : Ordinal + pn = find-p L C an (& p0) + pn+1 : Ordinal + pn+1 = find-p L C (Suc an) (& p0) + fd07 : odef (dense D) pn+1 + fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) + ... | yes _ = {!!} + ... | no _ = {!!} + fd09 : (i : Nat ) → odef L (find-p L C i (& p0)) + fd09 Zero = Lp0 + fd09 (Suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) + ... | yes _ = fd09 i + ... | no _ = {!!} + fd03 : odef (PDHOD L p0 C) pn+1 + fd03 = record { gr = Suc an ; pn<gr = λ y lt → lt ; x∈PP = fd09 (Suc an)} + fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) + fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ open GenericFilter open Filter -record Incompatible (P p : HOD ) (PP∋p : p ⊆ P ) : Set (suc (suc n)) where +record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where field - 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 )) + b : HOD + 0<b : ¬ o∅ ≡ & b + b<a : b ⊆ a -lemma725 : (P p : HOD ) (C : CountableModel ) - → (PP∋p : 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 ? 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 PP∋p 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 {!!} {!!} 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 ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) - ... | yes y = Incompatible.q (I x PP∋x) - ... | no n = Incompatible.r (I x PP∋x) - df¬⊆P : {x : HOD} → (lt : x ⊆ P) → df lt ⊆ P - df¬⊆P {x} PP∋x with ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) - ... | yes _ = Incompatible.PP∋q (I x PP∋x) - ... | no _ = Incompatible.PP∋r (I x PP∋x) - ¬G∋df : {x : HOD} → (lt : x ⊆ P) → ¬ G ∋ (df lt ) - ¬G∋df {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt ) ) - ... | no n = n - ... | yes y with Incompatible.incompatible (I x lt ) (Incompatible.q (I x lt )) (Incompatible.PP∋q (I x lt )) - ... | case1 ¬q⊆pn = λ _ → ¬q⊆pn refl-⊆ - ... | case2 ¬r⊆pn = {!!} - df-d : {x : HOD} → (lt : x ⊆ P) → D ∋ df lt - df-d {x} lt = ⟪ power← P _ (incl (df¬⊆P lt)) , ¬G∋df lt ⟫ - df-p : {x : HOD} → (lt : x ⊆ P) → x ⊆ df lt - df-p {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt) ) - ... | yes _ = Incompatible.p⊆q (I x lt) - ... | no _ = Incompatible.p⊆r (I x lt) - D-Dense : Dense {!!} - D-Dense = record { - dense = D - ; d⊆P = record { incl = λ {x} lt → {!!} } - ; dense-f = {!!} - ; dense-d = {!!} - ; dense-p = {!!} - } - D∩G=∅ : ( D ∩ G ) =h= od∅ - D∩G=∅ = ≡od∅→=od∅ ([a-b]∩b=0 {Power P} {G}) - D∩G≠∅ : ¬ (( D ∩ G ) =h= od∅ ) - D∩G≠∅ eq = generic (P-GenericFilter P {!!} {!!} {!!} {!!} C) D-Dense ( ==→o≡ eq ) - -open import PFOD O - --- HODω2 : HOD --- --- ω→2 : HOD --- ω→2 = Power infinite - -lemma725-1 : (p : HOD) → (PP∋p : p ⊆ HODω2 ) → Incompatible HODω2 p PP∋p -lemma725-1 = {!!} - -lemma726 : (C : CountableModel ) - → Union ( Replace' (Power (ω→2 ＼ HODω2)) (λ p lt → filter ( genf ( P-GenericFilter {!!} (ω→2 ＼ HODω2) p {!!} {!!} C )))) =h= ω→2 -- HODω2 ∋ p -lemma726 = {!!} +lemma232 : (P L p : HOD ) (C : CountableModel ) + → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) + → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) + → ¬ ( (ctl-M C) ∋ filter ( genf ( P-GenericFilter P L p LP Lp0 C )) ) +lemma232 P L p C LP Lp0 NA MG = {!!} -- -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } -- -record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (G : GenericFilter LP) : Set (suc n) where +record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (G : GenericFilter LP {!!} ) : Set (suc n) where field valx : HOD @@ -284,7 +240,7 @@ is-val : odef (* ox) ( & < * oy , * op > ) val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} - → (G : GenericFilter LP) + → (G : GenericFilter LP {!!} ) → HOD val x G = TransFinite {λ x → HOD } ind (& x) where ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD