Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1248:b1d024385208
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 15 Mar 2023 09:41:57 +0900 |
| parents | 0350fe03d73a |
| children | c57b8068f97c |
| files | src/generic-filter.agda |
| diffstat | 1 files changed, 192 insertions(+), 174 deletions(-) [+] |
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--- a/src/generic-filter.agda Tue Mar 14 14:41:39 2023 +0900 +++ b/src/generic-filter.agda Wed Mar 15 09:41:57 2023 +0900 @@ -61,6 +61,8 @@ ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) ctl← : (x : Ordinal )→ odef (ctl-M ) x → ℕ ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x + TC : {x y : Ordinal} → odef ctl-M x → odef (* x) y → odef ctl-M y + is-model : (x : HOD) → ctl-M ∋ (x ∩ ctl-M) -- we have no otherway round -- ctl-iso← : { x : ℕ } → ctl← (ctl→ x ) (ctl<M x) ≡ x -- @@ -72,11 +74,6 @@ open CountableModel -abs-minus : {p q : HOD} → (C : CountableModel) → ctl-M C ∋ (p ＼ q) -abs-minus {p} {q} C = ? where - p-q : {x : Ordinal } → odef (p ＼ q) x → ℕ - p-q pqx = ctl← C _ ? - ---- -- a(n) ∈ M -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q @@ -171,9 +168,12 @@ record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where field genf : Filter {L} {P} LP - generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Replace (Filter.filter genf) (λ x → P ＼ x )) ≡ od∅ ) rgen : HOD rgen = Replace (Filter.filter genf) (λ x → P ＼ x ) + field + generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Replace (Filter.filter genf) (λ x → P ＼ x )) ≡ od∅ ) + gfilter1 : {p q : HOD} → rgen ∋ p → q ⊆ p → rgen ∋ q + gfilter2 : {p q : HOD} → (rgen ∋ p ) ∧ (rgen ∋ q) → rgen ∋ (p ∪ q) P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) -- L is a Boolean Algebra @@ -183,165 +183,178 @@ P-GenericFilter P L p0 L⊆PP Lp0 CAP UNI NEG C = record { genf = record { filter = Replace (PDHOD L p0 C) (λ x → P ＼ x) ; f⊆L = gf01 ; filter1 = f1 ; filter2 = f2 } ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd ) + ; gfilter1 = gfilter1 + ; gfilter2 = gfilter2 } where - GP = Replace (PDHOD L p0 C) (λ x → P ＼ x) - f⊆PL : PDHOD L p0 C ⊆ L - f⊆PL lt = x∈PP lt - gf01 : Replace (PDHOD L p0 C) (λ x → P ＼ x) ⊆ L - gf01 {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef L k) (sym x=ψz) ( NEG (subst (λ k → odef L k) (sym &iso) (f⊆PL az)) ) - gf141 : {xp xq : Ordinal } → (Pp : PDN L p0 C xp) (Pq : PDN L p0 C xq) → (* xp ∪ * xq) ⊆ P - gf141 Pp Pq {x} (case1 xpx) = L⊆PP (PDN.x∈PP Pp) _ xpx - gf141 Pp Pq {x} (case2 xqx) = L⊆PP (PDN.x∈PP Pq) _ xqx - gf121 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → p ∩ q ≡ P ＼ * (& (* (Replaced.z gp) ∪ * (Replaced.z gq))) - gf121 {p} {q} gp gq = begin - p ∩ q ≡⟨ cong₂ (λ j k → j ∩ k ) (sym *iso) (sym *iso) ⟩ - (* (& p)) ∩ (* (& q)) ≡⟨ cong₂ (λ j k → ( * j ) ∩ ( * k)) (Replaced.x=ψz gp) (Replaced.x=ψz gq) ⟩ - * (& (P ＼ (* xp ))) ∩ (* (& (P ＼ (* xq )))) ≡⟨ cong₂ (λ j k → j ∩ k ) *iso *iso ⟩ - (P ＼ (* xp )) ∩ (P ＼ (* xq )) ≡⟨ gf02 {P} {* xp} {* xq} ⟩ - P ＼ ((* xp) ∪ (* xq)) ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ - P ＼ * (& (* xp ∪ * xq)) ∎ where - open ≡-Reasoning - xp = Replaced.z gp - xq = Replaced.z gq - gf131 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → P ＼ (p ∩ q) ≡ * (Replaced.z gp) ∪ * (Replaced.z gq) - gf131 {p} {q} gp gq = trans (cong (λ k → P ＼ k) (gf121 gp gq)) - (trans ( L＼Lx=x (subst (λ k → k ⊆ P) (sym *iso) (gf141 (Replaced.az gp) (Replaced.az gq))) ) *iso ) + GP = Replace (PDHOD L p0 C) (λ x → P ＼ x) + GPR = Replace GP (_＼_ P) + f⊆PL : PDHOD L p0 C ⊆ L + f⊆PL lt = x∈PP lt + gf01 : Replace (PDHOD L p0 C) (λ x → P ＼ x) ⊆ L + gf01 {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef L k) (sym x=ψz) ( NEG (subst (λ k → odef L k) (sym &iso) (f⊆PL az)) ) + gf141 : {xp xq : Ordinal } → (Pp : PDN L p0 C xp) (Pq : PDN L p0 C xq) → (* xp ∪ * xq) ⊆ P + gf141 Pp Pq {x} (case1 xpx) = L⊆PP (PDN.x∈PP Pp) _ xpx + gf141 Pp Pq {x} (case2 xqx) = L⊆PP (PDN.x∈PP Pq) _ xqx + gf121 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → p ∩ q ≡ P ＼ * (& (* (Replaced.z gp) ∪ * (Replaced.z gq))) + gf121 {p} {q} gp gq = begin + p ∩ q ≡⟨ cong₂ (λ j k → j ∩ k ) (sym *iso) (sym *iso) ⟩ + (* (& p)) ∩ (* (& q)) ≡⟨ cong₂ (λ j k → ( * j ) ∩ ( * k)) (Replaced.x=ψz gp) (Replaced.x=ψz gq) ⟩ + * (& (P ＼ (* xp ))) ∩ (* (& (P ＼ (* xq )))) ≡⟨ cong₂ (λ j k → j ∩ k ) *iso *iso ⟩ + (P ＼ (* xp )) ∩ (P ＼ (* xq )) ≡⟨ gf02 {P} {* xp} {* xq} ⟩ + P ＼ ((* xp) ∪ (* xq)) ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ + P ＼ * (& (* xp ∪ * xq)) ∎ where + open ≡-Reasoning + xp = Replaced.z gp + xq = Replaced.z gq + gf131 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → P ＼ (p ∩ q) ≡ * (Replaced.z gp) ∪ * (Replaced.z gq) + gf131 {p} {q} gp gq = trans (cong (λ k → P ＼ k) (gf121 gp gq)) + (trans ( L＼Lx=x (subst (λ k → k ⊆ P) (sym *iso) (gf141 (Replaced.az gp) (Replaced.az gq))) ) *iso ) - f1 : {p q : HOD} → L ∋ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ p → p ⊆ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ q - f1 {p} {q} L∋q record { z = z ; az = az ; x=ψz = x=ψz } p⊆q = record { z = _ - ; az = record { gr = gr az ; pn<gr = f04 ; x∈PP = NEG L∋q } ; x=ψz = f05 } where - open ≡-Reasoning - f04 : (y : Ordinal) → odef (* (& (P ＼ q))) y → odef (* (find-p L C (gr az ) (& p0))) y - f04 y qy = PDN.pn<gr az _ (subst (λ k → odef k y ) f06 (f03 qy )) where - f06 : * (& (P ＼ p)) ≡ * z - f06 = begin - * (& (P ＼ p)) ≡⟨ *iso ⟩ - P ＼ p ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ - P ＼ (* (& p)) ≡⟨ cong (λ k → P ＼ k) (cong (*) x=ψz) ⟩ - P ＼ (* (& (P ＼ * z))) ≡⟨ cong ( λ k → P ＼ k) *iso ⟩ - P ＼ (P ＼ * z) ≡⟨ L＼Lx=x (λ {x} lt → L⊆PP (x∈PP az) _ lt ) ⟩ - * z ∎ - f03 : odef (* (& (P ＼ q))) y → odef (* (& (P ＼ p))) y - f03 pqy with subst (λ k → odef k y ) *iso pqy - ... | ⟪ Py , nqy ⟫ = subst (λ k → odef k y ) (sym *iso) ⟪ Py , (λ py → nqy (p⊆q py) ) ⟫ - f05 : & q ≡ & (P ＼ * (& (P ＼ q))) - f05 = cong (&) ( begin - q ≡⟨ sym (L＼Lx=x (λ {x} lt → L⊆PP L∋q _ (subst (λ k → odef k x) (sym *iso) lt) )) ⟩ - P ＼ (P ＼ q ) ≡⟨ cong ( λ k → P ＼ k) (sym *iso) ⟩ - P ＼ * (& (P ＼ q)) ∎ ) - f2 : {p q : HOD} → GP ∋ p → GP ∋ q → L ∋ (p ∩ q) → GP ∋ (p ∩ q) - f2 {p} {q} record { z = xp ; az = Pp ; x=ψz = peq } - record { z = xq ; az = Pq ; x=ψz = qeq } L∋pq with <-cmp (gr Pp) (gr Pq) - ... | tri< a ¬b ¬c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq) } where - gp = record { z = xp ; az = Pp ; x=ψz = peq } - gq = record { z = xq ; az = Pq ; x=ψz = qeq } - gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) - gf10 = record { gr = PDN.gr Pq ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where - gf16 : gr Pp ≤ gr Pq - gf16 = <to≤ a - gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pq) (& p0))) y - gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy - ... | case1 xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y xpy ) - ... | case2 xqy = PDN.pn<gr Pq _ xqy - ... | tri≈ ¬a eq ¬c = record { z = & (* xp ∪ * xq) ; az = record { gr = gr Pp ; pn<gr = gf21 ; x∈PP = gf22 } ; x=ψz = gf23 } where - gp = record { z = xp ; az = Pp ; x=ψz = peq } - gq = record { z = xq ; az = Pq ; x=ψz = qeq } - gf22 : odef L (& (* xp ∪ * xq)) - gf22 = UNI (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pp)) (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pq)) - gf21 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y - gf21 y xpqy with subst (λ k → odef k y) *iso xpqy - ... | case1 xpy = PDN.pn<gr Pp _ xpy - ... | case2 xqy = subst (λ k → odef (* (find-p L C k (& p0))) y ) (sym eq) ( PDN.pn<gr Pq _ xqy ) - gf25 : odef L (& p) - gf25 = subst (λ k → odef L k ) (sym peq) ( NEG (subst (λ k → odef L k) (sym &iso) (PDN.x∈PP Pp) )) - gf27 : {x : Ordinal} → odef p x → odef (P ＼ * xp) x - gf27 {x} px = subst (λ k → odef k x) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) peq)) px - -- gf02 : {P a b : HOD } → (P ＼ a) ∩ (P ＼ b) ≡ ( P ＼ (a ∪ b) ) - gf23 : & (p ∩ q) ≡ & (P ＼ * (& (* xp ∪ * xq))) - gf23 = cong (&) (gf121 gp gq ) - ... | tri> ¬a ¬b c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq ) } where - gp = record { z = xp ; az = Pp ; x=ψz = peq } - gq = record { z = xq ; az = Pq ; x=ψz = qeq } - gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) - gf10 = record { gr = PDN.gr Pp ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where - gf16 : gr Pq ≤ gr Pp - gf16 = <to≤ c - gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y - gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy - ... | case1 xpy = PDN.pn<gr Pp _ xpy - ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y xqy ) - gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P ＼ x)) (_＼_ P) ≡ PDHOD L p0 C - gf00 = ==→o≡ record { eq→ = gf20 ; eq← = gf22 } where - gf20 : {x : Ordinal} → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x → PDN L p0 C x - gf20 {x} record { z = z₁ ; az = record { z = z ; az = az ; x=ψz = x=ψz₁ } ; x=ψz = x=ψz } = - subst (λ k → PDN L p0 C k ) (begin - z ≡⟨ sym &iso ⟩ - & (* z) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ - & (P ＼ ( P ＼ (* z) )) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ - & (P ＼ (* ( & (P ＼ (* z ))))) ≡⟨ cong (λ k → & (P ＼ (* k))) (sym x=ψz₁) ⟩ - & (P ＼ (* z₁)) ≡⟨ sym x=ψz ⟩ - x ∎ ) az where - open ≡-Reasoning - gf21 : {x : Ordinal } → odef (* z) x → odef P x - gf21 {x} lt = L⊆PP ( PDN.x∈PP az) _ lt - gf22 : {x : Ordinal} → PDN L p0 C x → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x - gf22 {x} pdx = record { z = _ ; az = record { z = _ ; az = pdx ; x=ψz = refl } ; x=ψz = ( begin - x ≡⟨ sym &iso ⟩ - & (* x) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ - & (P ＼ (P ＼ * x)) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ - & (P ＼ * (& (P ＼ * x))) ∎ ) } where - open ≡-Reasoning - gf21 : {z : Ordinal } → odef (* x) z → odef P z - gf21 {z} lt = L⊆PP ( PDN.x∈PP pdx ) z lt - fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (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 - fd09 : (i : ℕ ) → 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 not = fd17 where - fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) - fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 - fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) - fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) - fd17 = proj1 fd18 - an : ℕ - 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) - d=an : dense D ≡ * (ctl→ C an) - d=an = begin dense D ≡⟨ sym *iso ⟩ - * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ - * (ctl→ C an) ∎ where open ≡-Reasoning - fd07 : odef (dense D) pn+1 - fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) - ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where - L∋pn : L ∋ * (find-p L C an (& p0)) - L∋pn = subst (λ k → odef L k) (sym &iso) (fd09 an ) - L∋df : L ∋ ( dense-f D L∋pn ) - L∋df = (d⊆P D) ( dense-d D L∋pn ) - pn∋df : (* (ctl→ C an)) ∋ ( dense-f D L∋pn ) - pn∋df = subst (λ k → odef k (& ( dense-f D L∋pn ) )) d=an ( dense-d D L∋pn ) - pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (dense-f D L∋pn))) y - pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (dense-p D L∋pn py) - fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (dense-f D L∋pn)) - fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ - fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ - fd10 = ≡o∅→=od∅ y - ... | no not = fd27 where - fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) - fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 - fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) - fd27 : odef (dense D) (& fd29) - fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) - 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 ⟫ - + f1 : {p q : HOD} → L ∋ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ p → p ⊆ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ q + f1 {p} {q} L∋q record { z = z ; az = az ; x=ψz = x=ψz } p⊆q = record { z = _ + ; az = record { gr = gr az ; pn<gr = f04 ; x∈PP = NEG L∋q } ; x=ψz = f05 } where + open ≡-Reasoning + f04 : (y : Ordinal) → odef (* (& (P ＼ q))) y → odef (* (find-p L C (gr az ) (& p0))) y + f04 y qy = PDN.pn<gr az _ (subst (λ k → odef k y ) f06 (f03 qy )) where + f06 : * (& (P ＼ p)) ≡ * z + f06 = begin + * (& (P ＼ p)) ≡⟨ *iso ⟩ + P ＼ p ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ + P ＼ (* (& p)) ≡⟨ cong (λ k → P ＼ k) (cong (*) x=ψz) ⟩ + P ＼ (* (& (P ＼ * z))) ≡⟨ cong ( λ k → P ＼ k) *iso ⟩ + P ＼ (P ＼ * z) ≡⟨ L＼Lx=x (λ {x} lt → L⊆PP (x∈PP az) _ lt ) ⟩ + * z ∎ + f03 : odef (* (& (P ＼ q))) y → odef (* (& (P ＼ p))) y + f03 pqy with subst (λ k → odef k y ) *iso pqy + ... | ⟪ Py , nqy ⟫ = subst (λ k → odef k y ) (sym *iso) ⟪ Py , (λ py → nqy (p⊆q py) ) ⟫ + f05 : & q ≡ & (P ＼ * (& (P ＼ q))) + f05 = cong (&) ( begin + q ≡⟨ sym (L＼Lx=x (λ {x} lt → L⊆PP L∋q _ (subst (λ k → odef k x) (sym *iso) lt) )) ⟩ + P ＼ (P ＼ q ) ≡⟨ cong ( λ k → P ＼ k) (sym *iso) ⟩ + P ＼ * (& (P ＼ q)) ∎ ) + f2 : {p q : HOD} → GP ∋ p → GP ∋ q → L ∋ (p ∩ q) → GP ∋ (p ∩ q) + f2 {p} {q} record { z = xp ; az = Pp ; x=ψz = peq } + record { z = xq ; az = Pq ; x=ψz = qeq } L∋pq with <-cmp (gr Pp) (gr Pq) + ... | tri< a ¬b ¬c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq) } where + gp = record { z = xp ; az = Pp ; x=ψz = peq } + gq = record { z = xq ; az = Pq ; x=ψz = qeq } + gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) + gf10 = record { gr = PDN.gr Pq ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where + gf16 : gr Pp ≤ gr Pq + gf16 = <to≤ a + gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pq) (& p0))) y + gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy + ... | case1 xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y xpy ) + ... | case2 xqy = PDN.pn<gr Pq _ xqy + ... | tri≈ ¬a eq ¬c = record { z = & (* xp ∪ * xq) ; az = record { gr = gr Pp ; pn<gr = gf21 ; x∈PP = gf22 } ; x=ψz = gf23 } where + gp = record { z = xp ; az = Pp ; x=ψz = peq } + gq = record { z = xq ; az = Pq ; x=ψz = qeq } + gf22 : odef L (& (* xp ∪ * xq)) + gf22 = UNI (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pp)) (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pq)) + gf21 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y + gf21 y xpqy with subst (λ k → odef k y) *iso xpqy + ... | case1 xpy = PDN.pn<gr Pp _ xpy + ... | case2 xqy = subst (λ k → odef (* (find-p L C k (& p0))) y ) (sym eq) ( PDN.pn<gr Pq _ xqy ) + gf25 : odef L (& p) + gf25 = subst (λ k → odef L k ) (sym peq) ( NEG (subst (λ k → odef L k) (sym &iso) (PDN.x∈PP Pp) )) + gf27 : {x : Ordinal} → odef p x → odef (P ＼ * xp) x + gf27 {x} px = subst (λ k → odef k x) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) peq)) px + -- gf02 : {P a b : HOD } → (P ＼ a) ∩ (P ＼ b) ≡ ( P ＼ (a ∪ b) ) + gf23 : & (p ∩ q) ≡ & (P ＼ * (& (* xp ∪ * xq))) + gf23 = cong (&) (gf121 gp gq ) + ... | tri> ¬a ¬b c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq ) } where + gp = record { z = xp ; az = Pp ; x=ψz = peq } + gq = record { z = xq ; az = Pq ; x=ψz = qeq } + gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) + gf10 = record { gr = PDN.gr Pp ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where + gf16 : gr Pq ≤ gr Pp + gf16 = <to≤ c + gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y + gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy + ... | case1 xpy = PDN.pn<gr Pp _ xpy + ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y xqy ) + gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P ＼ x)) (_＼_ P) ≡ PDHOD L p0 C + gf00 = ==→o≡ record { eq→ = gf20 ; eq← = gf22 } where + gf20 : {x : Ordinal} → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x → PDN L p0 C x + gf20 {x} record { z = z₁ ; az = record { z = z ; az = az ; x=ψz = x=ψz₁ } ; x=ψz = x=ψz } = + subst (λ k → PDN L p0 C k ) (begin + z ≡⟨ sym &iso ⟩ + & (* z) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ + & (P ＼ ( P ＼ (* z) )) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ + & (P ＼ (* ( & (P ＼ (* z ))))) ≡⟨ cong (λ k → & (P ＼ (* k))) (sym x=ψz₁) ⟩ + & (P ＼ (* z₁)) ≡⟨ sym x=ψz ⟩ + x ∎ ) az where + open ≡-Reasoning + gf21 : {x : Ordinal } → odef (* z) x → odef P x + gf21 {x} lt = L⊆PP ( PDN.x∈PP az) _ lt + gf22 : {x : Ordinal} → PDN L p0 C x → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x + gf22 {x} pdx = record { z = _ ; az = record { z = _ ; az = pdx ; x=ψz = refl } ; x=ψz = ( begin + x ≡⟨ sym &iso ⟩ + & (* x) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ + & (P ＼ (P ＼ * x)) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ + & (P ＼ * (& (P ＼ * x))) ∎ ) } where + open ≡-Reasoning + gf21 : {z : Ordinal } → odef (* x) z → odef P z + gf21 {z} lt = L⊆PP ( PDN.x∈PP pdx ) z lt + fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (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 + fd09 : (i : ℕ ) → 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 not = fd17 where + fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) + fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 + fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) + fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) + fd17 = proj1 fd18 + an : ℕ + 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) + d=an : dense D ≡ * (ctl→ C an) + d=an = begin dense D ≡⟨ sym *iso ⟩ + * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ + * (ctl→ C an) ∎ where open ≡-Reasoning + fd07 : odef (dense D) pn+1 + fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) + ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where + L∋pn : L ∋ * (find-p L C an (& p0)) + L∋pn = subst (λ k → odef L k) (sym &iso) (fd09 an ) + L∋df : L ∋ ( dense-f D L∋pn ) + L∋df = (d⊆P D) ( dense-d D L∋pn ) + pn∋df : (* (ctl→ C an)) ∋ ( dense-f D L∋pn ) + pn∋df = subst (λ k → odef k (& ( dense-f D L∋pn ) )) d=an ( dense-d D L∋pn ) + pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (dense-f D L∋pn))) y + pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (dense-p D L∋pn py) + fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (dense-f D L∋pn)) + fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ + fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ + fd10 = ≡o∅→=od∅ y + ... | no not = fd27 where + fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) + fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 + fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) + fd27 : odef (dense D) (& fd29) + fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) + 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 ⟫ + gfilter1 : {p q : HOD} → GPR ∋ p → q ⊆ p → GPR ∋ q + gfilter1 {p} {q} record { z = z ; az = az ; x=ψz = x=ψz } q⊆p = record { z = _ ; az = gf30 ; x=ψz = ? } where + gf30 : GP ∋ (P ＼ q ) + gf30 = f1 ? ? ? + gfilter2 : {p q : HOD} → (GPR ∋ p) ∧ (GPR ∋ q) → Replace GP (_＼_ P) ∋ (p ∪ q) + gfilter2 {p} {q} ⟪ record { z = zp ; az = azp ; x=ψz = x=ψzp } , record { z = zq ; az = azq ; x=ψz = x=ψzq } ⟫ + = record { z = _ ; az = gf31 ; x=ψz = ? } where + gfp : GP ∋ (P ＼ p ) + gfp = ? + gf31 : GP ∋ ( (P ＼ p ) ∩ (P ＼ q ) ) + gf31 = f2 gfp ? ? open GenericFilter open Filter @@ -360,32 +373,37 @@ → (UNI : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q )) → (NEG : ({p : HOD} → L ∋ p → L ∋ ( P ＼ p))) → (C : CountableModel ) - → ( MP : ctl-M C ∋ P ) → ( {p : HOD} → (Lp : L ∋ p ) → NotCompatible L p Lp ) → ¬ ( ctl-M C ∋ rgen ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C )) -lemma232 P L p0 LPP Lp0 CAP UNI NEG C MP NC M∋gf = ¬gf∩D=0 record { eq→ = λ {x} gf∩D → ⊥-elim( proj2 (proj2 gf∩D) (proj1 gf∩D)) +lemma232 P L p0 LPP Lp0 CAP UNI NEG C NC MF = ¬rgf∩D=0 record { eq→ = λ {x} rgf∩D → ⊥-elim( proj2 (proj2 rgf∩D) (proj1 rgf∩D)) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where - gf = rgen ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C ) + GF = genf ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C ) + rgf = rgen ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C ) M = ctl-M C D : HOD - D = L ＼ gf - M∋D : M ∋ D - M∋D = subst (λ k → odef M k) ? (ctl<M C ?) + D = L ＼ rgf + M∋DM : M ∋ (D ∩ M ) + M∋DM = is-model C D D⊆PP : D ⊆ Power P D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx + ll01 : {q r : HOD } → (rgf ∋ q) ∧ (rgf ∋ r) → (q ⊆ rgf ) ∧ (r ⊆ rgf ) + ll01 {q} {r} rgfpq = ⟪ ll02 , ? ⟫ where + ll02 : {x : Ordinal } → odef q x → odef rgf x + ll02 {x} qx = record { z = ? ; az = record { z = ? ; az = ? ; x=ψz = ? } ; x=ψz = ? } + -- filter2 GF ? ? ? + -- with contra-position ? ? + -- ... | t = ? DD : Dense {L} {P} LPP Dense.dense DD = D Dense.d⊆P DD ⟪ Lx , _ ⟫ = Lx Dense.dense-f DD Lp = ? where ll00 : HOD ll00 with NotCompatible.¬compat (NC Lp) - ... | nc = ? where - ll01 : {q r : HOD } → (s : HOD) → ¬ ( (q ⊆ s) ∧ (r ⊆ s)) → (¬ (gf ∋ q)) ∨ (¬ (gf ∋ q)) - ll01 = ? + ... | nc = ? Dense.dense-d DD = ? Dense.dense-p DD = ? - ¬gf∩D=0 : ¬ ( (gf ∩ D) =h= od∅ ) - ¬gf∩D=0 = ? + ¬rgf∩D=0 : ¬ ( (rgf ∩ D) =h= od∅ ) + ¬rgf∩D=0 = ? -- -- P-Generic Filter defines a countable model D ⊂ C from P