Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1256:0b7e4eb68afc
change to Ideal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 16 Mar 2023 19:01:47 +0900 |
| parents | afecaee48825 |
| children | 004d8794697f |
| files | src/filter.agda src/generic-filter.agda |
| diffstat | 2 files changed, 98 insertions(+), 257 deletions(-) [+] |
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--- a/src/filter.agda Thu Mar 16 17:46:36 2023 +0900 +++ b/src/filter.agda Thu Mar 16 19:01:47 2023 +0900 @@ -156,7 +156,7 @@ ideal : HOD i⊆L : ideal ⊆ L ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q - ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → L ∋ (p ∩ q) → ideal ∋ (p ∪ q) + ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → L ∋ (p ∪ q) → ideal ∋ (p ∪ q) open Ideal
--- a/src/generic-filter.agda Thu Mar 16 17:46:36 2023 +0900 +++ b/src/generic-filter.agda Thu Mar 16 19:01:47 2023 +0900 @@ -179,265 +179,106 @@ d⊆P : dense ⊆ L has-expansion : {p : HOD} → (Lp : L ∋ p) → Expansion L dense Lp -record GenericFilter1 {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where +record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where field genf : Ideal {L} {P} LP generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Ideal.ideal genf ) ≡ od∅ ) -P-GenericFilter1 : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 - → (C : CountableModel ) → GenericFilter1 {L} {P} LP ( ctl-M C ) -P-GenericFilter1 P L p0 L⊆PP Lp0 C = record { - genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; ideal2 = ? } - ; generic = ? - } where - ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q - ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = - record { gr = gr ; pn<gr = λ y qy → pn<gr y ? ; x∈PP = ? } where - gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& q)) y - gf00 {y} qy = subst (λ k → odef k y ) ? (q⊆p (subst (λ k → odef k y) ? qy )) - -record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where - field - genf : Filter {L} {P} LP - 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 ∩ rgen ) ≡ od∅ ) - gideal1 : {p q : HOD} → rgen ∋ p → q ⊆ p → L ∋ ( P ＼ q) → rgen ∋ q - gideal2 : {p q : HOD} → (rgen ∋ p ) ∧ (rgen ∋ q) → rgen ∋ (p ∪ q) - P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) P-GenericFilter P L p0 L⊆PP Lp0 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 ) - ; gideal1 = gideal1 - ; gideal2 = gideal2 + genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; ideal2 = ? } + ; generic = fdense } where - 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 = ? } ; 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)) ? } 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 = ? } ; 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 = ? - 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)) ? } 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 - open Expansion - 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 ) - exp = has-expansion D L∋pn - L∋df : L ∋ ( expansion exp ) - L∋df = (d⊆P D) (dense∋exp exp) - pn∋df : (* (ctl→ C an)) ∋ ( expansion exp) - pn∋df = subst (λ k → odef k (& ( expansion exp))) d=an (dense∋exp exp ) - pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (expansion exp))) y - pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp exp py) - fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (expansion exp)) - 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 ⟫ - gpx→⊆P : {p : Ordinal } → odef GP p → (* p) ⊆ P - gpx→⊆P {p} record { z = z ; az = az ; x=ψz = x=ψz } {x} px with subst (λ k → odef k x ) - (trans (cong (*) x=ψz) *iso) px - ... | ⟪ Px , npz ⟫ = Px - L∋gpr : {p : HOD } → GPR ∋ p → (L ∋ p) ∧ ( L ∋ (P ＼ p)) - L∋gpr {p} record { z = zp ; az = record { z = z ; az = az ; x=ψz = x=ψzp } ; x=ψz = x=ψz } - = ⟪ subst (λ k → odef L k) fd40 (PDN.x∈PP az) , ? ⟫ where - fd41 : * z ⊆ P - fd41 {x} lt = L⊆PP ( PDN.x∈PP az ) _ lt - fd40 : z ≡ & p - fd40 = begin - z ≡⟨ sym &iso ⟩ - & (* z) ≡⟨ cong (&) (sym (L＼Lx=x fd41 )) ⟩ - & (P ＼ ( P ＼ * z ) ) ≡⟨ cong (λ k → & (P ＼ k)) (sym *iso) ⟩ - & (P ＼ * (& ( P ＼ * z ))) ≡⟨ cong (λ k → & (P ＼ * k )) (sym x=ψzp) ⟩ - & (P ＼ * zp) ≡⟨ sym x=ψz ⟩ - & p ∎ where open ≡-Reasoning - gpr→gp : {p : HOD} → GPR ∋ p → GP ∋ (P ＼ p ) - gpr→gp {p} record { z = zp ; az = azp ; x=ψz = x=ψzp } = gfp where - open ≡-Reasoning - gfp : GP ∋ (P ＼ p ) - gfp = subst (λ k → odef GP k) (begin - zp ≡⟨ sym &iso ⟩ - & (* zp) ≡⟨ cong (&) (sym (L＼Lx=x (gpx→⊆P azp) )) ⟩ - & (P ＼ (P ＼ (* zp) )) ≡⟨ cong (λ k → & ( P ＼ k)) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym x=ψzp))) ⟩ - & (P ＼ p) ∎ ) azp - gideal1 : {p q : HOD} → GPR ∋ p → q ⊆ p → L ∋ ( P ＼ q) → GPR ∋ q - gideal1 {p} {q} record { z = np ; az = record { z = z ; az = pdz ; x=ψz = x=ψznp } ; x=ψz = x=ψz } q⊆p Lpq - = record { z = _ ; az = gf30 ; x=ψz = cong (&) fd42 } where - gp = record { z = np ; az = record { z = z ; az = pdz ; x=ψz = x=ψznp } ; x=ψz = x=ψz } - open ≡-Reasoning - fd41 : * z ⊆ P - fd41 {x} lt = L⊆PP ( PDN.x∈PP pdz ) _ lt - p=*z : p ≡ * z - p=*z = trans (sym *iso) ( cong (*) (sym ( begin - z ≡⟨ sym &iso ⟩ - & (* z) ≡⟨ cong (&) (sym (L＼Lx=x fd41 )) ⟩ - & (P ＼ ( P ＼ * z ) ) ≡⟨ cong (λ k → & (P ＼ k)) (sym *iso) ⟩ - & (P ＼ * (& ( P ＼ * z ))) ≡⟨ cong (λ k → & (P ＼ * k )) (sym x=ψznp) ⟩ - & (P ＼ * np) ≡⟨ sym x=ψz ⟩ - & p ∎ ))) - q⊆P : q ⊆ P - q⊆P {x} lt = L⊆PP ( PDN.x∈PP pdz ) _ (subst (λ k → odef k x) p=*z (q⊆p lt)) - fd42 : q ≡ P ＼ * (& (P ＼ q)) - fd42 = trans (sym (L＼Lx=x q⊆P )) (cong (λ k → P ＼ k) (sym *iso) ) - gf32 : (P ＼ p) ⊆ (P ＼ q) - gf32 = proj1 (＼-⊆ {P} {q} {p} q⊆P ) q⊆p - gf30 : GP ∋ (P ＼ q ) - gf30 = f1 Lpq (gpr→gp gp) gf32 - gideal2 : {p q : HOD} → (GPR ∋ p) ∧ (GPR ∋ q) → Replace GP (_＼_ P) ∋ (p ∪ q) - gideal2 {p} {q} ⟪ gp , gq ⟫ - = record { z = _ ; az = gf31 ; x=ψz = cong (&) gf32 } where - open ≡-Reasoning - gf31 : GP ∋ ( (P ＼ p ) ∩ (P ＼ q ) ) - gf31 = f2 (gpr→gp gp) (gpr→gp gq) ? -- (CAP (proj2 (L∋gpr gp)) (proj2 (L∋gpr gq)) ) - gf33 : (p ∪ q) ⊆ P - gf33 {x} (case1 px) = L⊆PP (proj1 (L∋gpr gp)) _ (subst (λ k → odef k x) (sym *iso) px ) - gf33 {x} (case2 qx) = L⊆PP (proj1 (L∋gpr gq)) _ (subst (λ k → odef k x) (sym *iso) qx ) - gf32 : (p ∪ q) ≡ (P ＼ * (& ((P ＼ p) ∩ (P ＼ q)))) - gf32 = begin - p ∪ q ≡⟨ sym ( L＼Lx=x gf33 ) ⟩ - P ＼ (P ＼ (p ∪ q)) ≡⟨ cong (λ k → P ＼ k) (sym (gf02 {P} {p}{q} ) ) ⟩ - P ＼ ((P ＼ p) ∩ (P ＼ q)) ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ - P ＼ * (& ((P ＼ p) ∩ (P ＼ q))) ∎ + ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q + ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = + record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq } where + gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& p)) y + gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy )) + ideal2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∪ q) → PDHOD L p0 C ∋ (p ∪ q) + ideal2 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } + record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } Lpq = gf01 where + Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } + Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } + gf01 : PDHOD L p0 C ∋ (p ∪ q) + gf01 with <-cmp pgr qgr + ... | tri< a ¬b ¬c = record { gr = qgr ; pn<gr = gf03 ; x∈PP = Lpq } where + gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C qgr (& p0))) y + gf03 y pqy = gf15 y pqy where + gf16 : gr Pp ≤ gr Pq + gf16 = <to≤ a + gf15 : (y : Ordinal) → odef (* (& (p ∪ q))) 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 (subst (λ k → odef k y) (sym *iso) xpy) ) + ... | case2 xqy = PDN.pn<gr Pq _ (subst (λ k → odef k y) (sym *iso) xqy) + ... | tri≈ ¬a refl ¬c = record { gr = qgr ; pn<gr = gf03 ; x∈PP = Lpq } where + gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C qgr (& p0))) y + gf03 y pqy = gf15 y pqy where + gf16 : gr Pp ≤ gr Pq + gf16 = ≤-refl + gf15 : (y : Ordinal) → odef (* (& (p ∪ q))) 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 (subst (λ k → odef k y) (sym *iso) xpy) ) + ... | case2 xqy = PDN.pn<gr Pq _ (subst (λ k → odef k y) (sym *iso) xqy) + ... | tri> ¬a ¬b c = record { gr = pgr ; pn<gr = gf03 ; x∈PP = Lpq } where + gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C pgr (& p0))) y + gf03 y ppy = gf15 y ppy where + gf16 : gr Pq ≤ gr Pp + gf16 = <to≤ c + gf15 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C (gr Pp) (& p0))) y + gf15 y gppy with subst (λ k → odef k y ) *iso gppy + ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y (subst (λ k → odef k y) (sym *iso) xqy) ) + ... | case1 xpy = PDN.pn<gr Pp _ (subst (λ k → odef k y) (sym *iso) xpy) + 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 + open Expansion + 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 ) + exp = has-expansion D L∋pn + L∋df : L ∋ ( expansion exp ) + L∋df = (d⊆P D) (dense∋exp exp) + pn∋df : (* (ctl→ C an)) ∋ ( expansion exp) + pn∋df = subst (λ k → odef k (& ( expansion exp))) d=an (dense∋exp exp ) + pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (expansion exp))) y + pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp exp py) + fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (expansion exp)) + 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 ⟫ open GenericFilter open Filter @@ -455,12 +296,12 @@ → (C : CountableModel ) → ctl-M C ∋ L → ( {p : HOD} → (Lp : L ∋ p ) → NotCompatible L p Lp ) - → ¬ ( ctl-M C ∋ rgen ( P-GenericFilter P L p0 LPP Lp0 C )) + → ¬ ( ctl-M C ∋ Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C ))) lemma232 P L p0 LPP Lp0 C ML NC MF = ¬rgf∩D=0 record { eq→ = λ {x} rgf∩D → ⊥-elim( proj2 (proj1 rgf∩D) (proj2 rgf∩D)) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where PG = P-GenericFilter P L p0 LPP Lp0 C GF = genf PG - rgf = rgen PG + rgf = Ideal.ideal (genf PG) M = ctl-M C D : HOD D = L ＼ rgf @@ -493,9 +334,9 @@ ll05 : ¬ ( (q ⊆ (q ∪ r) ∧ (r ⊆ (q ∪ r)) )) ll05 = NotCompatible.¬compat (NC Lp ) (q ∪ r) ? ll03 : rgf ∋ p → rgf ∋ q → rgf ∋ (p ∪ q) - ll03 rp rq = gideal2 PG ⟪ rp , rq ⟫ + ll03 rp rq = ? -- Ideal.ideal2 GF ⟪ rp , rq ⟫ ll04 : rgf ∋ p → q ⊆ p → rgf ∋ q - ll04 rp q⊆p = gideal1 PG rp q⊆p ? + ll04 rp q⊆p = ? -- Ideal.ideal1 GF rp q⊆p ? ¬rgf∩D=0 : ¬ ( (D ∩ rgf ) =h= od∅ ) ¬rgf∩D=0 eq = generic PG DD M∋D (==→o≡ eq) @@ -526,5 +367,5 @@ → HOD val x G = TransFinite {λ x → HOD } ind (& x) where ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD - ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } + ind x valy = record { od = record { def = λ y → valS x y (& (Ideal.ideal (genf G))) } ; odmax = {!!} ; <odmax = {!!} }