Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison src/Topology.agda @ 1115:97f4a14d88ce

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 01 Jan 2023 17:02:54 +0900
parents ba3e053b85d4
children 6386019deef1
comparison
equal deleted inserted replaced
1114:ba3e053b85d4 1115:97f4a14d88ce
64 64
65 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y 65 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y
66 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ 66 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫
67 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) 67 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy))
68 68
69 -- OS = { U ⊂ X | ∀ x ∈ U → ∃ b ∈ B → x ∈ b ⊂ U } 69 -- OS = { U ⊂ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊂ U }
70 70
71 record aBase (P : HOD) {u x : Ordinal} (ux : odef (* u) x) : Set n where 71 record Base (L P : HOD) (u x : Ordinal) : Set n where
72 field 72 field
73 b : Ordinal 73 b : Ordinal
74 u⊂L : * u ⊂ L
74 sb : Subbase P b 75 sb : Subbase P b
75 b⊆u : * b ⊆ * u 76 b⊆u : * b ⊆ * u
76 bx : odef (* b) x 77 bx : odef (* b) x
77 78
78 record Base (P : HOD) (u : Ordinal) : Set n where 79 SO : (L P : HOD) → HOD
79 field 80 SO L P = record { od = record { def = λ u → {x : Ordinal } → odef (* u) x → Base L P u x } ; odmax = ? ; <odmax = ? }
80 ab : {x : Ordinal } → (ux : odef (* u) x ) → aBase P ux
81
82 SO : (P : HOD) → HOD
83 SO P = record { od = record { def = λ x → Base P x } ; odmax = ? ; <odmax = ? }
84 81
85 record IsSubBase (L P : HOD) : Set (suc n) where 82 record IsSubBase (L P : HOD) : Set (suc n) where
86 field 83 field
87 P⊆PL : P ⊆ Power L 84 P⊆PL : P ⊆ Power L
88 p : {x : HOD} → L ∋ x → HOD 85 p : {x : HOD} → L ∋ x → HOD
89 Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx 86 Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx
90 px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x 87 px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x
91 88
92 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L 89 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L
93 GeneratedTopogy L P isb = record { OS = SO P ; OS⊆PL = ? 90 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00
94 ; o∪ = ? ; o∩ = ? } 91 ; o∪ = tp02 ; o∩ = tp01 } where
92 tp00 : SO L P ⊆ Power L
93 tp00 {u} ou x ux with ou ux
94 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = proj2 u⊂L (b⊆u bx)
95 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q)
96 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊂ L) (sym *iso) ul
97 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = ? ; bx = ? } where
98 px : odef (* (& p)) x
99 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) )
100 qx : odef (* (& q)) x
101 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) )
102 b : Ordinal
103 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx)))
104 ul : (p ∩ q) ⊂ L
105 ul = subst (λ k → k ⊂ L ) *iso ⟪ ? , (λ {z} pq → IsSubBase.P⊆PL isb ? _ pq ) ⟫
106 tp02 : { q : HOD} → q ⊂ SO L P → SO L P ∋ Union q
107 tp02 {q} q⊂O {x} ux with subst (λ k → odef k x) *iso ux
108 ... | record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx
109 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = ? ; u⊂L = ? ; sb = ? ; b⊆u = ? ; bx = ? }
95 110
96 -- covers 111 -- covers
97 112
98 record _covers_ ( P q : HOD ) : Set (suc n) where 113 record _covers_ ( P q : HOD ) : Set (suc n) where
99 field 114 field