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comparison src/Topology.agda @ 1117:53ca3c609f0e

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generated topology from subbase done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 01 Jan 2023 20:07:04 +0900
parents 6386019deef1
children 441ad880a45d
comparison
equal deleted inserted replaced
1116:6386019deef1 1117:53ca3c609f0e
69 -- OS = { U ⊂ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊂ U } 69 -- OS = { U ⊂ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊂ U }
70 70
71 record Base (L P : HOD) (u x : Ordinal) : 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 u⊂L : * u ⊆ L
75 sb : Subbase P b 75 sb : Subbase P b
76 b⊆u : * b ⊆ * u 76 b⊆u : * b ⊆ * u
77 bx : odef (* b) x 77 bx : odef (* b) x
78 78
79 SO : (L P : HOD) → HOD 79 SO : (L P : HOD) → HOD
91 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L 91 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L
92 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00 92 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00
93 ; o∪ = tp02 ; o∩ = tp01 } where 93 ; o∪ = tp02 ; o∩ = tp01 } where
94 tp00 : SO L P ⊆ Power L 94 tp00 : SO L P ⊆ Power L
95 tp00 {u} ou x ux with ou ux 95 tp00 {u} ou x ux with ou ux
96 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = proj2 u⊂L (b⊆u bx) 96 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊂L (b⊆u bx)
97 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) 97 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q)
98 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊂ L) (sym *iso) ul 98 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) ul
99 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where 99 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where
100 px : odef (* (& p)) x 100 px : odef (* (& p)) x
101 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) 101 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) )
102 qx : odef (* (& q)) x 102 qx : odef (* (& q)) x
103 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) 103 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) )
113 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) 113 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11)
114 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q 114 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q
115 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) 115 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12)
116 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x 116 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x
117 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ 117 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫
118 ul : (p ∩ q) ⊂ L 118 ul : (p ∩ q) ⊆ L
119 ul = subst (λ k → k ⊂ L ) *iso ⟪ tp02 , (λ {z} pq → proj2 (Base.u⊂L (op px)) (pz pq) ) ⟫ where 119 ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊂L (op px)) (pz pq) ) where
120 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z 120 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z
121 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) 121 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) )
122 tp02 : & (* (& (p ∩ q))) o< & L
123 tp02 = subst ( λ k → k o< & L) (sym &iso) ?
124 tp02 : { q : HOD} → q ⊂ SO L P → SO L P ∋ Union q 122 tp02 : { q : HOD} → q ⊂ SO L P → SO L P ∋ Union q
125 tp02 {q} q⊂O {x} ux with subst (λ k → odef k x) *iso ux 123 tp02 {q} q⊂O {x} ux with subst (λ k → odef k x) *iso ux
126 ... | record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx 124 ... | record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx
127 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = b ; u⊂L = subst (λ k → k ⊂ L) (sym *iso) tp04 125 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) tp04
128 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where 126 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where
129 tp05 : Union q ⊆ L 127 tp05 : Union q ⊆ L
130 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx 128 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx
131 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } 129 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx }
132 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) 130 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx ))
133 tp04 : Union q ⊂ L 131 tp04 : Union q ⊆ L
134 tp04 = ⟪ ? , tp05 ⟫ 132 tp04 = tp05
135 tp06 : * b ⊆ Union q 133 tp06 : * b ⊆ Union q
136 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } 134 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz }
137 135
138 -- covers 136 -- covers
139 137