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

comparison filter.agda @ 269:30e419a2be24

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disjunction and conjunction
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 06 Oct 2019 16:42:42 +0900
parents 7b4a66710cdd
children fc3d4bc1dc5e
comparison
equal deleted inserted replaced
268:7b4a66710cdd 269:30e419a2be24
24 24
25 _∩_ : ( A B : OD ) → OD 25 _∩_ : ( A B : OD ) → OD
26 A ∩ B = record { def = λ x → def A x ∧ def B x } 26 A ∩ B = record { def = λ x → def A x ∧ def B x }
27 27
28 _∪_ : ( A B : OD ) → OD 28 _∪_ : ( A B : OD ) → OD
29 A ∪ B = Union (A , B) 29 A ∪ B = record { def = λ x → def A x ∨ def B x }
30
31 ∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B )
32 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
33 lemma1 : {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
34 lemma1 {x} lt = lemma3 lt where
35 lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) )
36 lemma4 {y} z with proj1 z
37 lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) )
38 lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) )
39 lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x
40 lemma3 not = double-neg-eilm (FExists _ lemma4 not)
41 lemma2 : {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x
42 lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
43 (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x}))
44 lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
45 (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
46
47 ∩-Select : { A B : OD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B )
48 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
49 lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
50 lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
51 lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
52 lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
53 record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
54
55 dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
56 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
57 lemma1 : {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x
58 lemma1 {x} lt with proj2 lt
59 lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } )
60 lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } )
61 lemma2 : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x
62 lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) }
63 lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) }
64
65 dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r )
66 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
67 lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x
68 lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }
69 lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) }
70 lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x
71 lemma2 {x} lt with proj1 lt | proj2 lt
72 lemma2 {x} lt | case1 cp | _ = case1 cp
73 lemma2 {x} lt | _ | case1 cp = case1 cp
74 lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } )
30 75
31 record Filter ( L : OD ) : Set (suc n) where 76 record Filter ( L : OD ) : Set (suc n) where
32 field 77 field
33 F1 : { p q : OD } → L ∋ p → ({ x : OD} → _⊆_ q p {x} ) → L ∋ q 78 F1 : { p q : OD } → L ∋ p → ({ x : OD} → _⊆_ p q {x} ) → L ∋ q
34 F2 : { p q : OD } → L ∋ p → L ∋ q → L ∋ (p ∩ q) 79 F2 : { p q : OD } → L ∋ p → L ∋ q → L ∋ (p ∩ q)
35 80
36 open Filter 81 open Filter
37 82
38 proper-filter : {L : OD} → Filter L → Set n 83 proper-filter : {L : OD} → Filter L → Set n
42 prime-filter {L} P {p} {q} = L ∋ ( p ∪ q) → ( L ∋ p ) ∨ ( L ∋ q ) 87 prime-filter {L} P {p} {q} = L ∋ ( p ∪ q) → ( L ∋ p ) ∨ ( L ∋ q )
43 88
44 ultra-filter : {L : OD} → Filter L → {p : OD } → Set n 89 ultra-filter : {L : OD} → Filter L → {p : OD } → Set n
45 ultra-filter {L} P {p} = ( L ∋ p ) ∨ ( ¬ ( L ∋ p )) 90 ultra-filter {L} P {p} = ( L ∋ p ) ∨ ( ¬ ( L ∋ p ))
46 91
47 postulate
48 dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
49 92
50 filter-lemma1 : {L : OD} → (P : Filter L) → {p q : OD } → ( (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} 93 filter-lemma1 : {L : OD} → (P : Filter L) → {p q : OD } → ( (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q}
51 filter-lemma1 {L} P {p} {q} u lt with u p | u q 94 filter-lemma1 {L} P {p} {q} u lt with u p | u q
52 filter-lemma1 {L} P {p} {q} u lt | case1 x | case1 y = case1 x 95 filter-lemma1 {L} P {p} {q} u lt | case1 x | case1 y = case1 x
53 filter-lemma1 {L} P {p} {q} u lt | case1 x | case2 y = case1 x 96 filter-lemma1 {L} P {p} {q} u lt | case1 x | case2 y = case1 x
59 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } ) 102 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } )
60 generated-filter {L} P p = record { 103 generated-filter {L} P p = record {
61 F1 = {!!} ; F2 = {!!} 104 F1 = {!!} ; F2 = {!!}
62 } 105 }
63 106
107 record Dense (P : OD ) : Set (suc n) where
108 field
109 dense : OD
110 incl : { x : OD} → _⊆_ dense P {x}
111 dense-f : OD → OD
112 dense-p : { p x : OD} → P ∋ p → _⊆_ p (dense-f p) {x}
113
64 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) 114 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω))
65 115
66 infinite = ZF.infinite OD→ZF 116 infinite = ZF.infinite OD→ZF
67 117
68 Hω2 : Filter (Power (Power infinite)) 118 module in-countable-ordinal {n : Level} where
69 Hω2 = record { F1 = {!!} ; F2 = {!!} }
70 119
120 import ordinal
121
122 open ordinal.C-Ordinal-with-choice
123
124 Hω2 : Filter (Power (Power infinite))
125 Hω2 = record { F1 = {!!} ; F2 = {!!} }
126