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separate ordered pair and Boolean Algebra
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 31 Dec 2019 11:22:52 +0900
parents 2169d948159b
children 6f10c47e4e7a
comparison
equal deleted inserted replaced
271:2169d948159b 272:985a1af11bce
29 A ∪ B = record { def = λ x → def A x ∨ def B x } 29 A ∪ B = record { def = λ x → def A x ∨ def B x }
30 30
31 _＼_ : ( A B : OD ) → OD 31 _＼_ : ( A B : OD ) → OD
32 A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) } 32 A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
33 33
34 ∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B )
35 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
36 lemma1 : {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
37 lemma1 {x} lt = lemma3 lt where
38 lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) )
39 lemma4 {y} z with proj1 z
40 lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) )
41 lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) )
42 lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x
43 lemma3 not = double-neg-eilm (FExists _ lemma4 not) -- choice
44 lemma2 : {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x
45 lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
46 (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x}))
47 lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
48 (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
49
50 ∩-Select : { A B : OD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B )
51 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
52 lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
53 lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
54 lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
55 lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
56 record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
57
58 dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
59 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
60 lemma1 : {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x
61 lemma1 {x} lt with proj2 lt
62 lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } )
63 lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } )
64 lemma2 : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x
65 lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) }
66 lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) }
67
68 dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r )
69 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
70 lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x
71 lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }
72 lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) }
73 lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x
74 lemma2 {x} lt with proj1 lt | proj2 lt
75 lemma2 {x} lt | case1 cp | _ = case1 cp
76 lemma2 {x} lt | _ | case1 cp = case1 cp
77 lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } )
78
79 record IsBooleanAlgebra ( L : Set n)
80 ( b1 : L )
81 ( b0 : L )
82 ( -_ : L → L )
83 ( _+_ : L → L → L )
84 ( _*_ : L → L → L ) : Set (suc n) where
85 field
86 +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c
87 *-assoc : {a b c : L } → a * ( b * c ) ≡ (a * b) * c
88 +-sym : {a b : L } → a + b ≡ b + a
89 -sym : {a b : L } → a * b ≡ b * a
90 -aab : {a b : L } → a + ( a * b ) ≡ a
91 *-aab : {a b : L } → a * ( a + b ) ≡ a
92 -dist : {a b c : L } → a + ( b * c ) ≡ ( a * b ) + ( a * c )
93 *-dist : {a b c : L } → a * ( b + c ) ≡ ( a + b ) * ( a + c )
94 a+0 : {a : L } → a + b0 ≡ a
95 a*1 : {a : L } → a * b1 ≡ a
96 a+-a1 : {a : L } → a + ( - a ) ≡ b1
97 a*-a0 : {a : L } → a * ( - a ) ≡ b0
98
99 record BooleanAlgebra ( L : Set n) : Set (suc n) where
100 field
101 b1 : L
102 b0 : L
103 -_ : L → L
104 _++_ : L → L → L
105 _**_ : L → L → L
106 isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _++_ _**_
107
108 34
109 record Filter ( L : OD ) : Set (suc n) where 35 record Filter ( L : OD ) : Set (suc n) where
110 field 36 field
111 filter : OD 37 filter : OD
112 proper : ¬ ( filter ∋ od∅ ) 38 proper : ¬ ( filter ∋ od∅ )