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annotate BAlgbra.agda @ 277:d9d3654baee1

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seperate choice from LEM
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 09 May 2020 09:38:21 +0900
parents 6f10c47e4e7a
children 5544f4921a44
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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272
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Ordinals
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 module BAlgbra {n : Level } (O : Ordinals {n}) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 open import zf
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 open import logic
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 import OD
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
8 import ODC
272
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open import Relation.Nullary
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 open import Relation.Binary
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import Data.Empty
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 open import Relation.Binary
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 open import Relation.Binary.Core
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 open import Relation.Binary.PropositionalEquality
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 open inOrdinal O
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 open OD O
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 open OD.OD
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
21 open ODAxiom odAxiom
272
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 open _∧_
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24 open _∨_
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 open Bool
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 _∩_ : ( A B : OD ) → OD
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 A ∩ B = record { def = λ x → def A x ∧ def B x }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 _∪_ : ( A B : OD ) → OD
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 A ∪ B = record { def = λ x → def A x ∨ def B x }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33 _＼_ : ( A B : OD ) → OD
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
34 A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 ∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 lemma1 : {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39 lemma1 {x} lt = lemma3 lt where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41 lemma4 {y} z with proj1 z
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42 lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
43 lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
44 lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
45 lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not) -- choice
272
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
46 lemma2 : {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47 lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
48 (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x}))
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
49 lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50 (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
52 ∩-Select : { A B : OD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
53 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54 lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55 lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
56 lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
57 lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
58 record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
60 dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
61 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
62 lemma1 : {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
63 lemma1 {x} lt with proj2 lt
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
64 lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
65 lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
66 lemma2 : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
67 lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
68 lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
69
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
70 dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
72 lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
73 lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
74 lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) }
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
75 lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
76 lemma2 {x} lt with proj1 lt | proj2 lt
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
77 lemma2 {x} lt | case1 cp | _ = case1 cp
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
78 lemma2 {x} lt | _ | case1 cp = case1 cp
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
79 lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
80
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
81 record IsBooleanAlgebra ( L : Set n)
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
82 ( b1 : L )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83 ( b0 : L )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
84 ( -_ : L → L )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
85 ( _+_ : L → L → L )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
86 ( _*_ : L → L → L ) : Set (suc n) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
87 field
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
88 +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
89 *-assoc : {a b c : L } → a * ( b * c ) ≡ (a * b) * c
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90 +-sym : {a b : L } → a + b ≡ b + a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
91 -sym : {a b : L } → a * b ≡ b * a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
92 -aab : {a b : L } → a + ( a * b ) ≡ a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
93 *-aab : {a b : L } → a * ( a + b ) ≡ a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
94 -dist : {a b c : L } → a + ( b * c ) ≡ ( a * b ) + ( a * c )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
95 *-dist : {a b c : L } → a * ( b + c ) ≡ ( a + b ) * ( a + c )
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96 a+0 : {a : L } → a + b0 ≡ a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97 a*1 : {a : L } → a * b1 ≡ a
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
98 a+-a1 : {a : L } → a + ( - a ) ≡ b1
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
99 a*-a0 : {a : L } → a * ( - a ) ≡ b0
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101 record BooleanAlgebra ( L : Set n) : Set (suc n) where
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
102 field
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 b1 : L
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104 b0 : L
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105 -_ : L → L
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
106 _++_ : L → L → L
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107 _**_ : L → L → L
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108 isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _++_ _**_
985a1af11bce separate ordered pair and Boolean Algebra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109