Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/BAlgebra.agda @ 1270:905311ffe7ec
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 26 Mar 2023 17:18:45 +0900 |
| parents | 5f1572d1f19a |
| children | a496dbb74a5f |
| rev | line source |
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| 431 | 1 open import Level |
| 2 open import Ordinals | |
| 1124 | 3 module BAlgebra {n : Level } (O : Ordinals {n}) where |
| 431 | 4 |
| 5 open import zf | |
| 6 open import logic | |
| 7 import OrdUtil | |
| 8 import OD | |
| 9 import ODUtil | |
| 10 import ODC | |
| 11 | |
| 12 open import Relation.Nullary | |
| 13 open import Relation.Binary | |
| 14 open import Data.Empty | |
| 15 open import Relation.Binary | |
| 16 open import Relation.Binary.Core | |
| 17 open import Relation.Binary.PropositionalEquality | |
| 18 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ; _+_ to _n+_ ) | |
| 19 | |
| 20 open inOrdinal O | |
| 21 open Ordinals.Ordinals O | |
| 22 open Ordinals.IsOrdinals isOrdinal | |
| 23 open Ordinals.IsNext isNext | |
| 24 open OrdUtil O | |
| 25 open ODUtil O | |
| 26 | |
| 27 open OD O | |
| 28 open OD.OD | |
| 29 open ODAxiom odAxiom | |
| 30 open HOD | |
| 31 | |
| 32 open _∧_ | |
| 33 open _∨_ | |
| 34 open Bool | |
| 35 | |
| 1123 | 36 L＼L=0 : { L : HOD } → L ＼ L ≡ od∅ |
| 37 L＼L=0 {L} = ==→o≡ ( record { eq→ = lem0 ; eq← = lem1 } ) where | |
| 38 lem0 : {x : Ordinal} → odef (L ＼ L) x → odef od∅ x | |
| 39 lem0 {x} ⟪ lx , ¬lx ⟫ = ⊥-elim (¬lx lx) | |
| 40 lem1 : {x : Ordinal} → odef od∅ x → odef (L ＼ L) x | |
| 41 lem1 {x} lt = ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt )) | |
| 42 | |
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43 L＼Lx=x : { L x : HOD } → x ⊆ L → L ＼ ( L ＼ x ) ≡ x |
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44 L＼Lx=x {L} {x} x⊆L = ==→o≡ ( record { eq→ = lem03 ; eq← = lem04 } ) where |
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45 lem03 : {z : Ordinal} → odef (L ＼ (L ＼ x)) z → odef x z |
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46 lem03 {z} ⟪ Lz , Lxz ⟫ with ODC.∋-p O x (* z) |
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47 ... | yes y = subst (λ k → odef x k ) &iso y |
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48 ... | no n = ⊥-elim ( Lxz ⟪ Lz , ( subst (λ k → ¬ odef x k ) &iso n ) ⟫ ) |
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49 lem04 : {z : Ordinal} → odef x z → odef (L ＼ (L ＼ x)) z |
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50 lem04 {z} xz with ODC.∋-p O L (* z) |
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51 ... | yes y = ⟪ subst (λ k → odef L k ) &iso y , ( λ p → proj2 p xz ) ⟫ |
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52 ... | no n = ⊥-elim ( n (subst (λ k → odef L k ) (sym &iso) ( x⊆L xz) )) |
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53 |
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54 L＼0=L : { L : HOD } → L ＼ od∅ ≡ L |
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55 L＼0=L {L} = ==→o≡ ( record { eq→ = lem05 ; eq← = lem06 } ) where |
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56 lem05 : {x : Ordinal} → odef (L ＼ od∅) x → odef L x |
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57 lem05 {x} ⟪ Lx , _ ⟫ = Lx |
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58 lem06 : {x : Ordinal} → odef L x → odef (L ＼ od∅) x |
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59 lem06 {x} Lx = ⟪ Lx , (λ lt → ¬x<0 lt) ⟫ |
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60 |
| 1182 | 61 ∨L＼X : { L X : HOD } → {x : Ordinal } → odef L x → odef X x ∨ odef (L ＼ X) x |
| 62 ∨L＼X {L} {X} {x} Lx with ODC.∋-p O X (* x) | |
| 63 ... | yes y = case1 ( subst (λ k → odef X k ) &iso y ) | |
| 64 ... | no n = case2 ⟪ Lx , subst (λ k → ¬ odef X k) &iso n ⟫ | |
| 65 | |
| 1241 | 66 ＼-⊆ : { P A B : HOD } → A ⊆ P → ( A ⊆ B → ( P ＼ B ) ⊆ ( P ＼ A )) ∧ (( P ＼ B ) ⊆ ( P ＼ A ) → A ⊆ B ) |
| 67 ＼-⊆ {P} {A} {B} A⊆P = ⟪ ( λ a<b {x} pbx → ⟪ proj1 pbx , (λ ax → proj2 pbx (a<b ax)) ⟫ ) , lem07 ⟫ where | |
| 68 lem07 : (P ＼ B) ⊆ (P ＼ A) → A ⊆ B | |
| 69 lem07 pba {x} ax with ODC.p∨¬p O (odef B x) | |
| 70 ... | case1 bx = bx | |
| 71 ... | case2 ¬bx = ⊥-elim ( proj2 ( pba ⟪ A⊆P ax , ¬bx ⟫ ) ax ) | |
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72 |
| 451 | 73 [a-b]∩b=0 : { A B : HOD } → (A ＼ B) ∩ B ≡ od∅ |
| 74 [a-b]∩b=0 {A} {B} = ==→o≡ record { eq← = λ lt → ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt )) | |
| 75 ; eq→ = λ {x} lt → ⊥-elim (proj2 (proj1 lt ) (proj2 lt)) } | |
| 76 | |
| 480 | 77 U-F=∅→F⊆U : {F U : HOD} → ((x : Ordinal) → ¬ ( odef F x ∧ ( ¬ odef U x ))) → F ⊆ U |
| 1096 | 78 U-F=∅→F⊆U {F} {U} not = gt02 where |
| 480 | 79 gt02 : { x : Ordinal } → odef F x → odef U x |
| 80 gt02 {x} fx with ODC.∋-p O U (* x) | |
| 81 ... | yes y = subst (λ k → odef U k ) &iso y | |
| 82 ... | no n = ⊥-elim ( not x ⟪ fx , subst (λ k → ¬ odef U k ) &iso n ⟫ ) | |
| 83 | |
| 431 | 84 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B ) |
| 85 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 86 lemma1 : {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x | |
| 1096 | 87 lemma1 {x} record { owner = owner ; ao = abo ; ox = ox } with pair=∨ (subst₂ (λ j k → odef (j , k ) owner) (sym *iso) (sym *iso) abo ) |
| 88 ... | case1 a=o = case1 (subst (λ k → odef k x ) ( begin | |
| 89 * owner ≡⟨ cong (*) (sym a=o) ⟩ | |
| 90 * (& A) ≡⟨ *iso ⟩ | |
| 91 A ∎ ) ox ) where open ≡-Reasoning | |
| 92 ... | case2 b=o = case2 (subst (λ k → odef k x ) (trans (cong (*) (sym b=o)) *iso ) ox) | |
| 431 | 93 lemma2 : {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x |
| 94 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) A | |
| 95 ⟪ case1 refl , d→∋ A A∋x ⟫ ) | |
| 96 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) B | |
| 97 ⟪ case2 refl , d→∋ B B∋x ⟫ ) | |
| 98 | |
| 99 ∩-Select : { A B : HOD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B ) | |
| 100 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 101 lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x | |
| 102 lemma1 {x} lt = ⟪ proj1 lt , subst (λ k → odef B k ) &iso (proj2 (proj2 lt)) ⟫ | |
| 103 lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x | |
| 104 lemma2 {x} lt = ⟪ proj1 lt , ⟪ d→∋ A (proj1 lt) , d→∋ B (proj2 lt) ⟫ ⟫ | |
| 105 | |
| 106 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r ) | |
| 107 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 108 lemma1 : {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x | |
| 109 lemma1 {x} lt with proj2 lt | |
| 110 lemma1 {x} lt | case1 q∋x = case1 ⟪ proj1 lt , q∋x ⟫ | |
| 111 lemma1 {x} lt | case2 r∋x = case2 ⟪ proj1 lt , r∋x ⟫ | |
| 112 lemma2 : {x : Ordinal} → odef ((p ∩ q) ∪ (p ∩ r)) x → odef (p ∩ (q ∪ r)) x | |
| 113 lemma2 {x} (case1 p∩q) = ⟪ proj1 p∩q , case1 (proj2 p∩q ) ⟫ | |
| 114 lemma2 {x} (case2 p∩r) = ⟪ proj1 p∩r , case2 (proj2 p∩r ) ⟫ | |
| 115 | |
| 116 dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r ) | |
| 117 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 118 lemma1 : {x : Ordinal} → odef (p ∪ (q ∩ r)) x → odef ((p ∪ q) ∩ (p ∪ r)) x | |
| 119 lemma1 {x} (case1 cp) = ⟪ case1 cp , case1 cp ⟫ | |
| 120 lemma1 {x} (case2 cqr) = ⟪ case2 (proj1 cqr) , case2 (proj2 cqr) ⟫ | |
| 121 lemma2 : {x : Ordinal} → odef ((p ∪ q) ∩ (p ∪ r)) x → odef (p ∪ (q ∩ r)) x | |
| 122 lemma2 {x} lt with proj1 lt | proj2 lt | |
| 123 lemma2 {x} lt | case1 cp | _ = case1 cp | |
| 124 lemma2 {x} lt | _ | case1 cp = case1 cp | |
| 125 lemma2 {x} lt | case2 cq | case2 cr = case2 ⟪ cq , cr ⟫ | |
| 126 | |
| 127 record IsBooleanAlgebra ( L : Set n) | |
| 128 ( b1 : L ) | |
| 129 ( b0 : L ) | |
| 130 ( -_ : L → L ) | |
| 131 ( _+_ : L → L → L ) | |
| 132 ( _x_ : L → L → L ) : Set (suc n) where | |
| 133 field | |
| 134 +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c | |
| 135 x-assoc : {a b c : L } → a x ( b x c ) ≡ (a x b) x c | |
| 136 +-sym : {a b : L } → a + b ≡ b + a | |
| 137 -sym : {a b : L } → a x b ≡ b x a | |
| 138 +-aab : {a b : L } → a + ( a x b ) ≡ a | |
| 139 x-aab : {a b : L } → a x ( a + b ) ≡ a | |
| 140 +-dist : {a b c : L } → a + ( b x c ) ≡ ( a x b ) + ( a x c ) | |
| 141 x-dist : {a b c : L } → a x ( b + c ) ≡ ( a + b ) x ( a + c ) | |
| 142 a+0 : {a : L } → a + b0 ≡ a | |
| 143 ax1 : {a : L } → a x b1 ≡ a | |
| 144 a+-a1 : {a : L } → a + ( - a ) ≡ b1 | |
| 145 ax-a0 : {a : L } → a x ( - a ) ≡ b0 | |
| 146 | |
| 147 record BooleanAlgebra ( L : Set n) : Set (suc n) where | |
| 148 field | |
| 149 b1 : L | |
| 150 b0 : L | |
| 151 -_ : L → L | |
| 152 _+_ : L → L → L | |
| 153 _x_ : L → L → L | |
| 154 isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _+_ _x_ | |
| 155 |