Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/BAlgebra.agda @ 1280:a496dbb74a5f
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 06 Apr 2023 11:50:41 +0900 |
| parents | 5f1572d1f19a |
| children | 437e48221ed5 |
| rev | line source |
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| 1280 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 431 | 3 open import Level |
| 4 open import Ordinals | |
| 1124 | 5 module BAlgebra {n : Level } (O : Ordinals {n}) where |
| 431 | 6 |
| 7 open import zf | |
| 8 open import logic | |
| 9 import OrdUtil | |
| 10 import OD | |
| 11 import ODUtil | |
| 12 import ODC | |
| 13 | |
| 14 open import Relation.Nullary | |
| 15 open import Relation.Binary | |
| 16 open import Data.Empty | |
| 17 open import Relation.Binary | |
| 18 open import Relation.Binary.Core | |
| 19 open import Relation.Binary.PropositionalEquality | |
| 20 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ; _+_ to _n+_ ) | |
| 21 | |
| 22 open inOrdinal O | |
| 23 open Ordinals.Ordinals O | |
| 24 open Ordinals.IsOrdinals isOrdinal | |
| 25 open Ordinals.IsNext isNext | |
| 26 open OrdUtil O | |
| 27 open ODUtil O | |
| 28 | |
| 29 open OD O | |
| 30 open OD.OD | |
| 31 open ODAxiom odAxiom | |
| 32 open HOD | |
| 33 | |
| 34 open _∧_ | |
| 35 open _∨_ | |
| 36 open Bool | |
| 37 | |
| 1123 | 38 L＼L=0 : { L : HOD } → L ＼ L ≡ od∅ |
| 39 L＼L=0 {L} = ==→o≡ ( record { eq→ = lem0 ; eq← = lem1 } ) where | |
| 40 lem0 : {x : Ordinal} → odef (L ＼ L) x → odef od∅ x | |
| 41 lem0 {x} ⟪ lx , ¬lx ⟫ = ⊥-elim (¬lx lx) | |
| 42 lem1 : {x : Ordinal} → odef od∅ x → odef (L ＼ L) x | |
| 43 lem1 {x} lt = ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt )) | |
| 44 | |
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45 L＼Lx=x : { L x : HOD } → x ⊆ L → L ＼ ( L ＼ x ) ≡ x |
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46 L＼Lx=x {L} {x} x⊆L = ==→o≡ ( record { eq→ = lem03 ; eq← = lem04 } ) where |
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47 lem03 : {z : Ordinal} → odef (L ＼ (L ＼ x)) z → odef x z |
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48 lem03 {z} ⟪ Lz , Lxz ⟫ with ODC.∋-p O x (* z) |
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49 ... | yes y = subst (λ k → odef x k ) &iso y |
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50 ... | no n = ⊥-elim ( Lxz ⟪ Lz , ( subst (λ k → ¬ odef x k ) &iso n ) ⟫ ) |
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51 lem04 : {z : Ordinal} → odef x z → odef (L ＼ (L ＼ x)) z |
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52 lem04 {z} xz with ODC.∋-p O L (* z) |
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53 ... | yes y = ⟪ subst (λ k → odef L k ) &iso y , ( λ p → proj2 p xz ) ⟫ |
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54 ... | no n = ⊥-elim ( n (subst (λ k → odef L k ) (sym &iso) ( x⊆L xz) )) |
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55 |
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56 L＼0=L : { L : HOD } → L ＼ od∅ ≡ L |
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57 L＼0=L {L} = ==→o≡ ( record { eq→ = lem05 ; eq← = lem06 } ) where |
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58 lem05 : {x : Ordinal} → odef (L ＼ od∅) x → odef L x |
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59 lem05 {x} ⟪ Lx , _ ⟫ = Lx |
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60 lem06 : {x : Ordinal} → odef L x → odef (L ＼ od∅) x |
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61 lem06 {x} Lx = ⟪ Lx , (λ lt → ¬x<0 lt) ⟫ |
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62 |
| 1182 | 63 ∨L＼X : { L X : HOD } → {x : Ordinal } → odef L x → odef X x ∨ odef (L ＼ X) x |
| 64 ∨L＼X {L} {X} {x} Lx with ODC.∋-p O X (* x) | |
| 65 ... | yes y = case1 ( subst (λ k → odef X k ) &iso y ) | |
| 66 ... | no n = case2 ⟪ Lx , subst (λ k → ¬ odef X k) &iso n ⟫ | |
| 67 | |
| 1241 | 68 ＼-⊆ : { P A B : HOD } → A ⊆ P → ( A ⊆ B → ( P ＼ B ) ⊆ ( P ＼ A )) ∧ (( P ＼ B ) ⊆ ( P ＼ A ) → A ⊆ B ) |
| 69 ＼-⊆ {P} {A} {B} A⊆P = ⟪ ( λ a<b {x} pbx → ⟪ proj1 pbx , (λ ax → proj2 pbx (a<b ax)) ⟫ ) , lem07 ⟫ where | |
| 70 lem07 : (P ＼ B) ⊆ (P ＼ A) → A ⊆ B | |
| 71 lem07 pba {x} ax with ODC.p∨¬p O (odef B x) | |
| 72 ... | case1 bx = bx | |
| 73 ... | case2 ¬bx = ⊥-elim ( proj2 ( pba ⟪ A⊆P ax , ¬bx ⟫ ) ax ) | |
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74 |
| 451 | 75 [a-b]∩b=0 : { A B : HOD } → (A ＼ B) ∩ B ≡ od∅ |
| 76 [a-b]∩b=0 {A} {B} = ==→o≡ record { eq← = λ lt → ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt )) | |
| 77 ; eq→ = λ {x} lt → ⊥-elim (proj2 (proj1 lt ) (proj2 lt)) } | |
| 78 | |
| 480 | 79 U-F=∅→F⊆U : {F U : HOD} → ((x : Ordinal) → ¬ ( odef F x ∧ ( ¬ odef U x ))) → F ⊆ U |
| 1096 | 80 U-F=∅→F⊆U {F} {U} not = gt02 where |
| 480 | 81 gt02 : { x : Ordinal } → odef F x → odef U x |
| 82 gt02 {x} fx with ODC.∋-p O U (* x) | |
| 83 ... | yes y = subst (λ k → odef U k ) &iso y | |
| 84 ... | no n = ⊥-elim ( not x ⟪ fx , subst (λ k → ¬ odef U k ) &iso n ⟫ ) | |
| 85 | |
| 431 | 86 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B ) |
| 87 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 88 lemma1 : {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x | |
| 1096 | 89 lemma1 {x} record { owner = owner ; ao = abo ; ox = ox } with pair=∨ (subst₂ (λ j k → odef (j , k ) owner) (sym *iso) (sym *iso) abo ) |
| 90 ... | case1 a=o = case1 (subst (λ k → odef k x ) ( begin | |
| 91 * owner ≡⟨ cong (*) (sym a=o) ⟩ | |
| 92 * (& A) ≡⟨ *iso ⟩ | |
| 93 A ∎ ) ox ) where open ≡-Reasoning | |
| 94 ... | case2 b=o = case2 (subst (λ k → odef k x ) (trans (cong (*) (sym b=o)) *iso ) ox) | |
| 431 | 95 lemma2 : {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x |
| 96 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) A | |
| 97 ⟪ case1 refl , d→∋ A A∋x ⟫ ) | |
| 98 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) B | |
| 99 ⟪ case2 refl , d→∋ B B∋x ⟫ ) | |
| 100 | |
| 101 ∩-Select : { A B : HOD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B ) | |
| 102 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 103 lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x | |
| 104 lemma1 {x} lt = ⟪ proj1 lt , subst (λ k → odef B k ) &iso (proj2 (proj2 lt)) ⟫ | |
| 105 lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x | |
| 106 lemma2 {x} lt = ⟪ proj1 lt , ⟪ d→∋ A (proj1 lt) , d→∋ B (proj2 lt) ⟫ ⟫ | |
| 107 | |
| 108 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r ) | |
| 109 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 110 lemma1 : {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x | |
| 111 lemma1 {x} lt with proj2 lt | |
| 112 lemma1 {x} lt | case1 q∋x = case1 ⟪ proj1 lt , q∋x ⟫ | |
| 113 lemma1 {x} lt | case2 r∋x = case2 ⟪ proj1 lt , r∋x ⟫ | |
| 114 lemma2 : {x : Ordinal} → odef ((p ∩ q) ∪ (p ∩ r)) x → odef (p ∩ (q ∪ r)) x | |
| 115 lemma2 {x} (case1 p∩q) = ⟪ proj1 p∩q , case1 (proj2 p∩q ) ⟫ | |
| 116 lemma2 {x} (case2 p∩r) = ⟪ proj1 p∩r , case2 (proj2 p∩r ) ⟫ | |
| 117 | |
| 118 dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r ) | |
| 119 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 120 lemma1 : {x : Ordinal} → odef (p ∪ (q ∩ r)) x → odef ((p ∪ q) ∩ (p ∪ r)) x | |
| 121 lemma1 {x} (case1 cp) = ⟪ case1 cp , case1 cp ⟫ | |
| 122 lemma1 {x} (case2 cqr) = ⟪ case2 (proj1 cqr) , case2 (proj2 cqr) ⟫ | |
| 123 lemma2 : {x : Ordinal} → odef ((p ∪ q) ∩ (p ∪ r)) x → odef (p ∪ (q ∩ r)) x | |
| 124 lemma2 {x} lt with proj1 lt | proj2 lt | |
| 125 lemma2 {x} lt | case1 cp | _ = case1 cp | |
| 126 lemma2 {x} lt | _ | case1 cp = case1 cp | |
| 127 lemma2 {x} lt | case2 cq | case2 cr = case2 ⟪ cq , cr ⟫ | |
| 128 | |
| 1280 | 129 record IsBooleanAlgebra {n : Level} ( L : Set n) |
| 130 ( _==_ : L → L → Set n ) | |
| 431 | 131 ( b1 : L ) |
| 132 ( b0 : L ) | |
| 133 ( -_ : L → L ) | |
| 134 ( _+_ : L → L → L ) | |
| 135 ( _x_ : L → L → L ) : Set (suc n) where | |
| 136 field | |
| 1280 | 137 +-assoc : {a b c : L } → (a + ( b + c )) == ((a + b) + c) |
| 138 x-assoc : {a b c : L } → (a x ( b x c )) == ((a x b) x c) | |
| 139 +-sym : {a b : L } → (a + b) == (b + a) | |
| 140 -sym : {a b : L } → (a x b) == (b x a) | |
| 141 +-aab : {a b : L } → (a + ( a x b )) == a | |
| 142 x-aab : {a b : L } → (a x ( a + b )) == a | |
| 143 +-dist : {a b c : L } → (a + ( b x c )) == (( a + b ) x ( a + c )) | |
| 144 x-dist : {a b c : L } → (a x ( b + c )) == (( a x b ) + ( a x c )) | |
| 145 a+0 : {a : L } → (a + b0) == a | |
| 146 ax1 : {a : L } → (a x b1) == a | |
| 147 a+-a1 : {a : L } → (a + ( - a )) == b1 | |
| 148 ax-a0 : {a : L } → (a x ( - a )) == b0 | |
| 431 | 149 |
| 1280 | 150 record BooleanAlgebra {n : Level} ( L : Set n) : Set (suc n) where |
| 431 | 151 field |
| 1280 | 152 _=b=_ : L → L → Set n |
| 431 | 153 b1 : L |
| 154 b0 : L | |
| 155 -_ : L → L | |
| 156 _+_ : L → L → L | |
| 157 _x_ : L → L → L | |
| 1280 | 158 isBooleanAlgebra : IsBooleanAlgebra L _=b=_ b1 b0 -_ _+_ _x_ |
| 159 | |
| 160 record PowerP (P : HOD) : Set (suc n) where | |
| 161 constructor ⟦_,_⟧ | |
| 162 field | |
| 163 hod : HOD | |
| 164 x⊆P : hod ⊆ P | |
| 165 | |
| 166 open PowerP | |
| 167 | |
| 168 HODBA : (P : HOD) → BooleanAlgebra (PowerP P) | |
| 169 HODBA P = record { _=b=_ = λ x y → hod x ≡ hod y ; b1 = ⟦ P , (λ x → x) ⟧ ; b0 = ⟦ od∅ , (λ x → ⊥-elim (¬x<0 x)) ⟧ | |
| 170 ; -_ = λ x → ⟦ P ＼ hod x , proj1 ⟧ | |
| 171 ; _+_ = λ x y → ⟦ hod x ∪ hod y , ? ⟧ ; _x_ = λ x y → ⟦ hod x ∩ hod y , (λ lt → x⊆P x (proj1 lt)) ⟧ | |
| 172 ; isBooleanAlgebra = record { | |
| 173 +-assoc = λ {a} {b} {c} → ==→o≡ record { eq→ = ? ; eq← = ? } | |
| 174 ; x-assoc = λ {a} {b} {c} → ==→o≡ record { eq→ = λ lt → ⟪ ⟪ proj1 lt , proj1 (proj2 lt) ⟫ , proj2 (proj2 lt) ⟫ | |
| 175 ; eq← = λ lt → ⟪ proj1 (proj1 lt) , ⟪ proj2 (proj1 lt) , proj2 lt ⟫ ⟫ } | |
| 176 ; +-sym = ? | |
| 177 ; -sym = λ {a} {b} → ==→o≡ record { eq→ = λ lt → ⟪ proj2 lt , proj1 lt ⟫ ; eq← = λ lt → ⟪ proj2 lt , proj1 lt ⟫ } | |
| 178 ; +-aab = λ {a} {b} → ==→o≡ record { eq→ = ? ; eq← = case1 } | |
| 179 ; x-aab = λ {a} {b} → ==→o≡ record { eq→ = proj1 ; eq← = λ ax → ⟪ ax , case1 ax ⟫ } | |
| 180 ; +-dist = dist-ord2 | |
| 181 ; x-dist = dist-ord | |
| 182 ; a+0 = ? | |
| 183 ; ax1 = ? | |
| 184 ; a+-a1 = ? | |
| 185 ; ax-a0 = ? } } | |
| 186 |