Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff filter.agda @ 272:985a1af11bce
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 |
line wrap: on
line diff
--- a/filter.agda Mon Dec 30 23:45:59 2019 +0900 +++ b/filter.agda Tue Dec 31 11:22:52 2019 +0900 @@ -31,80 +31,6 @@ _\_ : ( A B : OD ) → OD A \ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) } -∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B ) -∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x - lemma1 {x} lt = lemma3 lt where - lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) ) - lemma4 {y} z with proj1 z - lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) ) - lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) ) - lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x - lemma3 not = double-neg-eilm (FExists _ lemma4 not) -- choice - lemma2 : {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x - lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A - (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x})) - lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B - (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x})) - -∩-Select : { A B : OD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B ) -∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x - lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) } - lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x - lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 = - record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } } - -dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r ) -dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x - lemma1 {x} lt with proj2 lt - lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } ) - lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } ) - lemma2 : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x - lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) } - lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) } - -dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r ) -dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x - lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp } - lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) } - lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x - lemma2 {x} lt with proj1 lt | proj2 lt - lemma2 {x} lt | case1 cp | _ = case1 cp - lemma2 {x} lt | _ | case1 cp = case1 cp - lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } ) - -record IsBooleanAlgebra ( L : Set n) - ( b1 : L ) - ( b0 : L ) - ( -_ : L → L ) - ( _+_ : L → L → L ) - ( _*_ : L → L → L ) : Set (suc n) where - field - +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c - *-assoc : {a b c : L } → a * ( b * c ) ≡ (a * b) * c - +-sym : {a b : L } → a + b ≡ b + a - -sym : {a b : L } → a * b ≡ b * a - -aab : {a b : L } → a + ( a * b ) ≡ a - *-aab : {a b : L } → a * ( a + b ) ≡ a - -dist : {a b c : L } → a + ( b * c ) ≡ ( a * b ) + ( a * c ) - *-dist : {a b c : L } → a * ( b + c ) ≡ ( a + b ) * ( a + c ) - a+0 : {a : L } → a + b0 ≡ a - a*1 : {a : L } → a * b1 ≡ a - a+-a1 : {a : L } → a + ( - a ) ≡ b1 - a*-a0 : {a : L } → a * ( - a ) ≡ b0 - -record BooleanAlgebra ( L : Set n) : Set (suc n) where - field - b1 : L - b0 : L - -_ : L → L - _++_ : L → L → L - _**_ : L → L → L - isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _++_ _**_ - record Filter ( L : OD ) : Set (suc n) where field