Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison BAlgbra.agda @ 369:17adeeee0c2a

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fix Select and Replace
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 19 Jul 2020 10:02:43 +0900
parents 2a8a51375e49
children 6c72bee25653
comparison
equal deleted inserted replaced
368:30de2d9b93c1 369:17adeeee0c2a
1 {-# OPTIONS --allow-unsolved-metas #-}
1 open import Level 2 open import Level
2 open import Ordinals 3 open import Ordinals
3 module BAlgbra {n : Level } (O : Ordinals {n}) where 4 module BAlgbra {n : Level } (O : Ordinals {n}) where
4 5
5 open import zf 6 open import zf
53 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A 54 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
54 (record { proj1 = case1 refl ; proj2 = subst (λ k → odef A k) (sym diso) A∋x})) 55 (record { proj1 = case1 refl ; proj2 = subst (λ k → odef A k) (sym diso) A∋x}))
55 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B 56 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
56 (record { proj1 = case2 refl ; proj2 = subst (λ k → odef B k) (sym diso) B∋x})) 57 (record { proj1 = case2 refl ; proj2 = subst (λ k → odef B k) (sym diso) B∋x}))
57 58
58 ∩-Select : { A B : HOD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B ) 59 ∩-Select : { A B : HOD } → Select A ( λ x _ → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B )
59 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 60 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
60 lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x 61 lemma1 : {x : Ordinal} → odef (Select A (λ x₁ _ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x
61 lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → odef B k ) diso (proj2 (proj2 lt)) } 62 lemma1 {x} lt = record { proj1 = proj1 {!!} ; proj2 = subst (λ k → odef B k ) diso (proj2 (proj2 {!!} )) }
62 lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x 63 lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ _ → (A ∋ x₁) ∧ (B ∋ x₁))) x
63 lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 = 64 lemma2 {x} lt = {!!} -- record { proj1 = proj1 lt ; proj2 =
64 record { proj1 = subst (λ k → odef A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → odef B k ) (sym diso) (proj2 lt) } } 65 -- record { proj1 = subst (λ k → odef A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → odef B k ) (sym diso) (proj2 lt) } }
65 66
66 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r ) 67 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
67 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 68 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
68 lemma1 : {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x 69 lemma1 : {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x
69 lemma1 {x} lt with proj2 lt 70 lemma1 {x} lt with proj2 lt