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

comparison src/BAlgbra.agda @ 480:6c22ee73ff06

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 03 Apr 2022 17:53:13 +0900
parents 31f0a5a745a5
children 55ab5de1ae02
comparison
equal deleted inserted replaced
479:fea0c2454b85 480:6c22ee73ff06
54 ¬∅∋ {x} = ¬x<0 54 ¬∅∋ {x} = ¬x<0
55 55
56 [a-b]∩b=0 : { A B : HOD } → (A ＼ B) ∩ B ≡ od∅ 56 [a-b]∩b=0 : { A B : HOD } → (A ＼ B) ∩ B ≡ od∅
57 [a-b]∩b=0 {A} {B} = ==→o≡ record { eq← = λ lt → ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt )) 57 [a-b]∩b=0 {A} {B} = ==→o≡ record { eq← = λ lt → ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt ))
58 ; eq→ = λ {x} lt → ⊥-elim (proj2 (proj1 lt ) (proj2 lt)) } 58 ; eq→ = λ {x} lt → ⊥-elim (proj2 (proj1 lt ) (proj2 lt)) }
59
60 U-F=∅→F⊆U : {F U : HOD} → ((x : Ordinal) → ¬ ( odef F x ∧ ( ¬ odef U x ))) → F ⊆ U
61 U-F=∅→F⊆U {F} {U} not = record { incl = gt02 } where
62 gt02 : { x : Ordinal } → odef F x → odef U x
63 gt02 {x} fx with ODC.∋-p O U (* x)
64 ... | yes y = subst (λ k → odef U k ) &iso y
65 ... | no n = ⊥-elim ( not x ⟪ fx , subst (λ k → ¬ odef U k ) &iso n ⟫ )
59 66
60 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B ) 67 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
61 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 68 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
62 lemma1 : {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x 69 lemma1 : {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x
63 lemma1 {x} lt = lemma3 lt where 70 lemma1 {x} lt = lemma3 lt where