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

comparison src/BAlgbra.agda @ 1096:55ab5de1ae02

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 23 Dec 2022 12:54:05 +0900
parents 6c22ee73ff06
children 256a3ba634f6
comparison
equal deleted inserted replaced
1095:08b6aa6870d9 1096:55ab5de1ae02
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 59
60 U-F=∅→F⊆U : {F U : HOD} → ((x : Ordinal) → ¬ ( odef F x ∧ ( ¬ odef U x ))) → F ⊆ U 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 61 U-F=∅→F⊆U {F} {U} not = gt02 where
62 gt02 : { x : Ordinal } → odef F x → odef U x 62 gt02 : { x : Ordinal } → odef F x → odef U x
63 gt02 {x} fx with ODC.∋-p O U (* x) 63 gt02 {x} fx with ODC.∋-p O U (* x)
64 ... | yes y = subst (λ k → odef U k ) &iso y 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 ⟫ ) 65 ... | no n = ⊥-elim ( not x ⟪ fx , subst (λ k → ¬ odef U k ) &iso n ⟫ )
66 66
67 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B ) 67 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
68 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 68 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
69 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
70 lemma1 {x} lt = lemma3 lt where 70 lemma1 {x} record { owner = owner ; ao = abo ; ox = ox } with pair=∨ (subst₂ (λ j k → odef (j , k ) owner) (sym *iso) (sym *iso) abo )
71 lemma4 : {y : Ordinal} → odef (A , B) y ∧ odef (* y) x → ¬ (¬ ( odef A x ∨ odef B x) ) 71 ... | case1 a=o = case1 (subst (λ k → odef k x ) ( begin
72 lemma4 {y} z with proj1 z 72 * owner ≡⟨ cong (*) (sym a=o) ⟩
73 lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → odef k x ) *iso (proj2 z)) ) 73 * (& A) ≡⟨ *iso ⟩
74 lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → odef k x ) *iso (proj2 z)) ) 74 A ∎ ) ox ) where open ≡-Reasoning
75 lemma3 : (((u : Ordinal ) → ¬ odef (A , B) u ∧ odef (* u) x) → ⊥) → odef (A ∪ B) x 75 ... | case2 b=o = case2 (subst (λ k → odef k x ) (trans (cong (*) (sym b=o)) *iso ) ox)
76 lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not) -- choice
77 lemma2 : {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x 76 lemma2 : {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x
78 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) A 77 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) A
79 ⟪ case1 refl , d→∋ A A∋x ⟫ ) 78 ⟪ case1 refl , d→∋ A A∋x ⟫ )
80 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) B 79 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) B
81 ⟪ case2 refl , d→∋ B B∋x ⟫ ) 80 ⟪ case2 refl , d→∋ B B∋x ⟫ )