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

comparison cardinal.agda @ 227:a4cdfc84f65f

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 11 Aug 2019 18:37:33 +0900
parents 176ff97547b4
children 49736efc822b
comparison
equal deleted inserted replaced
226:176ff97547b4 227:a4cdfc84f65f
46 lemma : Ordinal → Ordinal 46 lemma : Ordinal → Ordinal
47 lemma y with p∨¬p ( _⊗_.π1 lt ≡ x ) 47 lemma y with p∨¬p ( _⊗_.π1 lt ≡ x )
48 lemma y | case1 refl = _⊗_.π2 lt 48 lemma y | case1 refl = _⊗_.π2 lt
49 lemma y | case2 not = o∅ 49 lemma y | case2 not = o∅
50 50
51 -- contra position of sup-o<
52 --
53
54 record Sup ( ψ : Ordinal → Ordinal ) : Set n where
55 field
56 sup-x : Ordinal
57 sup-lb : {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ sup-x )
58
59 sup-o> : ( ψ : Ordinal → Ordinal ) → Sup ψ
60 sup-o> ψ = record {
61 sup-x = od→ord ( minimul (Ord (osuc (sup-o ψ))) lemma )
62 ; sup-lb = λ {z} z<sψ → lemma1 z<sψ
63 } where
64 lemma0 : {x : Ordinal} → o∅ o< osuc x
65 lemma0 {x} with trio< o∅ (osuc x)
66 lemma0 {x} | tri< a ¬b ¬c = a
67 lemma0 {x} | tri≈ ¬a refl ¬c = ⊥-elim (¬x<0 <-osuc )
68 lemma0 {x} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
69 lemma : ¬ (Ord (osuc (sup-o ψ)) == od∅)
70 lemma record { eq→ = eq→ ; eq← = eq← } = ¬x<0 {o∅} ( eq→ lemma0 )
71 lemma1 : {z : Ordinal} → z o< sup-o ψ → z o< osuc (ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma)))
72 lemma1 {z} lt with trio< z (ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma)))
73 lemma1 {z} lt | tri< a ¬b ¬c = ordtrans a <-osuc
74 lemma1 {z} lt | tri≈ ¬a refl ¬c = <-osuc
75 lemma1 {z} lt | tri> ¬a ¬b c = ⊥-elim (o<> c lemma2 ) where
76 lemma2 : z o< ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma))
77 lemma2 = ordtrans sup-o< ( o<-subst (x∋minimul (Ord (osuc (sup-o ψ))) lemma ) ? ?)
78
51 ------------ 79 ------------
52 -- 80 --
53 -- Onto map 81 -- Onto map
54 -- def X x -> xmap 82 -- def X x -> xmap
55 -- X ---------------------------> Y 83 -- X ---------------------------> Y