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

comparison cardinal.agda @ 230:1b1620e2053c

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we need ordered pair
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 12 Aug 2019 08:58:51 +0900
parents 5e36744b8dce
children af60c40298a4
comparison
equal deleted inserted replaced
229:5e36744b8dce 230:1b1620e2053c
19 19
20 open _∧_ 20 open _∧_
21 open _∨_ 21 open _∨_
22 open Bool 22 open Bool
23 23
24 -- we have to work on Ordinal to keep OD Level n
25 -- since we use p∨¬p which works only on Level n
24 26
25 func : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD 27 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD
26 func f dom = Replace dom ( λ x → x , (ord→od (f (od→ord x) ))) 28 func→od f dom = Replace dom ( λ x → x , (ord→od (f (od→ord x) )))
27 29
28 record _⊗_ (A B : Ordinal) : Set n where 30 record _⊗_ (A B : Ordinal) : Set n where
29 field 31 field
30 π1 : Ordinal 32 π1 : Ordinal
31 π2 : Ordinal 33 π2 : Ordinal
32 A∋π1 : def (ord→od A) π1 34 A∋π1 : def (ord→od A) π1
33 B∋π2 : def (ord→od B) π2 35 B∋π2 : def (ord→od B) π2
34 36
37 -- Clearly wrong. We need ordered pair
35 Func : ( A B : OD ) → OD 38 Func : ( A B : OD ) → OD
36 Func A B = record { def = λ x → (od→ord A) ⊗ (od→ord B) } 39 Func A B = record { def = λ x → (od→ord A) ⊗ (od→ord B) }
37 40
38 π1 : { A B x : OD } → Func A B ∋ x → OD 41 open _⊗_
39 π1 {A} {B} {x} p = ord→od (_⊗_.π1 p)
40 42
41 π2 : { A B x : OD } → Func A B ∋ x → OD 43 func←od : { dom cod : OD } → (f : OD ) → Func dom cod ∋ f → (Ordinal → Ordinal )
42 π2 {A} {B} {x} p = ord→od (_⊗_.π2 p) 44 func←od {dom} {cod} f lt x = sup-o ( λ y → lemma y ) where
43
44 Func→func : { dom cod : OD } → (f : OD ) → Func dom cod ∋ f → (Ordinal → Ordinal )
45 Func→func {dom} {cod} f lt x = sup-o ( λ y → lemma y ) where
46 lemma : Ordinal → Ordinal 45 lemma : Ordinal → Ordinal
47 lemma y with p∨¬p ( _⊗_.π1 lt ≡ x ) 46 lemma y with p∨¬p ( _⊗_.π1 lt ≡ x )
48 lemma y | case1 refl = _⊗_.π2 lt 47 lemma y | case1 refl = _⊗_.π2 lt
49 lemma y | case2 not = o∅ 48 lemma y | case2 not = o∅
50 49
67 field 66 field
68 xmap : Ordinal 67 xmap : Ordinal
69 ymap : Ordinal 68 ymap : Ordinal
70 xfunc : def (Func X Y) xmap 69 xfunc : def (Func X Y) xmap
71 yfunc : def (Func Y X) ymap 70 yfunc : def (Func Y X) ymap
72 onto-iso : {y : Ordinal } → (lty : def Y y ) → Func→func (ord→od xmap) xfunc ( Func→func (ord→od ymap) yfunc y ) ≡ y 71 onto-iso : {y : Ordinal } → (lty : def Y y ) → func←od (ord→od xmap) xfunc ( func←od (ord→od ymap) yfunc y ) ≡ y
72
73 open Onto
74
75 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z
76 onto-restrict {X} {Y} {Z} onto Z⊆Y = record {
77 xmap = xmap1
78 ; ymap = zmap
79 ; xfunc = xfunc1
80 ; yfunc = zfunc
81 ; onto-iso = onto-iso1
82 } where
83 xmap1 : Ordinal
84 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} )
85 zmap : Ordinal
86 zmap = {!!}
87 xfunc1 : def (Func X Z) xmap1
88 xfunc1 = {!!}
89 zfunc : def (Func Z X) zmap
90 zfunc = {!!}
91 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func←od (ord→od xmap1) xfunc1 ( func←od (ord→od zmap) zfunc z ) ≡ z
92 onto-iso1 = {!!}
93
73 94
74 record Cardinal (X : OD ) : Set n where 95 record Cardinal (X : OD ) : Set n where
75 field 96 field
76 cardinal : Ordinal 97 cardinal : Ordinal
77 conto : Onto (Ord cardinal) X 98 conto : Onto X (Ord cardinal)
78 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto (Ord y) X 99 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y)
79 100
80 cardinal : (X : OD ) → Cardinal X 101 cardinal : (X : OD ) → Cardinal X
81 cardinal X = record { 102 cardinal X = record {
82 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) 103 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) )
83 ; conto = onto 104 ; conto = onto
84 ; cmax = cmax 105 ; cmax = cmax
85 } where 106 } where
86 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto (Ord x) X) ) 107 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) )
87 cardinal-p x with p∨¬p ( Onto (Ord x) X ) 108 cardinal-p x with p∨¬p ( Onto X (Ord x) )
88 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } 109 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True }
89 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } 110 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False }
90 S = sup-o (λ x → proj1 (cardinal-p x)) 111 S = sup-o (λ x → proj1 (cardinal-p x))
91 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto (Ord y) X)) → 112 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) →
92 Lift (suc n) (x o< (osuc S) → Onto (Ord x) X) 113 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) )
93 lemma1 x prev with trio< x (osuc S) 114 lemma1 x prev with trio< x (osuc S)
94 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a 115 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a
95 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = {!!} 116 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} )
96 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = {!!} 117 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where
118 lemma2 : Onto X (Ord x)
119 lemma2 with prev {!!} {!!}
120 ... | lift t = t {!!}
97 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) 121 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt ))
98 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) 122 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt ))
99 onto : Onto (Ord S) X 123 onto : Onto X (Ord S)
100 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto (Ord x) X ) } lemma1 S 124 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S
101 ... | lift t = t <-osuc where 125 ... | lift t = t <-osuc
102 cmax : (y : Ordinal) → S o< y → ¬ Onto (Ord y) X 126 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y)
103 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} 127 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S}
104 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where 128 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where
105 lemma : proj1 (cardinal-p y) ≡ y 129 lemma : proj1 (cardinal-p y) ≡ y
106 lemma with p∨¬p ( Onto (Ord y) X ) 130 lemma with p∨¬p ( Onto X (Ord y) )
107 lemma | case1 x = refl 131 lemma | case1 x = refl
108 lemma | case2 not = ⊥-elim ( not ontoy ) 132 lemma | case2 not = ⊥-elim ( not ontoy )
109 133
110 134
111 ----- 135 -----