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

comparison OPair.agda @ 329:5544f4921a44

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 05 Jul 2020 12:32:09 +0900
parents d9d3654baee1
children 2a8a51375e49
comparison
equal deleted inserted replaced
328:72f3e3b44c27 329:5544f4921a44
15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) 15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
16 16
17 open inOrdinal O 17 open inOrdinal O
18 open OD O 18 open OD O
19 open OD.OD 19 open OD.OD
20 open OD.HOD
20 open ODAxiom odAxiom 21 open ODAxiom odAxiom
21 22
22 open _∧_ 23 open _∧_
23 open _∨_ 24 open _∨_
24 open Bool 25 open Bool
25 26
26 open _==_ 27 open _==_
27 28
28 <_,_> : (x y : OD) → OD 29 _=h=_ : (x y : HOD) → Set n
30 x =h= y = od x == od y
31
32 <_,_> : (x y : HOD) → HOD
29 < x , y > = (x , x ) , (x , y ) 33 < x , y > = (x , x ) , (x , y )
30 34
31 exg-pair : { x y : OD } → (x , y ) == ( y , x ) 35 exg-pair : { x y : HOD } → (x , y ) =h= ( y , x )
32 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where 36 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
33 left : {z : Ordinal} → def (x , y) z → def (y , x) z 37 left : {z : Ordinal} → odef (x , y) z → odef (y , x) z
34 left (case1 t) = case2 t 38 left (case1 t) = case2 t
35 left (case2 t) = case1 t 39 left (case2 t) = case1 t
36 right : {z : Ordinal} → def (y , x) z → def (x , y) z 40 right : {z : Ordinal} → odef (y , x) z → odef (x , y) z
37 right (case1 t) = case2 t 41 right (case1 t) = case2 t
38 right (case2 t) = case1 t 42 right (case2 t) = case1 t
39 43
40 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y 44 ord≡→≡ : { x y : HOD } → od→ord x ≡ od→ord y → x ≡ y
41 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) 45 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
42 46
43 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y 47 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y
44 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq ) 48 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq )
45 49
46 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > 50 eq-prod : { x x' y y' : HOD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
47 eq-prod refl refl = refl 51 eq-prod refl refl = refl
48 52
49 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) 53 prod-eq : { x x' y y' : HOD } → < x , y > =h= < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
50 prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where 54 prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where
51 lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y 55 lemma0 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → x ≡ y
52 lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) 56 lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y)
53 lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) 57 lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl)
54 lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a ) 58 lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
55 lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a ) 59 lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
56 lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b 60 lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
57 lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl) 61 lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl)
58 lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c ) 62 lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
59 lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c ) 63 lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
60 lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y 64 lemma2 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → z ≡ y
61 lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where 65 lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where
62 lemma3 : ( x , x ) == ( y , z ) 66 lemma3 : ( x , x ) =h= ( y , z )
63 lemma3 = ==-trans eq exg-pair 67 lemma3 = ==-trans eq exg-pair
64 lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y 68 lemma1 : {x y : HOD } → ( x , x ) =h= ( y , y ) → x ≡ y
65 lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl) 69 lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl)
66 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s) 70 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
67 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s) 71 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
68 lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z 72 lemma4 : {x y z : HOD } → ( x , y ) =h= ( x , z ) → y ≡ z
69 lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl) 73 lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl)
70 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z 74 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
71 ... | refl with lemma2 (==-sym eq ) 75 ... | refl with lemma2 (==-sym eq )
72 ... | refl = refl 76 ... | refl = refl
73 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z 77 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
79 lemmay : y ≡ y' 83 lemmay : y ≡ y'
80 lemmay with lemmax 84 lemmay with lemmax
81 ... | refl with lemma4 eq -- with (x,y)≡(x,y') 85 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
82 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) 86 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
83 87
88 --
89 -- unlike ordered pair, ZFProduct is not a HOD
90
84 data ord-pair : (p : Ordinal) → Set n where 91 data ord-pair : (p : Ordinal) → Set n where
85 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) 92 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
86 93
87 ZFProduct : OD 94 ZFProduct : OD
88 ZFProduct = record { def = λ x → ord-pair x } 95 ZFProduct = record { def = λ x → ord-pair x }
92 -- eq-pair refl refl = HE.refl 99 -- eq-pair refl refl = HE.refl
93 100
94 pi1 : { p : Ordinal } → ord-pair p → Ordinal 101 pi1 : { p : Ordinal } → ord-pair p → Ordinal
95 pi1 ( pair x y) = x 102 pi1 ( pair x y) = x
96 103
97 π1 : { p : OD } → ZFProduct ∋ p → OD 104 π1 : { p : HOD } → def ZFProduct (od→ord p) → HOD
98 π1 lt = ord→od (pi1 lt ) 105 π1 lt = ord→od (pi1 lt )
99 106
100 pi2 : { p : Ordinal } → ord-pair p → Ordinal 107 pi2 : { p : Ordinal } → ord-pair p → Ordinal
101 pi2 ( pair x y ) = y 108 pi2 ( pair x y ) = y
102 109
103 π2 : { p : OD } → ZFProduct ∋ p → OD 110 π2 : { p : HOD } → def ZFProduct (od→ord p) → HOD
104 π2 lt = ord→od (pi2 lt ) 111 π2 lt = ord→od (pi2 lt )
105 112
106 op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy > 113 op-cons : { ox oy : Ordinal } → def ZFProduct (od→ord ( < ord→od ox , ord→od oy > ))
107 op-cons {ox} {oy} = pair ox oy 114 op-cons {ox} {oy} = pair ox oy
108 115
109 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > 116 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
110 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( 117 def-subst df refl refl = df
111 let open ≡-Reasoning in begin 118
112 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > 119 p-cons : ( x y : HOD ) → def ZFProduct (od→ord ( < x , y >))
113 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ 120 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
114 od→ord < x , y > 121 let open ≡-Reasoning in begin
115 ∎ ) 122 od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
123 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
124 od→ord < x , y >
125 ∎ )
116 126
117 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op 127 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op
118 op-iso (pair ox oy) = refl 128 op-iso (pair ox oy) = refl
119 129
120 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x 130 p-iso : { x : HOD } → (p : def ZFProduct (od→ord x) ) → < π1 p , π2 p > ≡ x
121 p-iso {x} p = ord≡→≡ (op-iso p) 131 p-iso {x} p = ord≡→≡ (op-iso p)
122 132
123 p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x 133 p-pi1 : { x y : HOD } → (p : def ZFProduct (od→ord < x , y >) ) → π1 p ≡ x
124 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) )) 134 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
125 135
126 p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y 136 p-pi2 : { x y : HOD } → (p : def ZFProduct (od→ord < x , y >) ) → π2 p ≡ y
127 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p))) 137 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
128 138