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comparison OPair.agda @ 272:985a1af11bce

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separate ordered pair and Boolean Algebra
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 31 Dec 2019 11:22:52 +0900
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children d9d3654baee1
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271:2169d948159b 272:985a1af11bce
1 open import Level
2 open import Ordinals
3 module OPair {n : Level } (O : Ordinals {n}) where
4
5 open import zf
6 open import logic
7 import OD
8
9 open import Relation.Nullary
10 open import Relation.Binary
11 open import Data.Empty
12 open import Relation.Binary
13 open import Relation.Binary.Core
14 open import Relation.Binary.PropositionalEquality
15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
16
17 open inOrdinal O
18 open OD O
19 open OD.OD
20
21 open _∧_
22 open _∨_
23 open Bool
24
25 open _==_
26
27 <_,_> : (x y : OD) → OD
28 < x , y > = (x , x ) , (x , y )
29
30 exg-pair : { x y : OD } → (x , y ) == ( y , x )
31 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
32 left : {z : Ordinal} → def (x , y) z → def (y , x) z
33 left (case1 t) = case2 t
34 left (case2 t) = case1 t
35 right : {z : Ordinal} → def (y , x) z → def (x , y) z
36 right (case1 t) = case2 t
37 right (case2 t) = case1 t
38
39 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y
40 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
41
42 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y
43 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq )
44
45 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
46 eq-prod refl refl = refl
47
48 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
49 prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where
50 lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y
51 lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y)
52 lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl)
53 lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
54 lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
55 lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
56 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 | case1 s = ⊥-elim ( o<¬≡ s c )
58 lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
59 lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y
60 lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where
61 lemma3 : ( x , x ) == ( y , z )
62 lemma3 = ==-trans eq exg-pair
63 lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y
64 lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl)
65 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
66 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
67 lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z
68 lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl)
69 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
70 ... | refl with lemma2 (==-sym eq )
71 ... | refl = refl
72 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
73 lemmax : x ≡ x'
74 lemmax with eq→ eq {od→ord (x , x)} (case1 refl)
75 lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x')
76 lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
77 ... | refl = lemma1 (ord→== s )
78 lemmay : y ≡ y'
79 lemmay with lemmax
80 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
81 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
82
83 data ord-pair : (p : Ordinal) → Set n where
84 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
85
86 ZFProduct : OD
87 ZFProduct = record { def = λ x → ord-pair x }
88
89 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
90 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
91 -- eq-pair refl refl = HE.refl
92
93 pi1 : { p : Ordinal } → ord-pair p → Ordinal
94 pi1 ( pair x y) = x
95
96 π1 : { p : OD } → ZFProduct ∋ p → OD
97 π1 lt = ord→od (pi1 lt )
98
99 pi2 : { p : Ordinal } → ord-pair p → Ordinal
100 pi2 ( pair x y ) = y
101
102 π2 : { p : OD } → ZFProduct ∋ p → OD
103 π2 lt = ord→od (pi2 lt )
104
105 op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
106 op-cons {ox} {oy} = pair ox oy
107
108 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y >
109 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
110 let open ≡-Reasoning in begin
111 od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
112 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
113 od→ord < x , y >
114 ∎ )
115
116 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op
117 op-iso (pair ox oy) = refl
118
119 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x
120 p-iso {x} p = ord≡→≡ (op-iso p)
121
122 p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x
123 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
124
125 p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y
126 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
127