|
431
|
1 {-# OPTIONS --allow-unsolved-metas #-}
|
|
|
2
|
|
|
3 open import Level
|
|
|
4 open import Ordinals
|
|
|
5 module OPair {n : Level } (O : Ordinals {n}) where
|
|
|
6
|
|
|
7 open import zf
|
|
|
8 open import logic
|
|
|
9 import OD
|
|
|
10 import ODUtil
|
|
|
11 import OrdUtil
|
|
|
12
|
|
|
13 open import Relation.Nullary
|
|
|
14 open import Relation.Binary
|
|
|
15 open import Data.Empty
|
|
|
16 open import Relation.Binary
|
|
|
17 open import Relation.Binary.Core
|
|
|
18 open import Relation.Binary.PropositionalEquality
|
|
|
19 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
|
|
|
20
|
|
|
21 open OD O
|
|
|
22 open OD.OD
|
|
|
23 open OD.HOD
|
|
|
24 open ODAxiom odAxiom
|
|
|
25
|
|
|
26 open Ordinals.Ordinals O
|
|
|
27 open Ordinals.IsOrdinals isOrdinal
|
|
|
28 open Ordinals.IsNext isNext
|
|
|
29 open OrdUtil O
|
|
|
30 open ODUtil O
|
|
|
31
|
|
|
32 open _∧_
|
|
|
33 open _∨_
|
|
|
34 open Bool
|
|
|
35
|
|
|
36 open _==_
|
|
|
37
|
|
|
38 <_,_> : (x y : HOD) → HOD
|
|
|
39 < x , y > = (x , x ) , (x , y )
|
|
|
40
|
|
|
41 exg-pair : { x y : HOD } → (x , y ) =h= ( y , x )
|
|
|
42 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
|
|
|
43 left : {z : Ordinal} → odef (x , y) z → odef (y , x) z
|
|
|
44 left (case1 t) = case2 t
|
|
|
45 left (case2 t) = case1 t
|
|
|
46 right : {z : Ordinal} → odef (y , x) z → odef (x , y) z
|
|
|
47 right (case1 t) = case2 t
|
|
|
48 right (case2 t) = case1 t
|
|
|
49
|
|
|
50 ord≡→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
|
|
|
51 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong ( λ k → * k ) eq )
|
|
|
52
|
|
|
53 od≡→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
|
|
|
54 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong ( λ k → & k ) eq )
|
|
|
55
|
|
|
56 eq-prod : { x x' y y' : HOD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
|
|
|
57 eq-prod refl refl = refl
|
|
|
58
|
|
|
59 xx=zy→x=y : {x y z : HOD } → ( x , x ) =h= ( z , y ) → x ≡ y
|
|
|
60 xx=zy→x=y {x} {y} eq with trio< (& x) (& y)
|
|
|
61 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c with eq← eq {& y} (case2 refl)
|
|
|
62 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
|
|
|
63 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
|
|
|
64 xx=zy→x=y {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
|
|
|
65 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c with eq← eq {& y} (case2 refl)
|
|
|
66 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
|
|
|
67 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
|
|
|
68
|
|
|
69 prod-eq : { x x' y y' : HOD } → < x , y > =h= < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
|
|
|
70 prod-eq {x} {x'} {y} {y'} eq = ⟪ lemmax , lemmay ⟫ where
|
|
|
71 lemma2 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → z ≡ y
|
|
|
72 lemma2 {x} {y} {z} eq = trans (sym (xx=zy→x=y lemma3 )) ( xx=zy→x=y eq ) where
|
|
|
73 lemma3 : ( x , x ) =h= ( y , z )
|
|
|
74 lemma3 = ==-trans eq exg-pair
|
|
|
75 lemma1 : {x y : HOD } → ( x , x ) =h= ( y , y ) → x ≡ y
|
|
|
76 lemma1 {x} {y} eq with eq← eq {& y} (case2 refl)
|
|
|
77 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
|
|
|
78 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
|
|
|
79 lemma4 : {x y z : HOD } → ( x , y ) =h= ( x , z ) → y ≡ z
|
|
|
80 lemma4 {x} {y} {z} eq with eq← eq {& z} (case2 refl)
|
|
|
81 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
|
|
|
82 ... | refl with lemma2 (==-sym eq )
|
|
|
83 ... | refl = refl
|
|
|
84 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
|
|
|
85 lemmax : x ≡ x'
|
|
|
86 lemmax with eq→ eq {& (x , x)} (case1 refl)
|
|
|
87 lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x')
|
|
|
88 lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
|
|
|
89 ... | refl = lemma1 (ord→== s )
|
|
|
90 lemmay : y ≡ y'
|
|
|
91 lemmay with lemmax
|
|
|
92 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
|
|
|
93 ... | eq1 = lemma4 (ord→== (cong (λ k → & k ) eq1 ))
|
|
|
94
|
|
1098
|
95 prod-≡ : { x x' y y' : HOD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
|
|
|
96 prod-≡ eq = prod-eq (ord→== (cong (&) eq ))
|
|
|
97
|
|
431
|
98 --
|
|
1098
|
99 -- unlike ordered pair, ZFPair is not a HOD
|
|
431
|
100
|
|
|
101 data ord-pair : (p : Ordinal) → Set n where
|
|
|
102 pair : (x y : Ordinal ) → ord-pair ( & ( < * x , * y > ) )
|
|
|
103
|
|
1098
|
104 ZFPair : OD
|
|
|
105 ZFPair = record { def = λ x → ord-pair x }
|
|
431
|
106
|
|
|
107 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
|
|
|
108 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
|
|
|
109 -- eq-pair refl refl = HE.refl
|
|
|
110
|
|
|
111 pi1 : { p : Ordinal } → ord-pair p → Ordinal
|
|
|
112 pi1 ( pair x y) = x
|
|
|
113
|
|
1098
|
114 π1 : { p : HOD } → def ZFPair (& p) → HOD
|
|
431
|
115 π1 lt = * (pi1 lt )
|
|
|
116
|
|
|
117 pi2 : { p : Ordinal } → ord-pair p → Ordinal
|
|
|
118 pi2 ( pair x y ) = y
|
|
|
119
|
|
1098
|
120 π2 : { p : HOD } → def ZFPair (& p) → HOD
|
|
431
|
121 π2 lt = * (pi2 lt )
|
|
|
122
|
|
1098
|
123 op-cons : ( ox oy : Ordinal ) → def ZFPair (& ( < * ox , * oy > ))
|
|
|
124 op-cons ox oy = pair ox oy
|
|
431
|
125
|
|
|
126 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
|
|
|
127 def-subst df refl refl = df
|
|
|
128
|
|
1098
|
129 p-cons : ( x y : HOD ) → def ZFPair (& ( < x , y >))
|
|
|
130 p-cons x y = def-subst {_} {_} {ZFPair} {& (< x , y >)} (pair (& x) ( & y )) refl (
|
|
431
|
131 let open ≡-Reasoning in begin
|
|
|
132 & < * (& x) , * (& y) >
|
|
|
133 ≡⟨ cong₂ (λ j k → & < j , k >) *iso *iso ⟩
|
|
|
134 & < x , y >
|
|
|
135 ∎ )
|
|
|
136
|
|
|
137 op-iso : { op : Ordinal } → (q : ord-pair op ) → & < * (pi1 q) , * (pi2 q) > ≡ op
|
|
|
138 op-iso (pair ox oy) = refl
|
|
|
139
|
|
1098
|
140 p-iso : { x : HOD } → (p : def ZFPair (& x) ) → < π1 p , π2 p > ≡ x
|
|
431
|
141 p-iso {x} p = ord≡→≡ (op-iso p)
|
|
|
142
|
|
1098
|
143 p-pi1 : { x y : HOD } → (p : def ZFPair (& < x , y >) ) → π1 p ≡ x
|
|
431
|
144 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
|
|
|
145
|
|
1098
|
146 p-pi2 : { x y : HOD } → (p : def ZFPair (& < x , y >) ) → π2 p ≡ y
|
|
431
|
147 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
|
|
|
148
|
|
|
149 ω-pair : {x y : HOD} → {m : Ordinal} → & x o< next m → & y o< next m → & (x , y) o< next m
|
|
|
150 ω-pair lx ly = next< (omax<nx lx ly ) ho<
|
|
|
151
|
|
|
152 ω-opair : {x y : HOD} → {m : Ordinal} → & x o< next m → & y o< next m → & < x , y > o< next m
|
|
|
153 ω-opair {x} {y} {m} lx ly = lemma0 where
|
|
|
154 lemma0 : & < x , y > o< next m
|
|
|
155 lemma0 = osucprev (begin
|
|
|
156 osuc (& < x , y >)
|
|
|
157 <⟨ osuc<nx ho< ⟩
|
|
|
158 next (omax (& (x , x)) (& (x , y)))
|
|
|
159 ≡⟨ cong (λ k → next k) (sym ( omax≤ _ _ pair-xx<xy )) ⟩
|
|
|
160 next (osuc (& (x , y)))
|
|
|
161 ≡⟨ sym (nexto≡) ⟩
|
|
|
162 next (& (x , y))
|
|
|
163 ≤⟨ x<ny→≤next (ω-pair lx ly) ⟩
|
|
|
164 next m
|
|
|
165 ∎ ) where
|
|
|
166 open o≤-Reasoning O
|
|
|
167
|
|
|
168 _⊗_ : (A B : HOD) → HOD
|
|
|
169 A ⊗ B = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) ))
|
|
|
170
|
|
|
171 product→ : {A B a b : HOD} → A ∋ a → B ∋ b → ( A ⊗ B ) ∋ < a , b >
|
|
1096
|
172 product→ {A} {B} {a} {b} A∋a B∋b = record { owner = _ ; ao = lemma1 ; ox = subst (λ k → odef k _) (sym *iso) lemma2 } where
|
|
431
|
173 lemma1 : odef (Replace B (λ b₁ → Replace A (λ a₁ → < a₁ , b₁ >))) (& (Replace A (λ a₁ → < a₁ , b >)))
|
|
|
174 lemma1 = replacement← B b B∋b
|
|
|
175 lemma2 : odef (Replace A (λ a₁ → < a₁ , b >)) (& < a , b >)
|
|
|
176 lemma2 = replacement← A a A∋a
|
|
|
177
|
|
|
178 x<nextA : {A x : HOD} → A ∋ x → & x o< next (odmax A)
|
|
|
179 x<nextA {A} {x} A∋x = ordtrans (c<→o< {x} {A} A∋x) ho<
|
|
|
180
|
|
|
181 A<Bnext : {A B x : HOD} → & A o< & B → A ∋ x → & x o< next (odmax B)
|
|
|
182 A<Bnext {A} {B} {x} lt A∋x = osucprev (begin
|
|
|
183 osuc (& x)
|
|
|
184 <⟨ osucc (c<→o< A∋x) ⟩
|
|
|
185 osuc (& A)
|
|
|
186 <⟨ osucc lt ⟩
|
|
|
187 osuc (& B)
|
|
|
188 <⟨ osuc<nx ho< ⟩
|
|
|
189 next (odmax B)
|
|
|
190 ∎ ) where open o≤-Reasoning O
|
|
|
191
|
|
1098
|
192 data ZFProduct (A B : HOD) : (p : Ordinal) → Set n where
|
|
|
193 ab-pair : {a b : Ordinal } → odef A a → odef B b → ZFProduct A B ( & ( < * a , * b > ) )
|
|
|
194
|
|
431
|
195 ZFP : (A B : HOD) → HOD
|
|
1098
|
196 ZFP A B = record { od = record { def = λ x → ZFProduct A B x }
|
|
|
197 ; odmax = omax (next (odmax A)) (next (odmax B)) ; <odmax = λ {y} px → lemma y px }
|
|
431
|
198 where
|
|
1098
|
199 lemma : (y : Ordinal) → ZFProduct A B y → y o< omax (next (odmax A)) (next (odmax B))
|
|
|
200 lemma p ( ab-pair {x} {y} ax by ) with trio< (& A) (& B)
|
|
|
201 lemma p ( ab-pair {x} {y} ax by ) | tri< a ¬b ¬c = ordtrans (ω-opair (A<Bnext a (subst (λ k → odef A k ) (sym &iso)
|
|
|
202 ax )) (lemma1 by )) (omax-y _ _ ) where
|
|
431
|
203 lemma1 : odef B y → & (* y) o< next (HOD.odmax B)
|
|
|
204 lemma1 lt = x<nextA {B} (d→∋ B lt)
|
|
1098
|
205 lemma p ( ab-pair {x} {y} ax by ) | tri≈ ¬a b ¬c = ordtrans (ω-opair (x<nextA {A}
|
|
|
206 (d→∋ A ax)) lemma2 ) (omax-x _ _ ) where
|
|
431
|
207 lemma2 : & (* y) o< next (HOD.odmax A)
|
|
1098
|
208 lemma2 = ordtrans ( subst (λ k → & (* y) o< k ) (sym b) (c<→o< (d→∋ B by))) ho<
|
|
|
209 lemma p ( ab-pair {x} {y} ax by ) | tri> ¬a ¬b c = ordtrans (ω-opair (x<nextA {A} (d→∋ A ax ))
|
|
|
210 (A<Bnext c (subst (λ k → odef B k ) (sym &iso) by))) (omax-x _ _ )
|
|
|
211
|
|
|
212 ZFP→ : {A B a b : HOD} → A ∋ a → B ∋ b → ZFP A B ∋ < a , b >
|
|
|
213 ZFP→ {A} {B} {a} {b} aa bb = subst (λ k → ZFProduct A B k ) (cong₂ (λ j k → & < j , k >) *iso *iso ) ( ab-pair aa bb )
|
|
431
|
214
|
|
|
215 ZFP⊆⊗ : {A B : HOD} {x : Ordinal} → odef (ZFP A B) x → odef (A ⊗ B) x
|
|
1098
|
216 ZFP⊆⊗ {A} {B} {px} ( ab-pair {a} {b} ax by ) = product→ (d→∋ A ax) (d→∋ B by)
|
|
|
217
|
|
|
218 ⊗⊆ZFPair : {A B x : HOD} → ( A ⊗ B ) ∋ x → def ZFPair (& x)
|
|
|
219 ⊗⊆ZFPair {A} {B} {x} record { owner = owner ; ao = record { z = a ; az = aa ; x=ψz = x=ψa } ; ox = ox } = zfp01 where
|
|
|
220 zfp02 : Replace A (λ z → < z , * a >) ≡ * owner
|
|
|
221 zfp02 = subst₂ ( λ j k → j ≡ k ) *iso refl (sym (cong (*) x=ψa ))
|
|
|
222 zfp01 : def ZFPair (& x)
|
|
|
223 zfp01 with subst (λ k → odef k (& x) ) (sym zfp02) ox
|
|
|
224 ... | record { z = b ; az = ab ; x=ψz = x=ψb } = subst (λ k → def ZFPair k) (cong (&) zfp00) (op-cons b a ) where
|
|
|
225 zfp00 : < * b , * a > ≡ x
|
|
|
226 zfp00 = sym ( subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) x=ψb) )
|
|
431
|
227
|
|
1098
|
228 ⊗⊆ZFP : {A B x : HOD} → ( A ⊗ B ) ∋ x → odef (ZFP A B) (& x)
|
|
|
229 ⊗⊆ZFP {A} {B} {x} record { owner = owner ; ao = record { z = a ; az = ba ; x=ψz = x=ψa } ; ox = ox } = zfp01 where
|
|
|
230 zfp02 : Replace A (λ z → < z , * a >) ≡ * owner
|
|
|
231 zfp02 = subst₂ ( λ j k → j ≡ k ) *iso refl (sym (cong (*) x=ψa ))
|
|
|
232 zfp01 : odef (ZFP A B) (& x)
|
|
|
233 zfp01 with subst (λ k → odef k (& x) ) (sym zfp02) ox
|
|
|
234 ... | record { z = b ; az = ab ; x=ψz = x=ψb } = subst (λ k → ZFProduct A B k ) (sym x=ψb) (ab-pair ab ba)
|
|
431
|
235
|
|
1098
|
236
|
|
|
237
|
|
|
238
|
