Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 189:540b845ea2de
Axiom of choies implies p ∨ ( ¬ p )
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 25 Jul 2019 14:42:19 +0900 |
parents | 1f2c8b094908 |
children | 6e778b0a7202 |
rev | line source |
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16 | 1 open import Level |
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2 module OD where |
3 | 3 |
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4 open import zf |
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5 open import ordinal |
23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
6 | 9 open import Data.Empty |
10 open import Relation.Nullary | |
11 open import Relation.Binary | |
12 open import Relation.Binary.Core | |
13 | |
27 | 14 -- Ordinal Definable Set |
11 | 15 |
141 | 16 record OD {n : Level} : Set (suc n) where |
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17 field |
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18 def : (x : Ordinal {n} ) → Set n |
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19 |
141 | 20 open OD |
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21 |
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22 open Ordinal |
120 | 23 open _∧_ |
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24 |
141 | 25 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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26 field |
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27 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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28 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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29 |
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30 id : {n : Level} {A : Set n} → A → A |
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31 id x = x |
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32 |
141 | 33 eq-refl : {n : Level} { x : OD {n} } → x == x |
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34 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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35 |
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36 open _==_ |
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37 |
141 | 38 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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39 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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40 |
141 | 41 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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42 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) } |
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43 |
141 | 44 ⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y |
120 | 45 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m |
46 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m | |
47 | |
179 | 48 -- Ordinal in OD ( and ZFSet ) Transitive Set |
141 | 49 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} |
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50 Ord {n} a = record { def = λ y → y o< a } |
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51 |
141 | 52 od∅ : {n : Level} → OD {n} |
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53 od∅ {n} = Ord o∅ |
40 | 54 |
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55 postulate |
141 | 56 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
57 od→ord : {n : Level} → OD {n} → Ordinal {n} | |
58 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
166 | 59 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
60 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x | |
113 | 61 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
150 | 62 -- we should prove this in agda, but simply put here |
141 | 63 ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y |
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64 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
159 | 65 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
66 -- ord→od x ≡ Ord x results the same | |
100 | 67 -- supermum as Replacement Axiom |
95 | 68 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
98 | 69 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
111 | 70 -- contra-position of mimimulity of supermum required in Power Set Axiom |
165 | 71 -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
72 -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
183 | 73 -- mimimul and x∋minimul is an Axiom of choice |
141 | 74 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
117 | 75 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) |
141 | 76 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) |
187 | 77 -- minimulity (may proved by ε-induction ) |
141 | 78 minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) |
123 | 79 |
141 | 80 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
95 | 81 _∋_ {n} a x = def a ( od→ord x ) |
82 | |
141 | 83 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n |
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84 x c< a = a ∋ x |
103 | 85 |
141 | 86 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
95 | 87 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
88 | |
141 | 89 cseq : {n : Level} → OD {n} → OD {n} |
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90 cseq x = record { def = λ y → def x (osuc y) } where |
113 | 91 |
141 | 92 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x |
95 | 93 def-subst df refl refl = df |
94 | |
141 | 95 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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96 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
95 | 97 |
141 | 98 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) |
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99 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )} |
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100 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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101 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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102 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
28 | 103 |
187 | 104 otrans : {n : Level} {a x y : Ordinal {n} } → def (Ord a) x → def (Ord x) y → def (Ord a) y |
105 otrans x<a y<x = ordtrans y<x x<a | |
123 | 106 |
37 | 107 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
81 | 108 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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109 c0 : Nat → Ordinal {n} → Set n |
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110 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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111 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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112 c2 Zero not = refl |
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113 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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114 ... | t with t (case1 ≤-refl ) |
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115 c2 (Suc lx) not | t | () |
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116 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) |
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117 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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118 ... | t with t (case2 Φ< ) |
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119 c3 lx (Φ .lx) d not | t | () |
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120 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 121 ... | t with t (case2 (s< s<refl ) ) |
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122 c3 lx (OSuc .lx x₁) d not | t | () |
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123 |
57 | 124 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
125 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
126 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
127 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
37 | 128 |
46 | 129 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } |
130 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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131 |
51 | 132 -- avoiding lv != Zero error |
141 | 133 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y |
51 | 134 orefl refl = refl |
135 | |
141 | 136 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
51 | 137 ==-iso {n} {x} {y} eq = record { |
138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
140 where | |
141 | 141 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z |
51 | 142 lemma {x} {z} d = def-subst d oiso refl |
143 | |
141 | 144 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
57 | 145 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) |
146 | |
141 | 147 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
51 | 148 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
149 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
150 lemma ox ox refl = eq-refl | |
151 | |
152 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
153 o≡→== {n} {x} {.x} refl = eq-refl | |
154 | |
155 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
156 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
157 | |
141 | 158 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x |
51 | 159 c≤-refl x = case1 refl |
160 | |
150 | 161 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
162 o∅≡od∅ {n} = ==→o≡ lemma where | |
163 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x | |
164 lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso | |
165 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x | |
166 lemma1 (case1 ()) | |
167 lemma1 (case2 ()) | |
168 lemma : ord→od o∅ == od∅ | |
169 lemma = record { eq→ = lemma0 ; eq← = lemma1 } | |
170 | |
171 ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n} | |
172 ord-od∅ {n} = sym ( subst (λ k → k ≡ od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) | |
80 | 173 |
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174 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} |
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175 eq→ ∅0 {w} (lift ()) |
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176 eq← ∅0 {w} (case1 ()) |
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177 eq← ∅0 {w} (case2 ()) |
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178 |
141 | 179 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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180 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
60 | 181 ∅< {n} {x} {y} d eq | lift () |
57 | 182 |
141 | 183 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
120 | 184 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) |
51 | 185 |
141 | 186 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
76 | 187 def-iso refl t = t |
188 | |
57 | 189 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
190 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
191 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
192 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
193 | |
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194 ppp : { n : Level } → { p : Set (suc n) } { a : OD {suc n} } → record { def = λ x → p } ∋ a → p |
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195 ppp {n} {p} {a} d = d |
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196 |
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197 -- |
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198 -- Axiom of choice in intutionistic logic implies the exclude middle |
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199 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog |
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200 -- |
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201 p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p ) |
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202 p∨¬p {n} p with is-o∅ ( od→ord ( record { def = λ x → p } )) |
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203 p∨¬p {n} p | yes eq = case2 (¬p eq) where |
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204 ps = record { def = λ x → p } |
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205 lemma : ps == od∅ → p → ⊥ |
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206 lemma eq p0 = ¬x<0 {n} {od→ord ps} (eq→ eq p0 ) |
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207 ¬p : (od→ord ps ≡ o∅) → p → ⊥ |
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208 ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) |
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209 p∨¬p {n} p | no ¬p = case1 (ppp {n} {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where |
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210 ps = record { def = λ x → p } |
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211 eqo∅ : ps == od∅ {suc n} → od→ord ps ≡ o∅ {suc n} |
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212 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) |
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213 lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) |
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214 lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) |
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215 |
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216 ∋-p : { n : Level } → ( p : Set (suc n) ) → Dec p |
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217 ∋-p {n} p with p∨¬p p |
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218 ∋-p {n} p | case1 x = yes x |
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219 ∋-p {n} p | case2 x = no x |
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220 |
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221 double-neg-eilm : {n : Level } {A : Set (suc n)} → ¬ ¬ A → A -- we don't have this in intutionistic logic |
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222 double-neg-eilm {n} {A} notnot with ∋-p A |
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223 ... | yes p = p |
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224 ... | no ¬p = ⊥-elim ( notnot ¬p ) |
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225 |
167 | 226 OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y ) |
227 OrdP {n} x y with trio< x (od→ord y) | |
228 OrdP {n} x y | tri< a ¬b ¬c = no ¬c | |
229 OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) | |
230 OrdP {n} x y | tri> ¬a ¬b c = yes c | |
119 | 231 |
79 | 232 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
94 | 233 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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234 |
141 | 235 in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n} |
148 | 236 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } |
141 | 237 |
96 | 238 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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239 |
141 | 240 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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241 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where |
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242 |
141 | 243 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
154 | 244 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→ |
96 | 245 |
246 -- Constructible Set on α | |
122 | 247 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } |
248 -- L (Φ 0) = Φ | |
249 -- L (OSuc lv n) = { Def ( L n ) } | |
250 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) | |
141 | 251 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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252 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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253 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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254 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
121 | 255 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) |
89 | 256 |
167 | 257 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α |
141 | 258 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x |
122 | 259 |
170 | 260 |
141 | 261 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
262 OD→ZF {n} = record { | |
263 ZFSet = OD {suc n} | |
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264 ; _∋_ = _∋_ |
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265 ; _≈_ = _==_ |
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266 ; ∅ = od∅ |
28 | 267 ; _,_ = _,_ |
268 ; Union = Union | |
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269 ; Power = Power |
28 | 270 ; Select = Select |
271 ; Replace = Replace | |
161 | 272 ; infinite = infinite |
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273 ; isZF = isZF |
28 | 274 } where |
144 | 275 ZFSet = OD {suc n} |
141 | 276 Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n} |
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277 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
141 | 278 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
279 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x } | |
280 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
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281 x , y = Ord (omax (od→ord x) (od→ord y)) |
144 | 282 _∩_ : ( A B : ZFSet ) → ZFSet |
145 | 283 A ∩ B = record { def = λ x → def A x ∧ def B x } |
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284 Union : OD {suc n} → OD {suc n} |
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285 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } |
54 | 286 _∈_ : ( A B : ZFSet ) → Set (suc n) |
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287 A ∈ B = B ∋ A |
54 | 288 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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289 _⊆_ A B {x} = A ∋ x → B ∋ x |
141 | 290 Power : OD {suc n} → OD {suc n} |
129 | 291 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
103 | 292 {_} : ZFSet → ZFSet |
293 { x } = ( x , x ) | |
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294 |
161 | 295 data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where |
296 iφ : infinite-d o∅ | |
297 isuc : {x : Ordinal {suc n} } → infinite-d x → | |
298 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
299 | |
300 infinite : OD {suc n} | |
301 infinite = record { def = λ x → infinite-d x } | |
302 | |
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303 infixr 200 _∈_ |
96 | 304 -- infixr 230 _∩_ _∪_ |
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305 infixr 220 _⊆_ |
161 | 306 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite |
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307 isZF = record { |
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308 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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309 ; pair = pair |
72 | 310 ; union→ = union→ |
311 ; union← = union← | |
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312 ; empty = empty |
129 | 313 ; power→ = power→ |
76 | 314 ; power← = power← |
186 | 315 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
183 | 316 ; ε-induction = ε-induction |
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317 ; infinity∅ = infinity∅ |
160 | 318 ; infinity = infinity |
116 | 319 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
135 | 320 ; replacement← = replacement← |
321 ; replacement→ = replacement→ | |
183 | 322 ; choice-func = choice-func |
323 ; choice = choice | |
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324 } where |
129 | 325 |
141 | 326 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
87 | 327 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
328 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
129 | 329 |
167 | 330 empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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331 empty x (case1 ()) |
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332 empty x (case2 ()) |
129 | 333 |
154 | 334 o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z} |
155 | 335 o<→c< lt lt1 = ordtrans lt1 lt |
336 | |
337 ⊆→o< : {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y | |
338 ⊆→o< {x} {y} lt with trio< x y | |
339 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
340 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
341 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) | |
342 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
151 | 343 |
144 | 344 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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345 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
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346 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) |
159 | 347 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
166 | 348 union← X z UX∋z = TransFiniteExists _ lemma UX∋z where |
165 | 349 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
350 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
144 | 351 |
352 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y | |
353 ψiso {ψ} t refl = t | |
354 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
355 selection {ψ} {X} {y} = record { | |
356 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
357 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
358 } | |
359 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x | |
360 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where | |
361 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
150 | 362 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
144 | 363 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) |
150 | 364 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
365 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
366 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) | |
144 | 367 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
150 | 368 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) |
144 | 369 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) |
150 | 370 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) |
371 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) | |
144 | 372 |
373 --- | |
374 --- Power Set | |
375 --- | |
376 --- First consider ordinals in OD | |
100 | 377 --- |
378 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
379 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A | |
380 -- | |
381 -- | |
142 | 382 ∩-≡ : { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) |
383 ∩-≡ {a} {b} inc = record { | |
384 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
385 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; | |
386 eq← = λ {x} x<a∩b → proj2 x<a∩b } | |
100 | 387 -- |
388 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
389 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
390 -- | |
141 | 391 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
129 | 392 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t} |
127 | 393 lemma refl (lemma1 lemma-eq )where |
129 | 394 lemma-eq : ZFSubset (Ord a) t == t |
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395 eq→ lemma-eq {z} w = proj2 w |
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396 eq← lemma-eq {z} w = record { proj2 = w ; |
129 | 397 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z} |
126 | 398 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
141 | 399 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}} |
129 | 400 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t |
150 | 401 lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) |
129 | 402 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x))) |
98 | 403 lemma = sup-o< |
129 | 404 |
144 | 405 -- |
406 -- Every set in OD is a subset of Ordinals | |
407 -- | |
142 | 408 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) |
166 | 409 |
410 -- we have oly double negation form because of the replacement axiom | |
411 -- | |
412 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
413 power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where | |
142 | 414 a = od→ord A |
415 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) | |
416 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t | |
166 | 417 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) |
418 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
142 | 419 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) |
420 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) | |
166 | 421 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) |
422 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
423 | |
142 | 424 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
425 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
426 a = od→ord A | |
427 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x | |
428 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
429 lemma3 : Def (Ord a) ∋ t | |
430 lemma3 = ord-power← a t lemma0 | |
152 | 431 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
432 lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t} | |
433 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
434 lemma4 = let open ≡-Reasoning in begin | |
435 A ∩ ord→od (od→ord t) | |
436 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
437 A ∩ t | |
438 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
439 t | |
440 ∎ | |
142 | 441 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
152 | 442 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x))) |
443 lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}) | |
142 | 444 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
151 | 445 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
446 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
447 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) | |
142 | 448 |
141 | 449 regularity : (x : OD) (not : ¬ (x == od∅)) → |
115 | 450 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
117 | 451 proj1 (regularity x not ) = x∋minimul x not |
452 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where | |
453 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ | |
454 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where | |
455 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) | |
142 | 456 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso) |
457 ; proj2 = proj2 (proj2 s) } | |
117 | 458 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ |
459 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) | |
129 | 460 |
186 | 461 extensionality0 : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
462 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
463 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
464 | |
465 extensionality : {A B w : OD {suc n} } → ((z : OD {suc n}) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) | |
466 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d | |
467 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
129 | 468 |
161 | 469 infinity∅ : infinite ∋ od∅ {suc n} |
470 infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where | |
471 lemma : o∅ ≡ od→ord od∅ | |
472 lemma = let open ≡-Reasoning in begin | |
473 o∅ | |
474 ≡⟨ sym diso ⟩ | |
475 od→ord ( ord→od o∅ ) | |
476 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
477 od→ord od∅ | |
478 ∎ | |
479 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
480 infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
481 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
482 ≡ od→ord (Union (x , (x , x))) | |
483 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
484 | |
179 | 485 -- Axiom of choice ( is equivalent to the existence of minimul in our case ) |
162 | 486 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
487 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD | |
488 choice-func X {x} not X∋x = minimul x not | |
489 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A | |
490 choice X {A} X∋A not = x∋minimul A not | |
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492 -- choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → OD {suc n} |
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493 -- |
176 | 494 -- another form of regularity |
495 -- | |
496 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} | |
497 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) | |
498 → (x : OD {suc n} ) → ψ x | |
499 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where | |
500 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } | |
501 → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) | |
502 ε-induction-ord Zero (Φ 0) (case1 ()) | |
503 ε-induction-ord Zero (Φ 0) (case2 ()) | |
504 ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = | |
505 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where | |
506 lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } | |
507 lemma y lt with osuc-≡< y<x | |
508 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso | |
509 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 | |
510 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = | |
511 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where | |
179 | 512 -- |
513 -- if lv of z if less than x Ok | |
514 -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma | |
515 -- | |
516 -- lx Suc lx (1) lz(a) <lx by case1 | |
517 -- ly(1) ly(2) (2) lz(b) <lx by case1 | |
518 -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) | |
519 -- | |
176 | 520 lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ |
521 lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 | |
522 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly | |
523 lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin | |
524 lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) | |
525 ≡⟨ cong ( λ k → lv k ) diso ⟩ | |
526 lv (record { lv = ly ; ord = oy }) | |
527 ≡⟨⟩ | |
528 ly | |
529 ∎ | |
530 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z | |
531 lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt | |
532 lemma z lt | case1 lz<ly with <-cmp lx ly | |
533 lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
180 | 534 lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) |
176 | 535 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) |
180 | 536 lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) |
176 | 537 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) |
538 lemma z lt | case2 lz=ly with <-cmp lx ly | |
539 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
180 | 540 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) |
179 | 541 ... | eq = subst (λ k → ψ k ) oiso |
542 (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) | |
180 | 543 lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) |
179 | 544 ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where |
176 | 545 lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z |
546 lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) | |
547 lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → | |
179 | 548 lx ≡ ly → ly ≡ lv (od→ord z) → ψ z |
176 | 549 lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) |
550 |