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1 open import Level
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2 open import Ordinals
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433
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3 module PFOD {n : Level } (O : Ordinals {n}) where
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431
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4
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5 import filter
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6 open import zf
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7 open import logic
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8 -- open import partfunc {n} O
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9 import OD
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10
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11 open import Relation.Nullary
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12 open import Relation.Binary
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13 open import Data.Empty
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14 open import Relation.Binary
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15 open import Relation.Binary.Core
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16 open import Relation.Binary.PropositionalEquality
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17 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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18 import BAlgbra
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19
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20 open BAlgbra O
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21
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22 open inOrdinal O
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23 open OD O
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24 open OD.OD
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25 open ODAxiom odAxiom
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26 import OrdUtil
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27 import ODUtil
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28 open Ordinals.Ordinals O
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29 open Ordinals.IsOrdinals isOrdinal
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30 open Ordinals.IsNext isNext
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31 open OrdUtil O
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32 open ODUtil O
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33
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34
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35 import ODC
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36
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37 open filter O
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38
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39 open _∧_
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40 open _∨_
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41 open Bool
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42
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43
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44 open HOD
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45
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46 -------
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47 -- the set of finite partial functions from ω to 2
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48 --
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49 --
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50
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51 open import Data.List hiding (filter)
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52 open import Data.Maybe
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53
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54 import OPair
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55 open OPair O
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56
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57 data Hω2 : (i : Nat) ( x : Ordinal ) → Set n where
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58 hφ : Hω2 0 o∅
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59 h0 : {i : Nat} {x : Ordinal } → Hω2 i x →
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60 Hω2 (Suc i) (& (Union ((< nat→ω i , nat→ω 0 >) , * x )))
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61 h1 : {i : Nat} {x : Ordinal } → Hω2 i x →
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62 Hω2 (Suc i) (& (Union ((< nat→ω i , nat→ω 1 >) , * x )))
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63 he : {i : Nat} {x : Ordinal } → Hω2 i x →
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64 Hω2 (Suc i) x
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65
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66 record Hω2r (x : Ordinal) : Set n where
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67 field
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68 count : Nat
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69 hω2 : Hω2 count x
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70
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71 open Hω2r
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72
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73 HODω2 : HOD
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74 HODω2 = record { od = record { def = λ x → Hω2r x } ; odmax = next o∅ ; <odmax = odmax0 } where
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75 P : (i j : Nat) (x : Ordinal ) → HOD
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76 P i j x = ((nat→ω i , nat→ω i) , (nat→ω i , nat→ω j)) , * x
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77 nat1 : (i : Nat) (x : Ordinal) → & (nat→ω i) o< next x
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78 nat1 i x = next< nexto∅ ( <odmax infinite (ω∋nat→ω {i}))
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79 lemma1 : (i j : Nat) (x : Ordinal ) → osuc (& (P i j x)) o< next x
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80 lemma1 i j x = osuc<nx (pair-<xy (pair-<xy (pair-<xy (nat1 i x) (nat1 i x) ) (pair-<xy (nat1 i x) (nat1 j x) ) )
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81 (subst (λ k → k o< next x) (sym &iso) x<nx ))
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82 lemma : (i j : Nat) (x : Ordinal ) → & (Union (P i j x)) o< next x
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83 lemma i j x = next< (lemma1 i j x ) ho<
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84 odmax0 : {y : Ordinal} → Hω2r y → y o< next o∅
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85 odmax0 {y} r with hω2 r
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86 ... | hφ = x<nx
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87 ... | h0 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma i 0 x)
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88 ... | h1 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma i 1 x)
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89 ... | he {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) x<nx
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90
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91 3→Hω2 : List (Maybe Two) → HOD
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92 3→Hω2 t = list→hod t 0 where
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93 list→hod : List (Maybe Two) → Nat → HOD
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94 list→hod [] _ = od∅
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95 list→hod (just i0 ∷ t) i = Union (< nat→ω i , nat→ω 0 > , ( list→hod t (Suc i) ))
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96 list→hod (just i1 ∷ t) i = Union (< nat→ω i , nat→ω 1 > , ( list→hod t (Suc i) ))
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97 list→hod (nothing ∷ t) i = list→hod t (Suc i )
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98
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99 Hω2→3 : (x : HOD) → HODω2 ∋ x → List (Maybe Two)
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100 Hω2→3 x = lemma where
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101 lemma : { y : Ordinal } → Hω2r y → List (Maybe Two)
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102 lemma record { count = 0 ; hω2 = hφ } = []
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103 lemma record { count = (Suc i) ; hω2 = (h0 hω3) } = just i0 ∷ lemma record { count = i ; hω2 = hω3 }
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104 lemma record { count = (Suc i) ; hω2 = (h1 hω3) } = just i1 ∷ lemma record { count = i ; hω2 = hω3 }
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105 lemma record { count = (Suc i) ; hω2 = (he hω3) } = nothing ∷ lemma record { count = i ; hω2 = hω3 }
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106
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107 ω→2 : HOD
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108 ω→2 = Power infinite
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109
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110 ω→2f : (x : HOD) → ω→2 ∋ x → Nat → Two
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111 ω→2f x lt n with ODC.∋-p O x (nat→ω n)
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112 ω→2f x lt n | yes p = i1
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113 ω→2f x lt n | no ¬p = i0
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114
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115 fω→2-sel : ( f : Nat → Two ) (x : HOD) → Set n
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116 fω→2-sel f x = (infinite ∋ x) ∧ ( (lt : odef infinite (& x) ) → f (ω→nat x lt) ≡ i1 )
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117
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118 fω→2 : (Nat → Two) → HOD
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119 fω→2 f = Select infinite (fω→2-sel f)
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120
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121 open _==_
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122
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123 import Axiom.Extensionality.Propositional
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124 postulate f-extensionality : { n m : Level} → Axiom.Extensionality.Propositional.Extensionality n m
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125
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126 ω2∋f : (f : Nat → Two) → ω→2 ∋ fω→2 f
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127 ω2∋f f = power← infinite (fω→2 f) (λ {x} lt → proj1 ((proj2 (selection {fω→2-sel f} {infinite} )) lt))
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128
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129 ω→2f≡i1 : (X i : HOD) → (iω : infinite ∋ i) → (lt : ω→2 ∋ X ) → ω→2f X lt (ω→nat i iω) ≡ i1 → X ∋ i
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130 ω→2f≡i1 X i iω lt eq with ODC.∋-p O X (nat→ω (ω→nat i iω))
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131 ω→2f≡i1 X i iω lt eq | yes p = subst (λ k → X ∋ k ) (nat→ω-iso iω) p
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132
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133 open _⊆_
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134
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135 -- someday ...
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136 -- postulate
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137 -- ω→2f-iso : (X : HOD) → ( lt : ω→2 ∋ X ) → fω→2 ( ω→2f X lt ) =h= X
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138 -- fω→2-iso : (f : Nat → Two) → ω→2f ( fω→2 f ) (ω2∋f f) ≡ f
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139
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