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431
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1 {-# OPTIONS --allow-unsolved-metas #-}
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2 open import Level
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3 open import Ordinals
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4 module ODUtil {n : Level } (O : Ordinals {n} ) where
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5
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6 open import zf
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7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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8 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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9 open import Data.Nat.Properties
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10 open import Data.Empty
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11 open import Relation.Nullary
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12 open import Relation.Binary hiding ( _⇔_ )
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13
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14 open import logic
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15 open import nat
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16
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17 open Ordinals.Ordinals O
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18 open Ordinals.IsOrdinals isOrdinal
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19 open Ordinals.IsNext isNext
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20 import OrdUtil
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21 open OrdUtil O
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22
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23 import OD
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24 open OD O
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25 open OD.OD
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26 open ODAxiom odAxiom
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27
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28 open HOD
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29 open _⊆_
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30 open _∧_
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31 open _==_
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32
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33 cseq : HOD → HOD
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34 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where
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35 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x)
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36 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc )
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37
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38
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39 pair-xx<xy : {x y : HOD} → & (x , x) o< osuc (& (x , y) )
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40 pair-xx<xy {x} {y} = ⊆→o≤ lemma where
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41 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z
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42 lemma {z} (case1 refl) = case1 refl
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43 lemma {z} (case2 refl) = case1 refl
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44
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45 pair-<xy : {x y : HOD} → {n : Ordinal} → & x o< next n → & y o< next n → & (x , y) o< next n
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46 pair-<xy {x} {y} {o} x<nn y<nn with trio< (& x) (& y) | inspect (omax (& x)) (& y)
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47 ... | tri< a ¬b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx y<nn)) ho<
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48 ... | tri> ¬a ¬b c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx x<nn)) ho<
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49 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (omax≡ _ _ b) (subst (λ k → osuc k o< next o) b (osuc<nx x<nn))) ho<
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50
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51 -- another form of infinite
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52 -- pair-ord< : {x : Ordinal } → Set n
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53 pair-ord< : {x : HOD } → ( {y : HOD } → & y o< next (odmax y) ) → & ( x , x ) o< next (& x)
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54 pair-ord< {x} ho< = subst (λ k → & (x , x) o< k ) lemmab0 lemmab1 where
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55 lemmab0 : next (odmax (x , x)) ≡ next (& x)
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56 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡)
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57 lemmab1 : & (x , x) o< next ( odmax (x , x))
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58 lemmab1 = ho<
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59
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60 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C
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61 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) }
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62
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63 refl-⊆ : {A : HOD} → A ⊆ A
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64 refl-⊆ {A} = record { incl = λ x → x }
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65
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66 od⊆→o≤ : {x y : HOD } → x ⊆ y → & x o< osuc (& y)
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67 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) &iso (incl lt (d→∋ x x>z)))
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68
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480
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69 ⊆→= : {F U : HOD} → F ⊆ U → U ⊆ F → F =h= U
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70 ⊆→= {F} {U} FU UF = record { eq→ = λ {x} lt → subst (λ k → odef U k) &iso (incl FU (subst (λ k → odef F k) (sym &iso) lt) )
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71 ; eq← = λ {x} lt → subst (λ k → odef F k) &iso (incl UF (subst (λ k → odef U k) (sym &iso) lt) ) }
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72
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519
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73 ¬A∋x→A≡od∅ : (A : HOD) → {x : HOD} → A ∋ x → ¬ ( & A ≡ o∅ )
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74 ¬A∋x→A≡od∅ A {x} ax a=0 = ¬x<0 ( subst (λ k → & x o< k) a=0 (c<→o< ax ))
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75
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431
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76 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A )
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77 subset-lemma {A} {x} = record {
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78 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
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79 ; proj2 = λ x⊆A lt → ⟪ incl x⊆A lt , lt ⟫
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80 }
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81
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82 ω<next-o∅ : {y : Ordinal} → infinite-d y → y o< next o∅
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83 ω<next-o∅ {y} lt = <odmax infinite lt
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84
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85 nat→ω : Nat → HOD
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86 nat→ω Zero = od∅
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87 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y))
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88
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89 ω→nato : {y : Ordinal} → infinite-d y → Nat
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90 ω→nato iφ = Zero
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91 ω→nato (isuc lt) = Suc (ω→nato lt)
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92
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93 ω→nat : (n : HOD) → infinite ∋ n → Nat
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94 ω→nat n = ω→nato
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95
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96 ω∋nat→ω : {n : Nat} → def (od infinite) (& (nat→ω n))
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97 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) (sym ord-od∅) iφ
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98 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) lemma (isuc ( ω∋nat→ω {n})) where
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99 lemma : & (Union (* (& (nat→ω n)) , (* (& (nat→ω n)) , * (& (nat→ω n))))) ≡ & (nat→ω (Suc n))
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100 lemma = subst (λ k → & (Union (k , ( k , k ))) ≡ & (nat→ω (Suc n))) (sym *iso) refl
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101
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102 pair1 : { x y : HOD } → (x , y ) ∋ x
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103 pair1 = case1 refl
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104
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105 pair2 : { x y : HOD } → (x , y ) ∋ y
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106 pair2 = case2 refl
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107
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108 single : {x y : HOD } → (x , x ) ∋ y → x ≡ y
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109 single (case1 eq) = ==→o≡ ( ord→== (sym eq) )
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110 single (case2 eq) = ==→o≡ ( ord→== (sym eq) )
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111
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112 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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113 -- postulate f-extensionality : { n m : Level} → HE.Extensionality n m
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114
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115 ω-prev-eq1 : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → ¬ (x o< y)
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116 ω-prev-eq1 {x} {y} eq x<y = eq→ (ord→== eq) {& (* y)} (λ not2 → not2 (& (* y , * y))
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117 ⟪ case2 refl , subst (λ k → odef k (& (* y))) (sym *iso) (case1 refl)⟫ ) (λ u → lemma u ) where
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118 lemma : (u : Ordinal) → ¬ def (od (* x , (* x , * x))) u ∧ def (od (* u)) (& (* y))
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119 lemma u t with proj1 t
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120 lemma u t | case1 u=x = o<> (c<→o< {* y} {* u} (proj2 t)) (subst₂ (λ j k → j o< k )
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121 (trans (sym &iso) (trans (sym u=x) (sym &iso)) ) (sym &iso) x<y ) -- x ≡ & (* u)
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122 lemma u t | case2 u=xx = o<¬≡ (lemma1 (subst (λ k → odef k (& (* y)) ) (trans (cong (λ k → * k ) u=xx) *iso ) (proj2 t))) x<y where
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123 lemma1 : {x y : Ordinal } → (* x , * x ) ∋ * y → x ≡ y -- y = x ∈ ( x , x ) = u
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124 lemma1 (case1 eq) = subst₂ (λ j k → j ≡ k ) &iso &iso (sym eq)
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125 lemma1 (case2 eq) = subst₂ (λ j k → j ≡ k ) &iso &iso (sym eq)
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126
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127 ω-prev-eq : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → x ≡ y
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128 ω-prev-eq {x} {y} eq with trio< x y
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129 ω-prev-eq {x} {y} eq | tri< a ¬b ¬c = ⊥-elim (ω-prev-eq1 eq a)
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130 ω-prev-eq {x} {y} eq | tri≈ ¬a b ¬c = b
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131 ω-prev-eq {x} {y} eq | tri> ¬a ¬b c = ⊥-elim (ω-prev-eq1 (sym eq) c)
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132
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133 ω-∈s : (x : HOD) → Union ( x , (x , x)) ∋ x
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134 ω-∈s x not = not (& (x , x)) ⟪ case2 refl , subst (λ k → odef k (& x) ) (sym *iso) (case1 refl) ⟫
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135
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136 ωs≠0 : (x : HOD) → ¬ ( Union ( x , (x , x)) ≡ od∅ )
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137 ωs≠0 y eq = ⊥-elim ( ¬x<0 (subst (λ k → & y o< k ) ord-od∅ (c<→o< (subst (λ k → odef k (& y )) eq (ω-∈s y) ))) )
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138
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139 nat→ω-iso : {i : HOD} → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i
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140 nat→ω-iso {i} = ε-induction {λ i → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i } ind i where
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141 ind : {x : HOD} → ({y : HOD} → x ∋ y → (lt : infinite ∋ y) → nat→ω (ω→nat y lt) ≡ y) →
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142 (lt : infinite ∋ x) → nat→ω (ω→nat x lt) ≡ x
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143 ind {x} prev lt = ind1 lt *iso where
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144 ind1 : {ox : Ordinal } → (ltd : infinite-d ox ) → * ox ≡ x → nat→ω (ω→nato ltd) ≡ x
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145 ind1 {o∅} iφ refl = sym o∅≡od∅
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146 ind1 (isuc {x₁} ltd) ox=x = begin
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147 nat→ω (ω→nato (isuc ltd) )
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148 ≡⟨⟩
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149 Union (nat→ω (ω→nato ltd) , (nat→ω (ω→nato ltd) , nat→ω (ω→nato ltd)))
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150 ≡⟨ cong (λ k → Union (k , (k , k ))) lemma ⟩
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151 Union (* x₁ , (* x₁ , * x₁))
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152 ≡⟨ trans ( sym *iso) ox=x ⟩
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153 x
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154 ∎ where
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155 open ≡-Reasoning
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156 lemma0 : x ∋ * x₁
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157 lemma0 = subst (λ k → odef k (& (* x₁))) (trans (sym *iso) ox=x) (λ not → not
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158 (& (* x₁ , * x₁)) ⟪ pair2 , subst (λ k → odef k (& (* x₁))) (sym *iso) pair1 ⟫ )
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159 lemma1 : infinite ∋ * x₁
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160 lemma1 = subst (λ k → odef infinite k) (sym &iso) ltd
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161 lemma3 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ltd ≅ ltd1
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162 lemma3 iφ iφ refl = HE.refl
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163 lemma3 iφ (isuc {y} ltd1) eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso eq (c<→o< (ω-∈s (* y)) )))
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164 lemma3 (isuc {y} ltd) iφ eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso (sym eq) (c<→o< (ω-∈s (* y)) )))
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165 lemma3 (isuc {x} ltd) (isuc {y} ltd1) eq with lemma3 ltd ltd1 (ω-prev-eq (sym eq))
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166 ... | t = HE.cong₂ (λ j k → isuc {j} k ) (HE.≡-to-≅ (ω-prev-eq eq)) t
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167 lemma2 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ω→nato ltd ≡ ω→nato ltd1
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168 lemma2 {x} {y} ltd ltd1 eq = lemma6 eq (lemma3 {x} {y} ltd ltd1 eq) where
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169 lemma6 : {x y : Ordinal} → {ltd : infinite-d x } {ltd1 : infinite-d y } → y ≡ x → ltd ≅ ltd1 → ω→nato ltd ≡ ω→nato ltd1
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170 lemma6 refl HE.refl = refl
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171 lemma : nat→ω (ω→nato ltd) ≡ * x₁
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172 lemma = trans (cong (λ k → nat→ω k) (lemma2 {x₁} {_} ltd (subst (λ k → infinite-d k ) (sym &iso) ltd) &iso ) ) ( prev {* x₁} lemma0 lemma1 )
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173
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174 ω→nat-iso : {i : Nat} → ω→nat ( nat→ω i ) (ω∋nat→ω {i}) ≡ i
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175 ω→nat-iso {i} = lemma i (ω∋nat→ω {i}) *iso where
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176 lemma : {x : Ordinal } → ( i : Nat ) → (ltd : infinite-d x ) → * x ≡ nat→ω i → ω→nato ltd ≡ i
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177 lemma {x} Zero iφ eq = refl
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178 lemma {x} (Suc i) iφ eq = ⊥-elim ( ωs≠0 (nat→ω i) (trans (sym eq) o∅≡od∅ )) -- Union (nat→ω i , (nat→ω i , nat→ω i)) ≡ od∅
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179 lemma Zero (isuc {x} ltd) eq = ⊥-elim ( ωs≠0 (* x) (subst (λ k → k ≡ od∅ ) *iso eq ))
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180 lemma (Suc i) (isuc {x} ltd) eq = cong (λ k → Suc k ) (lemma i ltd (lemma1 eq) ) where -- * x ≡ nat→ω i
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181 lemma1 : * (& (Union (* x , (* x , * x)))) ≡ Union (nat→ω i , (nat→ω i , nat→ω i)) → * x ≡ nat→ω i
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182 lemma1 eq = subst (λ k → * x ≡ k ) *iso (cong (λ k → * k)
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183 ( ω-prev-eq (subst (λ k → _ ≡ k ) &iso (cong (λ k → & k ) (sym
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184 (subst (λ k → _ ≡ Union ( k , ( k , k ))) (sym *iso ) eq ))))))
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185
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