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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 zorn {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 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 import BAlgbra
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18
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19
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20 open inOrdinal O
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21 open OD O
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22 open OD.OD
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23 open ODAxiom odAxiom
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24 import OrdUtil
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25 import ODUtil
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26 open Ordinals.Ordinals O
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27 open Ordinals.IsOrdinals isOrdinal
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28 open Ordinals.IsNext isNext
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29 open OrdUtil O
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30 open ODUtil O
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31
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32
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33 import ODC
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34
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35
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36 open _∧_
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37 open _∨_
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38 open Bool
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39
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40
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41 open HOD
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42
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43 record Element (A : HOD) : Set (suc n) where
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44 field
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45 elm : HOD
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46 is-elm : A ∋ elm
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47
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48 open Element
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49
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50 IsPartialOrderSet : ( A : HOD ) → (_≤_ : (x y : HOD) → Set n ) → Set (suc n)
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51 IsPartialOrderSet A _≤_ = IsPartialOrder _≤A_ _≡A_ where
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52 _≤A_ : (x y : Element A ) → Set n
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53 x ≤A y = elm x ≤ elm y
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54 _≡A_ : (x y : Element A ) → Set (suc n)
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55 x ≡A y = elm x ≡ elm y
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56
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57 open _==_
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58 open _⊆_
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59
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60 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → {_≤_ : (x y : HOD) → Set n } → IsPartialOrderSet A _≤_ → IsPartialOrderSet B _≤_
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61 ⊆-IsPartialOrderSet {A} {B} B⊆A {_≤_} PA = record {
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62 isPreorder = record { isEquivalence = record { refl = ? ; sym = {!!} ; trans = {!!} } ; reflexive = {!!} ; trans = {!!} }
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63 ; antisym = {!!}
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64 }
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65
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66 IsTotalOrderSet : ( A : HOD ) → (_≤_ : (x y : HOD) → Set n ) → Set (suc n)
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67 IsTotalOrderSet A _≤_ = IsTotalOrder _≤A_ _≡A_ where
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68 _≤A_ : (x y : Element A ) → Set n
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69 x ≤A y = elm x ≤ elm y
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70 _≡A_ : (x y : Element A ) → Set (suc n)
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71 x ≡A y = elm x ≡ elm y
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72
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73 me : { A a : HOD } → A ∋ a → Element A
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74 me {A} {a} lt = record { elm = a ; is-elm = lt }
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75
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76 record SUP ( A B : HOD ) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
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77 field
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78 sup : HOD
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79 A∋maximal : A ∋ sup
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80 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x ≤ sup ) -- B is Total, use positive
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81
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82 record Maximal ( A : HOD ) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
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83 field
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84 maximal : HOD
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85 A∋maximal : A ∋ maximal
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86 ¬maximal≤x : {x : HOD} → A ∋ x → ¬ maximal ≤ x -- A is Partial, use negative
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87
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88
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89 record ZChain ( A : HOD ) (y : Ordinal) (_≤_ : (x y : HOD) → Set n ) : Set (suc n) where
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90 field
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91 fb : (x : Ordinal ) → HOD
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92 A∋fb : (ox : Ordinal ) → ox o≤ y → A ∋ fb ox
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93 total : {ox oz : Ordinal } → (x≤y : ox o≤ y ) → (z≤y : oz o≤ y ) → ( ox ≡ oz ) ∨ ( fb ox ≤ fb oz ) ∨ ( fb oz ≤ fb ox )
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94 monotonic : {ox oz : Ordinal } → (x≤y : ox o≤ y ) → (z≤y : oz o≤ y ) → ox o≤ oz → fb ox ≤ fb oz
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95
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96 Zorn-lemma : { A : HOD } → { _≤_ : (x y : HOD) → Set n }
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97 → o∅ o< & A
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98 → IsPartialOrderSet A _≤_
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99 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B _≤_ → SUP A B _≤_ ) -- SUP condition
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100 → Maximal A _≤_
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101 Zorn-lemma {A} {_≤_} 0<A PO supP = zorn00 where
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102 someA : HOD
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103 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
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104 isSomeA : A ∋ someA
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105 isSomeA = ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
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106 HasMaximal : HOD
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107 HasMaximal = record { od = record { def = λ y → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))} ; odmax = & A ; ≤odmax = z08 } where
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108 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))) → y o< & A
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109 z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
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110 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y ≤ * m))) → ⊥
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111 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
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112 Gtx : { x : HOD} → A ∋ x → HOD
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113 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x ≤ (* y)) } ; odmax = & A ; ≤odmax = z09 } where
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114 z09 : {y : Ordinal} → (odef A y ∧ (x ≤ (* y))) → y o< & A
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115 z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
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116 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a ≤ b ) → b ≤ a → ⊥
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117 z01 {a} {b} A∋a A∋b (case1 a=b) b≤a = {!!} -- proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b≤a
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118 z01 {a} {b} A∋a A∋b (case2 a≤b) b≤a = {!!} -- proj1 (proj1 (PO (me A∋b) (me A∋a)) b≤a) a≤b
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119 -- ZChain is not compatible with the SUP condition
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120 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where
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121 field
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122 bx : Ordinal
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123 bx≤y : bx o≤ y
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124 is-fb : x ≡ & (fb bx )
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125 bx≤A : (z : ZChain A (& A) _≤_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o≤ & A
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126 bx≤A z {x} bx = BX.bx≤y bx
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127 B : (z : ZChain A (& A) _≤_ ) → HOD
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128 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; ≤odmax = {!!} }
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129 z11 : (z : ZChain A (& A) _≤_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-elm x))
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130 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
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131 obx : (z : ZChain A (& A) _≤_ ) → {x : HOD} → B z ∋ x → Ordinal
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132 obx z {x} bx = BX.bx bx
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133 obx=fb : (z : ZChain A (& A) _≤_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z (BX.bx bx)
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134 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx))
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135 B⊆A : (z : ZChain A (& A) _≤_ ) → B z ⊆ A
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136 B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) (ZChain.A∋fb z (BX.bx bx) (BX.bx≤y bx) ) }
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137 PO-B : (z : ZChain A (& A) _≤_ ) → IsPartialOrderSet (B z) _≤_
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138 PO-B z = subst₂ (λ j k → IsPartialOrder j k ) {!!} {!!} {!!} where
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139 _≤B_ = {!!}
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140 _≡B_ = {!!}
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141 -- a b = {!!} -- PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) } record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) }
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142 bx-monotonic : (z : ZChain A (& A) _≤_ ) → {x y : Element (B z)} → obx z (is-elm x) o≤ obx z (is-elm y) → elm x ≤ elm y
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143 bx-monotonic z {x} {y} a = subst₂ (λ j k → j ≤ k ) (sym (z11 z x)) (sym (z11 z y)) (ZChain.monotonic z (bx≤A z (is-elm x)) (bx≤A z (is-elm y)) a )
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144 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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145 z12 : (z : ZChain A (& A) _≤_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z)) (y : BX (& b) (& A) (ZChain.fb z))
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146 → obx z x ≡ obx z y → bx≤A z x ≅ bx≤A z y
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147 z12 z {a} {b} x y eq = {!!}
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148 bx-inject : (z : ZChain A (& A) _≤_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y
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149 bx-inject z {x} {y} eq = begin
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150 elm x ≡⟨ {!!} ⟩
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151 {!!} ≡⟨ cong (λ k → {!!} ) {!!} ⟩
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152 {!!} ≡⟨ {!!} ⟩
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153 elm y ∎ where open ≡-Reasoning
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154 B-is-total : (z : ZChain A (& A) _≤_ ) → IsTotalOrderSet (B z) _≤_
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155 B-is-total = {!!}
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156 B-Tri : (z : ZChain A (& A) _≤_ ) → Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x ≤ elm y )
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157 B-Tri z x y with trio< (obx z {!!}) (obx z {!!})
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158 ... | tri< a ¬b ¬c = {!!} where -- tri≤ z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where
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159 z10 : elm x ≤ elm y
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160 z10 = {!!} -- bx-monotonic z {x} {y} a
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161 ... | tri≈ ¬a b ¬c = {!!} -- tri≈ {!!} (bx-inject z {x} {y} b) {!!}
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162 ... | tri> ¬a ¬b c = {!!} -- tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c)
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163 ZChain→¬SUP : (z : ZChain A (& A) _≤_ ) → ¬ (SUP A (B z) _≤_ )
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164 ZChain→¬SUP z sp = ⊥-elim {!!} where
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165 z03 : & (SUP.sup sp) o< osuc (& A)
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166 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
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167 z02 : (x : HOD) → B z ∋ x → SUP.sup sp ≤ x → ⊥
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168 z02 x xe s≤x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x≤sup sp xe) s≤x
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169 ind : HasMaximal =h= od∅
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170 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _≤_ )
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171 → ZChain A x _≤_
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172 ind nomx x prev with Oprev-p x
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173 ... | yes op with ODC.∋-p O A (* x)
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174 ... | no ¬Ax = {!!} where
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175 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
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176 px = Oprev.oprev op
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177 zc1 : ZChain A px _≤_
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178 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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179 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o≤ osuc x → sup ≤ ZChain.fb zc1 as
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180 z04 sup as s≤x with trio< (& sup) x
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181 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
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182 ... | tri< a ¬b ¬c = {!!} -- ZChain.¬x≤sup zc1 _ as ( subst (λ k → & sup o≤ k ) (sym (Oprev.oprev=x op)) a )
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183 ... | tri> ¬a ¬b c with osuc-≡< s≤x
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184 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
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185 ... | case2 lt = ⊥-elim (¬a lt )
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186 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
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187 px = Oprev.oprev op
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188 zc1 : ZChain A px _≤_
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189 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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190 z06 : ZChain A x _≤_
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191 z06 with is-o∅ (& (Gtx ax))
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192 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
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193 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x ≤ * m)
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194 x-is-maximal m am = ¬x≤m where
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195 ¬x≤m : ¬ (* x ≤ * m)
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196 ¬x≤m x≤m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x ≤ k ) (cong (*) (sym &iso)) x≤m ⟫ (≡o∅→=od∅ nogt)
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197 ... | no not = {!!} where
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198 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
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199 ... | tri< a ¬b ¬c = {!!} where
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200 zc1 : ZChain A (& A) _≤_
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201 zc1 = prev (& A) a
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202 ... | tri≈ ¬a b ¬c = {!!} where
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203 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
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204 ... | no ¬Ax = {!!} where
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205 ... | yes ax with is-o∅ (& (Gtx ax))
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206 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
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207 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x ≤ * m)
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208 x-is-maximal m am = ¬x≤m where
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209 ¬x≤m : ¬ (* x ≤ * m)
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210 ¬x≤m x≤m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x ≤ k ) (cong (*) (sym &iso)) x≤m ⟫ (≡o∅→=od∅ nogt)
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211 ... | no not = {!!} where
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212 zorn00 : Maximal A _≤_
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213 zorn00 with is-o∅ ( & HasMaximal )
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214 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal≤x = zorn02 } where
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215 -- yes we have the maximal
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216 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
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217 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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218 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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219 zorn01 = proj1 hasm
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220 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ≤ x)
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221 zorn02 {x} ax m≤x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j ≤ k) (sym *iso) (sym *iso) m≤x )
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222 ... | yes ¬Maximal = ⊥-elim {!!} where
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223 -- if we have no maximal, make ZChain, which contradict SUP condition
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224 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _≤_
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225 z x nomx = TransFinite (ind nomx) x
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226
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227 _⊆'_ : ( A B : HOD ) → Set n
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228 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
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229
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230 MaximumSubset : {L P : HOD}
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231 → o∅ o< & L → o∅ o< & P → P ⊆ L
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232 → IsPartialOrderSet P _⊆'_
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233 → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
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234 → Maximal P (_⊆'_)
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235 MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
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