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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 TotalOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
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51 TotalOrderSet A _<_ = Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x < elm y )
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52
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53 PartialOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
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54 PartialOrderSet A _<_ = (a b : Element A)
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55 → (elm a < elm b → ((¬ elm b < elm a) ∧ (¬ (elm a ≡ elm b) ))) ∧ (elm a ≡ elm b → (¬ elm a < elm b) ∧ (¬ elm b < elm a))
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56
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57 me : { A a : HOD } → A ∋ a → Element A
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58 me {A} {a} lt = record { elm = a ; is-elm = lt }
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59
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60 record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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61 field
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62 sup : HOD
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63 A∋maximal : A ∋ sup
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64 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total
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65
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66 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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67 field
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68 maximal : HOD
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69 A∋maximal : A ∋ maximal
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70 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
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71
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72 open _==_
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73 open _⊆_
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74
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75 record ZChain ( A : HOD ) (y : Ordinal) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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76 field
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77 fb : Ordinal → HOD
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78 A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox
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79 monotonic : (ox oy : Ordinal ) → ox o< y → ox o< oy → fb ox < fb oy
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80
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81 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
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82 → o∅ o< & A
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83 → PartialOrderSet A _<_
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84 → ( ( B : HOD) → (B⊆A : B ⊆ A) → TotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition
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85 → Maximal A _<_
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86 Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where
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87 someA : HOD
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88 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
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89 isSomeA : A ∋ someA
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90 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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91 HasMaximal : HOD
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92 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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93 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A
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94 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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95 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥
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96 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
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97 Gtx : { x : HOD} → A ∋ x → HOD
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98 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where
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99 z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A
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100 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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101 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
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102 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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103 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = {!!}
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104 -- proj1 (PO (me A∋b) (me A∋a)) b<a ⟪ a<b , (λ b=a → proj1 (proj2 (PO (me A∋b) (me A∋a)) b=a ) b<a ) ⟫
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105 -- ZChain is not compatible with the SUP condition
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106 B : (z : ZChain A (& A) _<_ ) → HOD
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107 B = {!!}
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108 ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ )
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109 ZChain→¬SUP z sp = ⊥-elim {!!} where
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110 z03 : & (SUP.sup sp) o< osuc (& A)
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111 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
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112 z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥
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113 z02 x xe s<x = {!!} -- z01 (incl (ZChain.B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x≤sup sp xe) s<x
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114 ind : HasMaximal =h= od∅
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115 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ )
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116 → ZChain A x _<_
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117 ind nomx x prev with Oprev-p x
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118 ... | yes op with ODC.∋-p O A (* x)
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119 ... | no ¬Ax = {!!} where
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120 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
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121 px = Oprev.oprev op
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122 zc1 : ZChain A px _<_
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123 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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124 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as
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125 z04 sup as s<x with trio< (& sup) x
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126 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
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127 ... | 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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128 ... | tri> ¬a ¬b c with osuc-≡< s<x
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129 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
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130 ... | case2 lt = ⊥-elim (¬a lt )
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131 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
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132 px = Oprev.oprev op
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133 zc1 : ZChain A px _<_
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134 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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135 z06 : ZChain A x _<_
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136 z06 with is-o∅ (& (Gtx ax))
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137 ... | 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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138 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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139 x-is-maximal m am = ¬x<m where
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140 ¬x<m : ¬ (* x < * m)
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141 ¬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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142 ... | no not = {!!} where
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143 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
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144 ... | tri< a ¬b ¬c = {!!} where
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145 zc1 : ZChain A (& A) _<_
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146 zc1 = prev (& A) a
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147 ... | tri≈ ¬a b ¬c = {!!} where
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148 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
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149 ... | no ¬Ax = {!!} where
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150 ... | yes ax with is-o∅ (& (Gtx ax))
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151 ... | 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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152 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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153 x-is-maximal m am = ¬x<m where
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154 ¬x<m : ¬ (* x < * m)
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155 ¬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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156 ... | no not = {!!} where
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157 zorn00 : Maximal A _<_
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158 zorn00 with is-o∅ ( & HasMaximal )
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159 ... | 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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160 -- yes we have the maximal
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161 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
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162 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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163 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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164 zorn01 = proj1 hasm
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165 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
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166 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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167 ... | yes ¬Maximal = ⊥-elim {!!} where
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168 -- if we have no maximal, make ZChain, which contradict SUP condition
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169 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_
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170 z x nomx = TransFinite (ind nomx) x
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171
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172 _⊆'_ : ( A B : HOD ) → Set n
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173 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
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174
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175 MaximumSubset : {L P : HOD}
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176 → o∅ o< & L → o∅ o< & P → P ⊆ L
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177 → PartialOrderSet P _⊆'_
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178 → ( (B : HOD) → B ⊆ P → TotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
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179 → Maximal P (_⊆'_)
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180 MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
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