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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 _<_ = IsStrictPartialOrder _≡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 isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A
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61 isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) }
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62
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63 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → {_<_ : (x y : HOD) → Set n } → IsPartialOrderSet A _<_ → IsPartialOrderSet B _<_
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64 ⊆-IsPartialOrderSet {A} {B} B⊆A {_<_} PA = record {
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65 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = λ eq → case1 eq ; trans = λ {x} {y} {z} → trans1 {x} {y} {z}
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66 ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; trans = trans1 ; <-resp-≈ = resp0
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67 } where
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68 _<B_ : (x y : Element B ) → Set n
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69 x <B y = elm x < elm y
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70 trans1 : {x y z : Element B} → x <B y → y <B z → x <B z
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71 trans1 {x} {y} {z} x<y y<z = IsStrictPartialOrder.trans PA {isA B⊆A x} {isA B⊆A y} {isA B⊆A z} x<y y<z
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72 irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y )
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73 irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y
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74 open import Data.Product
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75 resp0 : ({x y z : Element B} → elm y ≡ elm z → x <B y → x <B z) × ({x y z : Element B} → elm y ≡ elm z → y <B x → z <B x)
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76 resp0 = Data.Product._,_ (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })
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77 (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })
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78
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79 open import Relation.Binary.Properties.Poset as Poset
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80
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81 IsTotalOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
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82 IsTotalOrderSet A _<_ = IsStrictTotalOrder _≡A_ _<A_ where
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83 _<A_ : (x y : Element A ) → Set n
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84 x <A y = elm x < elm y
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85 _≡A_ : (x y : Element A ) → Set (suc n)
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86 x ≡A y = elm x ≡ elm y
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87
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88 me : { A a : HOD } → A ∋ a → Element A
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89 me {A} {a} lt = record { elm = a ; is-elm = lt }
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90
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91 record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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92 field
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93 sup : HOD
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94 A∋maximal : A ∋ sup
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95 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive
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96
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97 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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98 field
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99 maximal : HOD
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100 A∋maximal : A ∋ maximal
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101 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
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102
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103
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104 record ZChain ( A : HOD ) (y : Ordinal) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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105 field
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106 fb : (x : Ordinal ) → HOD
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107 A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox
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108 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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109 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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110
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111 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
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112 → o∅ o< & A
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113 → IsPartialOrderSet A _<_
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114 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition
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115 → Maximal A _<_
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116 Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where
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117 someA : HOD
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118 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
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119 isSomeA : A ∋ someA
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120 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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121 HasMaximal : HOD
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122 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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123 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A
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124 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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125 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥
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126 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
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127 Gtx : { x : HOD} → A ∋ x → HOD
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128 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where
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129 z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A
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130 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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131 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
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132 z01 {a} {b} A∋a A∋b (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋a} (sym a=b) b<a
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133 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b} b<a a<b)
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134 -- ZChain is not compatible with the SUP condition
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135 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where
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136 field
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137 bx : Ordinal
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138 bx<y : bx o< y
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139 is-fb : x ≡ & (fb bx )
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140 bx<A : (z : ZChain A (& A) _<_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o< & A
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141 bx<A z {x} bx = BX.bx<y bx
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142 z12 : (z : ZChain A (& A) _<_ ) → {y : Ordinal} → BX y (& A) (ZChain.fb z) → y o< & A
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143 z12 z {y} bx = subst (λ k → k o< & A) (sym (BX.is-fb bx)) (c<→o< (ZChain.A∋fb z (BX.bx bx) (BX.bx<y bx)))
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144 B : (z : ZChain A (& A) _<_ ) → HOD
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145 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; <odmax = z12 z }
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146 z11 : (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-elm x))
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147 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
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148 obx : (z : ZChain A (& A) _<_ ) → {x : HOD} → B z ∋ x → Ordinal
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149 obx z {x} bx = BX.bx bx
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150 obx=fb : (z : ZChain A (& A) _<_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z (BX.bx bx)
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151 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx))
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152 B⊆A : (z : ZChain A (& A) _<_ ) → B z ⊆ A
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153 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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154 PO-B : (z : ZChain A (& A) _<_ ) → IsPartialOrderSet (B z) _<_
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155 PO-B z = ⊆-IsPartialOrderSet (B⊆A z) PO
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156 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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157 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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158 bcmp : (z : ZChain A (& A) _<_ ) → Trichotomous (λ (x : Element (B z)) y → elm x ≡ elm y ) (λ x y → elm x < elm y )
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159 bcmp z x y with trio< (obx z (is-elm x)) (obx z (is-elm y))
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160 ... | tri< a ¬b ¬c = tri< z15 (λ eq → z01 (incl (B⊆A z) (is-elm y)) (incl (B⊆A z) (is-elm x))(case1 (sym eq)) z15 ) z17 where
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161 z15 : elm x < elm y
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162 z15 = bx-monotonic z {x} {y} a
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163 z17 : elm y < elm x → ⊥
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164 z17 lt = z01 (incl (B⊆A z) (is-elm y)) (incl (B⊆A z) (is-elm x))(case2 lt) z15
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165 ... | tri≈ ¬a b ¬c = tri≈ (IsStrictPartialOrder.irrefl PO {isA (B⊆A z) x} {isA (B⊆A z) y} z14) z14 z16 where
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166 z14 : elm x ≡ elm y
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167 z14 = begin
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168 elm x ≡⟨ obx=fb z (is-elm x) ⟩
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169 ZChain.fb z (BX.bx (is-elm x)) ≡⟨ cong (ZChain.fb z) b ⟩
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170 ZChain.fb z (BX.bx (is-elm y)) ≡⟨ sym ( obx=fb z (is-elm y)) ⟩
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171 elm y ∎ where open ≡-Reasoning
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172 z16 = IsStrictPartialOrder.irrefl PO {isA (B⊆A z) y} {isA (B⊆A z) x} (sym z14)
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173 ... | tri> ¬a ¬b c = tri> z17 (λ eq → z01 (incl (B⊆A z) (is-elm x)) (incl (B⊆A z) (is-elm y))(case1 eq) z15 ) z15 where
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174 z15 : elm y < elm x
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175 z15 = bx-monotonic z {y} {x} c
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176 z17 : elm x < elm y → ⊥
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177 z17 lt = z01 (incl (B⊆A z) (is-elm x)) (incl (B⊆A z) (is-elm y))(case2 lt) z15
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178 B-is-total : (z : ZChain A (& A) _<_ ) → IsTotalOrderSet (B z) _<_
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179 B-is-total zc = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
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180 ; trans = λ {x} {y} {z} x<y y<z → IsStrictPartialOrder.trans PO {isA (B⊆A zc) x} {isA (B⊆A zc) y} {isA (B⊆A zc) z} x<y y<z
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181 ; compare = bcmp zc }
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182 ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ )
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183 ZChain→¬SUP z sp = ⊥-elim {!!} where
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184 z03 : & (SUP.sup sp) o< osuc (& A)
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185 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
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186 z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥
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187 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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188 ind : HasMaximal =h= od∅
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189 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ )
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190 → ZChain A x _<_
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191 ind nomx x prev with Oprev-p x
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192 ... | yes op with ODC.∋-p O A (* x)
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193 ... | no ¬Ax = {!!} where
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194 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
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195 px = Oprev.oprev op
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196 zc1 : ZChain A px _<_
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197 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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198 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as
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199 z04 sup as s<x with trio< (& sup) x
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200 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
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201 ... | 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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202 ... | tri> ¬a ¬b c with osuc-≡< s<x
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203 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
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204 ... | case2 lt = ⊥-elim (¬a lt )
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205 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
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206 px = Oprev.oprev op
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207 zc1 : ZChain A px _<_
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208 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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209 z06 : ZChain A x _<_
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210 z06 with is-o∅ (& (Gtx ax))
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211 ... | 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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212 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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213 x-is-maximal m am = ¬x<m where
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214 ¬x<m : ¬ (* x < * m)
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215 ¬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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216 ... | no not = {!!} where
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217 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
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218 ... | tri< a ¬b ¬c = {!!} where
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219 zc1 : ZChain A (& A) _<_
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220 zc1 = prev (& A) a
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221 ... | tri≈ ¬a b ¬c = {!!} where
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222 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
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223 ... | no ¬Ax = {!!} where
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224 ... | yes ax with is-o∅ (& (Gtx ax))
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225 ... | 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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226 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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227 x-is-maximal m am = ¬x<m where
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228 ¬x<m : ¬ (* x < * m)
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229 ¬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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230 ... | no not = {!!} where
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231 zorn00 : Maximal A _<_
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467
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232 zorn00 with is-o∅ ( & HasMaximal )
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493
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233 ... | 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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478
|
234 -- yes we have the maximal
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483
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235 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
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236 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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477
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237 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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483
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238 zorn01 = proj1 hasm
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493
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239 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
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240 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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483
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241 ... | yes ¬Maximal = ⊥-elim {!!} where
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478
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242 -- if we have no maximal, make ZChain, which contradict SUP condition
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493
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243 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_
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471
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244 z x nomx = TransFinite (ind nomx) x
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|
464
|
245
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482
|
246 _⊆'_ : ( A B : HOD ) → Set n
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247 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
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248
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249 MaximumSubset : {L P : HOD}
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250 → o∅ o< & L → o∅ o< & P → P ⊆ L
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491
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251 → IsPartialOrderSet P _⊆'_
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252 → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
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482
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253 → Maximal P (_⊆'_)
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254 MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
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