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1 {-# OPTIONS --allow-unsolved-metas #-}
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2 open import Level hiding ( suc ; zero )
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3 open import Ordinals
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4 import OD
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5 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) where
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6
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7 open import zf
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8 open import logic
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9 -- open import partfunc {n} O
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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 (Level.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 _<A_ : {A : HOD} → (x y : Element A ) → Set n
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51 x <A y = elm x < elm y
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52 _≡A_ : {A : HOD} → (x y : Element A ) → Set (Level.suc n)
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53 x ≡A y = elm x ≡ elm y
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54
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55 IsPartialOrderSet : ( A : HOD ) → Set (Level.suc n)
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56 IsPartialOrderSet A = IsStrictPartialOrder (_≡A_ {A}) _<A_
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57
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58 open _==_
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59 open _⊆_
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60
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61 isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A
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62 isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) }
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63
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64 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → IsPartialOrderSet A → IsPartialOrderSet B
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65 ⊆-IsPartialOrderSet {A} {B} B⊆A PA = record {
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66 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {x} {y} {z} → trans1 {x} {y} {z}
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67 ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; <-resp-≈ = resp0
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68 } where
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69 _<B_ : (x y : Element B ) → Set n
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70 x <B y = elm x < elm y
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71 trans1 : {x y z : Element B} → x <B y → y <B z → x <B z
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72 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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73 irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y )
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74 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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75 open import Data.Product
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76 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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77 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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78 (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })
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79
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80 -- open import Relation.Binary.Properties.Poset as Poset
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81
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82 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n)
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83 IsTotalOrderSet A = IsStrictTotalOrder (_≡A_ {A}) _<A_
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84
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85 me : { A a : HOD } → A ∋ a → Element A
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86 me {A} {a} lt = record { elm = a ; is-elm = lt }
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87
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88 A∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (A ∋ x) ≡ (A ∋ y )
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89 A∋x-irr A {x} {y} refl = refl
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90
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91 me-elm-refl : (A : HOD) → (x : Element A) → elm (me {A} (d→∋ A (is-elm x))) ≡ elm x
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92 me-elm-refl A record { elm = ex ; is-elm = ax } = *iso
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93
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94 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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95
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96 -- postulate -- may be proved by transfinite induction and functional extentionality
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97 -- ∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (ax : A ∋ x) (ay : A ∋ y ) → ax ≅ ay
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98 -- odef-irr : (A : OD) {x y : Ordinal} → x ≡ y → (ax : def A x) (ay : def A y ) → ax ≅ ay
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99
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100 -- is-elm-irr : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y
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101 -- is-elm-irr A {x} {y} eq = ∋x-irr A eq (is-elm x) (is-elm y )
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102
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103 El-irr2 : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y → x ≡ y
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104 El-irr2 A {x} {y} refl HE.refl = refl
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105
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106 -- El-irr : {A : HOD} → {x y : Element A } → elm x ≡ elm y → x ≡ y
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107 -- El-irr {A} {x} {y} eq = El-irr2 A eq ( is-elm-irr A eq )
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108
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109 record ZChain ( A : HOD ) (y : Ordinal) : Set (Level.suc n) where
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110 field
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111 max : HOD
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112 A∋max : A ∋ max
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113 y<max : y o< & max
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114 chain : HOD
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115 chain⊆A : chain ⊆ A
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116 total : IsTotalOrderSet chain
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117 chain-max : (x : HOD) → chain ∋ x → (x ≡ max ) ∨ ( x < max )
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118
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119 data IChain (A : HOD) : Ordinal → Set n where
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120 ifirst : {ox : Ordinal} → odef A ox → IChain A ox
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121 inext : {ox oy : Ordinal} → odef A oy → * ox < * oy → IChain A ox → IChain A oy
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122
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123 -- * ox < .. < * oy
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124 ic-connect : {A : HOD} {oy : Ordinal} → (ox : Ordinal) → (iy : IChain A oy) → Set n
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125 ic-connect {A} ox (ifirst {oy} ay) = Lift n ⊥
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126 ic-connect {A} ox (inext {oy} {oz} ay y<z iz) = (ox ≡ oy) ∨ ic-connect ox iz
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127
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128 ic→odef : {A : HOD} {ox : Ordinal} → IChain A ox → odef A ox
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129 ic→odef {A} {ox} (ifirst ax) = ax
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130 ic→odef {A} {ox} (inext ax x<y ic) = ax
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131
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132 ic→< : {A : HOD} → IsPartialOrderSet A → (x : Ordinal) → odef A x → {y : Ordinal} → (iy : IChain A y) → ic-connect {A} {y} x iy → * x < * y
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133 ic→< {A} PO x ax {y} (ifirst ay) ()
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134 ic→< {A} PO x ax {y} (inext ay x<y iy) (case1 refl) = x<y
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135 ic→< {A} PO x ax {y} (inext {oy} ay x<y iy) (case2 ic) = IsStrictPartialOrder.trans PO
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136 {me (subst (λ k → odef A k) (sym &iso) ax )} {me (subst (λ k → odef A k) (sym &iso) (ic→odef {A} {oy} iy) ) } {me (subst (λ k → odef A k) (sym &iso) ay) }
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137 (ic→< {A} PO x ax iy ic ) x<y
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138
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139 record IChained (A : HOD) (x y : Ordinal) : Set n where
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140 field
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141 iy : IChain A y
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142 ic : ic-connect x iy
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143
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144 IChainSet : {A : HOD} → Element A → HOD
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145 IChainSet {A} ax = record { od = record { def = λ y → odef A y ∧ IChained A (& (elm ax)) y }
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146 ; odmax = & A ; <odmax = λ {y} lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k) (sym &iso) (proj1 lt))) }
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147
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148 IChainSet⊆A : {A : HOD} → (x : Element A ) → IChainSet x ⊆ A
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149 IChainSet⊆A {A} x = record { incl = λ {oy} y → proj1 y }
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150
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151 ¬IChained-refl : (A : HOD) {x : Ordinal} → IsPartialOrderSet A → ¬ IChained A x x
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152 ¬IChained-refl A {x} PO record { iy = iy ; ic = ic } = IsStrictPartialOrder.irrefl PO
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153 {me (subst (λ k → odef A k ) (sym &iso) ic0) } {me (subst (λ k → odef A k ) (sym &iso) ic0) } refl (ic→< {A} PO x ic0 iy ic ) where
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154 ic0 : odef A x
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155 ic0 = ic→odef {A} iy
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156
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157 -- there is a y, & y > & x
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158
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159 record OSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where
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160 field
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161 y : Ordinal
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162 icy : odef (IChainSet {A} (me ax)) y
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163 y>x : x o< y
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164
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165 record IChainSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where
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166 field
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167 y : Ordinal
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168 A∋y : odef A y
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169 y>x : * x < * y
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170
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171 -- finite IChain
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172
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173 ic→A∋y : (A : HOD) {x y : Ordinal} (ax : A ∋ * x) → odef (IChainSet {A} (me ax)) y → A ∋ * y
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174 ic→A∋y A {x} {y} ax ⟪ ay , _ ⟫ = subst (λ k → odef A k) (sym &iso) ay
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175
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176 record InFiniteIChain (A : HOD) (max : Ordinal) {x : Ordinal} (ax : A ∋ * x) : Set n where
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177 field
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178 chain<x : (y : Ordinal ) → odef (IChainSet {A} (me ax)) y → y o< max
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179 c-infinite : (y : Ordinal ) → (cy : odef (IChainSet {A} (me ax)) y )
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180 → IChainSup> A (ic→A∋y A ax cy)
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181
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182 open import Data.Nat hiding (_<_)
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183 import Data.Nat.Properties as NP
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184 open import nat
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185
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186 data Chain (A : HOD) (s : Ordinal) (next : Ordinal → Ordinal ) : ( x : Ordinal ) → Set n where
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187 cfirst : odef A s → Chain A s next s
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188 csuc : (x : Ordinal ) → (ax : odef A x ) → Chain A s next x → odef A (next x) → Chain A s next (next x )
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189
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190 ct∈A : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → {x : Ordinal} → Chain A s next x → odef A x
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191 ct∈A A s next {x} (cfirst x₁) = x₁
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192 ct∈A A s next {.(next x )} (csuc x ax t anx) = anx
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193
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194 --
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195 -- extract single chain from countable infinite chains
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196 --
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197 ChainClosure : (A : HOD) (s : Ordinal) → (next : Ordinal → Ordinal ) → HOD
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198 ChainClosure A s next = record { od = record { def = λ x → Chain A s next x } ; odmax = & A ; <odmax = cc01 } where
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199 cc01 : {y : Ordinal} → Chain A s next y → y o< & A
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200 cc01 {y} cy = subst (λ k → k o< & A ) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) ( ct∈A A s next cy ) ) )
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201
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202 cton0 : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) {y : Ordinal } → Chain A s next y → ℕ
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203 cton0 A s next (cfirst _) = zero
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204 cton0 A s next (csuc x ax z _) = suc (cton0 A s next z)
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205 cton : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → Element (ChainClosure A s next) → ℕ
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206 cton A s next y = cton0 A s next (is-elm y)
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207
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208 cinext : (A : HOD) {x max : Ordinal } → (ax : A ∋ * x ) → (ifc : InFiniteIChain A max ax ) → Ordinal → Ordinal
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209 cinext A ax ifc y with ODC.∋-p O (IChainSet (me ax)) (* y)
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210 ... | yes ics-y = IChainSup>.y ( InFiniteIChain.c-infinite ifc y (subst (λ k → odef (IChainSet (me ax)) k) &iso ics-y ))
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211 ... | no _ = o∅
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212
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213 InFCSet : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x)
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214 → (ifc : InFiniteIChain A max ax ) → HOD
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215 InFCSet A {x} ax ifc = ChainClosure (IChainSet {A} (me ax)) x (cinext A ax ifc )
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216
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217 InFCSet⊆A : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x) → (ifc : InFiniteIChain A max ax ) → InFCSet A ax ifc ⊆ A
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218 InFCSet⊆A A {x} ax ifc = record { incl = λ {y} lt → incl (IChainSet⊆A (me ax)) (
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219 ct∈A (IChainSet {A} (me ax)) x (cinext A ax ifc) lt ) }
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220
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221 ChainClosure-is-total : (A : HOD) → {x max : Ordinal} (ax : A ∋ * x)
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222 → IsPartialOrderSet A
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223 → (ifc : InFiniteIChain A max ax )
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224 → IsTotalOrderSet ( InFCSet A ax ifc )
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225 ChainClosure-is-total A {x} ax PO ifc = record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO
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226 ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z} ; compare = cmp } where
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227 IPO : IsPartialOrderSet (InFCSet A ax ifc )
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228 IPO = ⊆-IsPartialOrderSet record { incl = λ {y} lt → incl (InFCSet⊆A A {x} ax ifc) lt} PO
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229 B = IChainSet {A} (me ax)
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230 cnext = cinext A ax ifc
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231 ct02 : {oy : Ordinal} → (y : Chain B x cnext oy ) → A ∋ * oy
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232 ct02 y = incl (IChainSet⊆A {A} (me ax)) (subst (λ k → odef (IChainSet (me ax)) k) (sym &iso) (ct∈A B x cnext y) )
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233 ct-inject : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy )
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234 → (cton0 B x cnext x1) ≡ (cton0 B x cnext y) → ox ≡ oy
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235 ct-inject {ox} {ox} (cfirst x) (cfirst x₁) refl = refl
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236 ct-inject {.(cnext x₀ )} {.(cnext x₃ )} (csuc x₀ ax x₁ x₂) (csuc x₃ ax₁ y x₄) eq = cong cnext ct05 where
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237 ct06 : {x y : ℕ} → suc x ≡ suc y → x ≡ y
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238 ct06 refl = refl
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239 ct05 : x₀ ≡ x₃
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240 ct05 = ct-inject x₁ y (ct06 eq)
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241 ct-monotonic : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy )
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242 → (cton0 B x cnext x1) Data.Nat.< (cton0 B x cnext y) → * ox < * oy
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243 ct-monotonic {ox} {oy} x1 (csuc oy1 ay y _) (s≤s lt) with NP.<-cmp ( cton0 B x cnext x1 ) ( cton0 B x cnext y )
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244 ... | tri< a ¬b ¬c = ct07 where
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245 ct07 : * ox < * (cnext oy1)
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246 ct07 with ODC.∋-p O (IChainSet {A} (me ax)) (* oy1)
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247 ... | no not = ⊥-elim ( not (subst (λ k → odef (IChainSet {A} (me ax)) k ) (sym &iso) ay ) )
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248 ... | yes ay1 = IsStrictPartialOrder.trans PO {me (ct02 x1) } {me (ct02 y)} {me ct031 } (ct-monotonic x1 y a ) ct011 where
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249 ct031 : A ∋ * (IChainSup>.y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 ) ))
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250 ct031 = subst (λ k → odef A k ) (sym &iso) (
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251 IChainSup>.A∋y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) )
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252 ct011 : * oy1 < * ( IChainSup>.y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) )
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253 ct011 = IChainSup>.y>x (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 ))
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254 ... | tri≈ ¬a b ¬c = ct11 where
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255 ct11 : * ox < * (cnext oy1)
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256 ct11 with ODC.∋-p O (IChainSet {A} (me ax)) (* oy1)
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257 ... | no not = ⊥-elim ( not (subst (λ k → odef (IChainSet {A} (me ax)) k ) (sym &iso) ay ) )
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258 ... | yes ay1 = subst (λ k → * k < _) (sym (ct-inject _ _ b)) ct011 where
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259 ct011 : * oy1 < * ( IChainSup>.y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) )
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260 ct011 = IChainSup>.y>x (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 ))
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261 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> lt c )
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262 ct12 : {y z : Element (ChainClosure B x cnext) } → elm y ≡ elm z → elm y < elm z → ⊥
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263 ct12 {y} {z} y=z y<z = IsStrictPartialOrder.irrefl IPO {y} {z} y=z y<z
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264 ct13 : {y z : Element (ChainClosure B x cnext) } → elm y < elm z → elm z < elm y → ⊥
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265 ct13 {y} {z} y<z y>z = IsStrictPartialOrder.irrefl IPO {y} {y} refl ( IsStrictPartialOrder.trans IPO {y} {z} {y} y<z y>z )
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266 ct17 : (x1 : Element (ChainClosure B x cnext)) → Chain B x cnext (& (elm x1))
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267 ct17 x1 = is-elm x1
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509
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268 cmp : Trichotomous _ _
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513
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269 cmp x1 y with NP.<-cmp (cton B x cnext x1) (cton B x cnext y)
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517
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270 ... | tri< a ¬b ¬c = tri< ct04 ct14 ct15 where
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513
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271 ct04 : elm x1 < elm y
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272 ct04 = subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a)
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517
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273 ct14 : ¬ elm x1 ≡ elm y
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274 ct14 eq = ct12 {x1} {y} eq (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a ) )
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275 ct15 : ¬ (elm y < elm x1)
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276 ct15 lt = ct13 {y} {x1} lt (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a ) )
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277 ... | tri≈ ¬a b ¬c = tri≈ (ct12 {x1} {y} ct16) ct16 (ct12 {y} {x1} (sym ct16)) where
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278 ct16 : elm x1 ≡ elm y
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279 ct16 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (ct-inject {& (elm x1)} {& (elm y)} (is-elm x1) (is-elm y) b ))
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280 ... | tri> ¬a ¬b c = tri> ct15 ct14 ct04 where
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281 ct04 : elm y < elm x1
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282 ct04 = subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c)
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283 ct14 : ¬ elm x1 ≡ elm y
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284 ct14 eq = ct12 {y} {x1} (sym eq) (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c ) )
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285 ct15 : ¬ (elm x1 < elm y)
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286 ct15 lt = ct13 {x1} {y} lt (subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm y) (is-elm x1) c ) )
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509
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287
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501
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288
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502
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289 record IsFC (A : HOD) {x : Ordinal} (ax : A ∋ * x) (y : Ordinal) : Set n where
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501
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290 field
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291 icy : odef (IChainSet {A} (me ax)) y
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520
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292 c-finite : ¬ IChainSup> A (subst (λ k → odef A k ) (sym &iso) (proj1 icy) )
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497
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293
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508
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294 record Maximal ( A : HOD ) : Set (Level.suc n) where
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503
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295 field
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296 maximal : HOD
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297 A∋maximal : A ∋ maximal
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298 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
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299
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300 --
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301 -- possible three cases in a limit ordinal step
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302 --
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507
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303 -- case 1) < goes > x (will contradic in the transfinite induction )
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503
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304 -- case 2) no > x in some chain ( maximal )
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507
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305 -- case 3) countably infinite chain below x (will be prohibited by sup condtion )
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503
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306 --
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307 Zorn-lemma-3case : { A : HOD }
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498
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308 → o∅ o< & A
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309 → IsPartialOrderSet A
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522
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310 → (x : Element A) → OSup> A (d→∋ A (is-elm x)) ∨ Maximal A ∨ InFiniteIChain A (& (elm x)) (d→∋ A (is-elm x))
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503
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311 Zorn-lemma-3case {A} 0<A PO x = zc2 where
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499
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312 Gtx : HOD
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313 Gtx = record { od = record { def = λ y → odef ( IChainSet x ) y ∧ ( & (elm x) o< y ) } ; odmax = & A
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501
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314 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 (proj1 lt)))) }
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315 HG : HOD
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502
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316 HG = record { od = record { def = λ y → odef A y ∧ IsFC A (d→∋ A (is-elm x) ) y } ; odmax = & A
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501
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317 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 lt) )) }
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522
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318 zc2 : OSup> A (d→∋ A (is-elm x)) ∨ Maximal A ∨ InFiniteIChain A (& (elm x)) (d→∋ A (is-elm x))
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499
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319 zc2 with is-o∅ (& Gtx)
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504
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320 ... | no not = case1 record { y = & y ; icy = zc4 ; y>x = proj2 zc3 } where
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321 y : HOD
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322 y = ODC.minimal O Gtx (λ eq → not (=od∅→≡o∅ eq))
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323 zc3 : odef ( IChainSet x ) (& y) ∧ ( & (elm x) o< (& y ))
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324 zc3 = ODC.x∋minimal O Gtx (λ eq → not (=od∅→≡o∅ eq))
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325 zc4 : odef (IChainSet (me (subst (OD.def (od A)) (sym &iso) (is-elm x)))) (& y)
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506
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326 zc4 = ⟪ proj1 (proj1 zc3) , subst (λ k → IChained A (& k) (& y) ) (sym *iso) (proj2 (proj1 zc3)) ⟫
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501
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327 ... | yes nogt with is-o∅ (& HG)
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505
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328 ... | no finite-chain = case2 (case1 record { maximal = y ; A∋maximal = proj1 zc3 ; ¬maximal<x = zc4 } ) where
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504
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329 y : HOD
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505
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330 y = ODC.minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq))
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504
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331 zc3 : odef A (& y) ∧ IsFC A (d→∋ A (is-elm x) ) (& y)
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505
|
332 zc3 = ODC.x∋minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq))
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504
|
333 zc5 : odef (IChainSet {A} (me (d→∋ A (is-elm x) ))) (& y)
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334 zc5 = IsFC.icy (proj2 zc3)
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335 zc4 : {z : HOD} → A ∋ z → ¬ (y < z)
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520
|
336 zc4 {z} az y<z = IsFC.c-finite (proj2 zc3) record { y = & z ; A∋y = az ; y>x = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) y<z } where
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506
|
337 zc8 : ic-connect (& (* (& (elm x)))) (IChained.iy (proj2 zc5))
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338 zc8 = IChained.ic (proj2 zc5)
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339 zc7 : elm x < y
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340 zc7 = subst₂ (λ j k → j < k ) *iso *iso ( ic→< {A} PO (& (elm x)) (is-elm x) (IChained.iy (proj2 zc5))
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341 (subst (λ k → ic-connect (& k) (IChained.iy (proj2 zc5)) ) (me-elm-refl A x) (IChained.ic (proj2 zc5)) ) )
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505
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342 zc6 : elm x < z
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506
|
343 zc6 = IsStrictPartialOrder.trans PO {x} {me (proj1 zc3)} {me az} zc7 y<z
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521
|
344 ... | yes inifite = case2 (case2 record { c-infinite = zc9 ; chain<x = zc10} ) where
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518
|
345 B : HOD
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346 B = IChainSet {A} (me (subst (OD.def (od A)) (sym &iso) (is-elm x)))
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347 Nx : (y : Ordinal) → odef A y → HOD
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520
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348 Nx y ay = record { od = record { def = λ x → odef A x ∧ ( * y < * x ) } ; odmax = & A
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518
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349 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 lt))) }
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521
|
350 zc10 : (oy : Ordinal) → odef (IChainSet {A} (me (subst (OD.def (od A)) (sym &iso) (is-elm x)))) oy → oy o< & (elm x)
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351 zc10 oy icsy = zc21 where
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352 zc20 : (y : HOD) → (IChainSet x) ∋ y → & (elm x) o< & y → ⊥
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353 zc20 y icsy lt = ¬A∋x→A≡od∅ Gtx ⟪ icsy , lt ⟫ nogt
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354 zc22 : IChainSet x ∋ * oy
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355 zc22 = ⟪ subst (λ k → odef A k) (sym &iso) (proj1 icsy)
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356 , subst₂ (λ j k → IChained A j k) (cong (&) (me-elm-refl A x)) (sym &iso) (proj2 icsy) ⟫
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357 zc21 : oy o< & (elm x)
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358 zc21 with trio< oy (& (elm x) )
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359 ... | tri< a ¬b ¬c = a
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360 ... | tri≈ ¬a b ¬c = ⊥-elim (¬IChained-refl A PO (subst₂ (λ j k → IChained A j k ) zc23 b (proj2 icsy)) ) where
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361 zc23 : & (* (& (elm x))) ≡ & (elm x)
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362 zc23 = cong (&) *iso
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363 ... | tri> ¬a ¬b c = ⊥-elim ( zc20 (* oy) zc22 (subst (λ k → & (elm x) o< k) (sym &iso) c ))
|
|
520
|
364 zc9 : (y : Ordinal) (cy : odef B y) →
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|
518
|
365 IChainSup> A (ic→A∋y A (subst (OD.def (od A)) (sym &iso) (is-elm x)) cy)
|
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521
|
366 zc9 y cy with is-o∅ (& (Nx y (proj1 cy) ))
|
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519
|
367 ... | yes no-next = ⊥-elim zc16 where
|
|
520
|
368 zc18 : ¬ IChainSup> A (subst (odef A) (sym &iso) (proj1 (subst (λ k → odef (IChainSet (me (d→∋ A (is-elm x)))) k) (sym &iso) cy)))
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369 zc18 ics = ¬A∋x→A≡od∅ (Nx y (proj1 cy) ) ⟪ subst (λ k → odef A k ) (sym &iso) (IChainSup>.A∋y ics)
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370 , subst₂ (λ j k → j < k ) *iso (cong (*) (sym &iso))( IChainSup>.y>x ics) ⟫ no-next
|
|
519
|
371 zc17 : IsFC A {& (elm x)} (d→∋ A (is-elm x)) (& (* y))
|
|
520
|
372 zc17 = record { icy = subst (λ k → odef (IChainSet (me (d→∋ A (is-elm x)))) k ) (sym &iso) cy ; c-finite = zc18 }
|
|
519
|
373 zc16 : ⊥
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|
374 zc16 = ¬A∋x→A≡od∅ HG ⟪ subst (λ k → odef A k ) (sym &iso) (proj1 cy ) , zc17 ⟫ inifite
|
|
520
|
375 ... | no not = record { y = & zc13 ; A∋y = proj1 zc12 ; y>x = proj2 zc12 } where
|
|
519
|
376 zc13 = ODC.minimal O (Nx y (proj1 cy)) (λ eq → not (=od∅→≡o∅ eq ))
|
|
520
|
377 zc12 : odef A (& zc13 ) ∧ ( * y < * ( & zc13 ))
|
|
519
|
378 zc12 = ODC.x∋minimal O (Nx y (proj1 cy)) (λ eq → not (=od∅→≡o∅ eq ))
|
|
499
|
379
|
|
517
|
380 all-climb-case : { A : HOD } → (0<A : o∅ o< & A) → IsPartialOrderSet A
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|
381 → (( x : Element A) → OSup> A (d→∋ A (is-elm x) ))
|
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522
|
382 → InFiniteIChain A (& A) (d→∋ A (ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ))
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|
383 all-climb-case {A} 0<A PO climb = record { c-infinite = λ y cy → ac00 (& x) y (proj1 cy) (subst (λ k → IChained A k y )
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384 (cong (&) (me-elm-refl A (me ax))) (proj2 cy))
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|
385 ; chain<x = ac01 } where
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|
386 x = ODC.minimal O A (λ eq → ¬x<0 (subst (_o<_ o∅) (=od∅→≡o∅ eq) 0<A))
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387 ax = ODC.x∋minimal O A (λ eq → ¬x<0 (subst (_o<_ o∅) (=od∅→≡o∅ eq) 0<A))
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|
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388 B = IChainSet {A} (me (d→∋ A ax))
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389 ac01 : (y : Ordinal) → odef (IChainSet {A} (me (d→∋ A ax))) y → y o< & A
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390 ac01 y ⟪ ay , _ ⟫ = subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) ay) )
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391 ac00 : (x y : Ordinal) (ay : odef A y) (cy : IChained A x y) → IChainSup> A (subst (λ k → odef A k ) (sym &iso) ay )
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|
392 ac00 x y ay cy = record { y = z ; A∋y = az ; y>x = {!!} } where
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393 z : Ordinal
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394 z = OSup>.y ( climb (me (subst (λ k → odef A k ) (sym &iso) ay) ) )
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395 az : odef A z
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396 icy : odef (IChainSet {A} (me (subst (λ k → odef A k ) {!!} ay))) z
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397 icy = OSup>.icy ( climb (me (subst (λ k → odef A k ) (sym &iso) ay) ) )
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398 az = {!!}
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399 -- incl (IChainSet⊆A {A} ? ) (subst (λ k → odef (IChainSet {A} ? ) k ) ? (OSup>.icy ( climb (me (subst (λ k → odef A k ) (sym &iso) ay) ) )))
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400 - = OSup>.y ( climb (me (subst (λ k → odef A k ) (sym &iso) ay) ) )
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401 -- iy0 : IChained A (& (* (& (ODC.minimal O A (λ eq → ¬x<0 (subst (_o<_ o∅) (=od∅→≡o∅ eq) 0<A)))))) ?
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402 -- iy0 = iy
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403
|
|
|
404
|
|
498
|
405
|
|
508
|
406 record SUP ( A B : HOD ) : Set (Level.suc n) where
|
|
503
|
407 field
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|
|
408 sup : HOD
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|
409 A∋maximal : A ∋ sup
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410 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive
|
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411
|
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497
|
412 Zorn-lemma : { A : HOD }
|
|
464
|
413 → o∅ o< & A
|
|
497
|
414 → IsPartialOrderSet A
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|
|
415 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition
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|
416 → Maximal A
|
|
507
|
417 Zorn-lemma {A} 0<A PO supP = zorn04 where
|
|
493
|
418 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
|
|
496
|
419 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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420 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)
|
|
522
|
421 z02 : {x : Ordinal } → (ax : A ∋ * x ) → InFiniteIChain A x ax → ⊥
|
|
507
|
422 z02 {x} ax ic = zc5 ic where
|
|
|
423 FC : HOD
|
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|
424 FC = IChainSet {A} (me ax)
|
|
522
|
425 zc6 : InFiniteIChain A x ax → ¬ SUP A FC
|
|
507
|
426 zc6 inf = {!!}
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|
|
427 FC-is-total : IsTotalOrderSet FC
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|
|
428 FC-is-total = {!!}
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|
|
429 FC⊆A : FC ⊆ A
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|
|
430 FC⊆A = record { incl = λ {x} lt → proj1 lt }
|
|
522
|
431 zc5 : InFiniteIChain A x ax → ⊥
|
|
507
|
432 zc5 x = zc6 x ( supP FC FC⊆A FC-is-total )
|
|
478
|
433 -- ZChain is not compatible with the SUP condition
|
|
497
|
434 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y ∨ Maximal A )
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|
435 → ZChain A x ∨ Maximal A
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|
|
436 ind x prev with Oprev-p x
|
|
477
|
437 ... | yes op with ODC.∋-p O A (* x)
|
|
498
|
438 ... | no ¬Ax = zc1 where
|
|
476
|
439 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
|
|
471
|
440 px = Oprev.oprev op
|
|
498
|
441 zc1 : ZChain A x ∨ Maximal A
|
|
497
|
442 zc1 with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
|
|
498
|
443 ... | case2 x = case2 x -- we have the Maximal
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|
444 ... | case1 z with trio< x (& (ZChain.max z))
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|
445 ... | tri< a ¬b ¬c = case1 record { max = ZChain.max z ; y<max = a }
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|
|
446 ... | tri≈ ¬a b ¬c = {!!} -- x = max so ¬ A ∋ max
|
|
|
447 ... | tri> ¬a ¬b c = {!!} -- can't happen
|
|
503
|
448 ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x
|
|
472
|
449 px = Oprev.oprev op
|
|
503
|
450 zc1 : OSup> A (subst (OD.def (od A)) (sym &iso) ax) → ZChain A x ∨ Maximal A
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|
|
451 zc1 os with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
|
|
498
|
452 ... | case2 x = case2 x
|
|
507
|
453 ... | case1 x = {!!}
|
|
503
|
454 zc4 : ZChain A x ∨ Maximal A
|
|
|
455 zc4 with Zorn-lemma-3case 0<A PO (me ax)
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|
|
456 ... | case1 y>x = zc1 y>x
|
|
|
457 ... | case2 (case1 x) = case2 x
|
|
522
|
458 ... | case2 (case2 ex) = ⊥-elim (zc5 {!!} ) where
|
|
503
|
459 FC : HOD
|
|
|
460 FC = IChainSet {A} (me ax)
|
|
522
|
461 B : InFiniteIChain A x ax → HOD
|
|
511
|
462 B ifc = InFCSet A ax ifc
|
|
522
|
463 zc6 : (ifc : InFiniteIChain A x ax ) → ¬ SUP A (B ifc)
|
|
503
|
464 zc6 = {!!}
|
|
522
|
465 FC-is-total : (ifc : InFiniteIChain A x ax) → IsTotalOrderSet (B ifc)
|
|
511
|
466 FC-is-total ifc = ChainClosure-is-total A ax PO ifc
|
|
522
|
467 B⊆A : (ifc : InFiniteIChain A x ax) → B ifc ⊆ A
|
|
511
|
468 B⊆A = {!!}
|
|
522
|
469 ifc : InFiniteIChain A x (subst (OD.def (od A)) (sym &iso) ax) → InFiniteIChain A x ax
|
|
517
|
470 ifc record { c-infinite = c-infinite } = record { c-infinite = {!!} } where
|
|
512
|
471 ifc01 : {!!} -- me (subst (OD.def (od A)) (sym &iso) ax)
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472 ifc01 = {!!}
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473 -- (y : Ordinal) → odef (IChainSet (me (subst (OD.def (od A)) (sym &iso) ax))) y → y o< & (* x₁)
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474 -- (y : Ordinal) → odef (IChainSet (me ax)) y → y o< x₁
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522
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475 zc5 : InFiniteIChain A x (subst (OD.def (od A)) (sym &iso) ax) → ⊥
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511
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476 zc5 x = zc6 (ifc x) ( supP (B (ifc x)) (B⊆A (ifc x)) (FC-is-total (ifc x) ))
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497
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477 ind x prev | no ¬ox with trio< (& A) x --- limit ordinal case
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483
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478 ... | tri< a ¬b ¬c = {!!} where
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497
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479 zc1 : ZChain A (& A)
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480 zc1 with prev (& A) a
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481 ... | t = {!!}
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483
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482 ... | tri≈ ¬a b ¬c = {!!} where
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478
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483 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
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484 ... | no ¬Ax = {!!} where
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507
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485 ... | yes ax = {!!}
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497
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486 zorn03 : (x : Ordinal) → ZChain A x ∨ Maximal A
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507
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487 zorn03 x = TransFinite ind x
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497
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488 zorn04 : Maximal A
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489 zorn04 with zorn03 (& A)
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507
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490 ... | case1 chain = ⊥-elim ( o<> (c<→o< {ZChain.max chain} {A} (ZChain.A∋max chain)) (ZChain.y<max chain) )
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497
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491 ... | case2 m = m
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464
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492
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516
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493 -- usage (see filter.agda )
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494 --
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497
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495 -- _⊆'_ : ( A B : HOD ) → Set n
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496 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
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482
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497
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497
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498 -- MaximumSubset : {L P : HOD}
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499 -- → o∅ o< & L → o∅ o< & P → P ⊆ L
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500 -- → IsPartialOrderSet P _⊆'_
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501 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
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502 -- → Maximal P (_⊆'_)
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503 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
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