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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 IsPartialOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
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58 IsPartialOrderSet A _<_ = IsPartialOrder _<A_ _≡A_ where
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59 _<A_ : (x y : Element A ) → Set n
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60 x <A y = elm x < elm y
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61 _≡A_ : (x y : Element A ) → Set (suc n)
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62 x ≡A y = elm x ≡ elm y
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63
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64 IsTotalOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
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65 IsTotalOrderSet A _<_ = IsTotalOrder _<A_ _≡A_ where
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66 _<A_ : (x y : Element A ) → Set n
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67 x <A y = elm x < elm y
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68 _≡A_ : (x y : Element A ) → Set (suc n)
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69 x ≡A y = elm x ≡ elm y
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70
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71 me : { A a : HOD } → A ∋ a → Element A
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72 me {A} {a} lt = record { elm = a ; is-elm = lt }
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73
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74 record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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75 field
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76 sup : HOD
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77 A∋maximal : A ∋ sup
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78 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive
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79
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80 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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81 field
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82 maximal : HOD
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83 A∋maximal : A ∋ maximal
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84 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
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85
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86 open _==_
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87 open _⊆_
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88
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89 record ZChain ( A : HOD ) (y : Ordinal) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
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90 field
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91 fb : (x : Ordinal ) → x o< y → HOD
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92 A∋fb : (ox : Ordinal ) → (x<y : ox o< y ) → A ∋ fb ox x<y
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93 monotonic : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ox o< oz → fb ox x<y < fb oz z<y
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94
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95 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
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96 → o∅ o< & A
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97 → PartialOrderSet A _<_
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98 → ( ( B : HOD) → (B⊆A : B ⊆ A) → TotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition
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99 → Maximal A _<_
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100 Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where
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101 someA : HOD
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102 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
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103 isSomeA : A ∋ someA
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104 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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105 HasMaximal : HOD
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106 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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107 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A
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108 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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109 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥
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110 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
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111 Gtx : { x : HOD} → A ∋ x → HOD
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112 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where
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113 z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A
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114 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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115 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
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116 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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117 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = proj1 (proj1 (PO (me A∋b) (me A∋a)) b<a) a<b
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118 -- ZChain is not compatible with the SUP condition
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119 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → (x o< y ) → HOD ) : Set n where
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120 field
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121 bx : Ordinal
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122 bx<y : bx o< y
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123 is-fb : x ≡ & (fb bx bx<y )
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124 bx<A : (z : ZChain A (& A) _<_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o< & A
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125 bx<A z {x} bx = BX.bx<y bx
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126 B : (z : ZChain A (& A) _<_ ) → HOD
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127 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; <odmax = {!!} }
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128 z11 : (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-elm x)) (bx<A z (is-elm x))
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129 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
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130 obx : (z : ZChain A (& A) _<_ ) → {x : HOD} → B z ∋ x → Ordinal
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131 obx z {x} bx = BX.bx bx
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132 obx=fb : (z : ZChain A (& A) _<_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z ( obx z bx ) (bx<A z bx )
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133 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx))
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134 B⊆A : (z : ZChain A (& A) _<_ ) → B z ⊆ A
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135 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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136 PO-B : (z : ZChain A (& A) _<_ ) → PartialOrderSet (B z) _<_
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137 PO-B z a b = PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) } record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) }
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138 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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139 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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140 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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141 z12 : (z : ZChain A (& A) _<_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z)) (y : BX (& b) (& A) (ZChain.fb z))
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142 → obx z x ≡ obx z y → bx<A z x ≅ bx<A z y
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143 z12 z {a} {b} x y eq = {!!}
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144 bx-inject : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y
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145 bx-inject z {x} {y} eq = begin
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146 elm x ≡⟨ {!!} ⟩
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147 {!!} ≡⟨ cong (λ k → {!!} ) {!!} ⟩
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148 {!!} ≡⟨ {!!} ⟩
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149 elm y ∎ where open ≡-Reasoning
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150 B-is-total : (z : ZChain A (& A) _<_ ) → TotalOrderSet (B z) _<_
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151 B-is-total z x y with trio< (obx z (is-elm x)) (obx z (is-elm y))
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152 ... | tri< a ¬b ¬c = tri< z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where
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153 z10 : elm x < elm y
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154 z10 = bx-monotonic z {x} {y} a
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155 ... | tri≈ ¬a b ¬c = tri≈ {!!} (bx-inject z {x} {y} b) {!!}
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156 ... | tri> ¬a ¬b c = tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c)
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157 ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ )
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158 ZChain→¬SUP z sp = ⊥-elim {!!} where
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159 z03 : & (SUP.sup sp) o< osuc (& A)
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160 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
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161 z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥
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162 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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163 ind : HasMaximal =h= od∅
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164 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ )
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165 → ZChain A x _<_
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166 ind nomx x prev with Oprev-p x
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167 ... | yes op with ODC.∋-p O A (* x)
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168 ... | no ¬Ax = {!!} where
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169 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
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170 px = Oprev.oprev op
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171 zc1 : ZChain A px _<_
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172 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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173 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as
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174 z04 sup as s<x with trio< (& sup) x
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175 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
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176 ... | 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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177 ... | tri> ¬a ¬b c with osuc-≡< s<x
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178 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
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179 ... | case2 lt = ⊥-elim (¬a lt )
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180 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
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181 px = Oprev.oprev op
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182 zc1 : ZChain A px _<_
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183 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
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184 z06 : ZChain A x _<_
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185 z06 with is-o∅ (& (Gtx ax))
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186 ... | 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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187 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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188 x-is-maximal m am = ¬x<m where
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189 ¬x<m : ¬ (* x < * m)
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190 ¬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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191 ... | no not = {!!} where
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192 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
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193 ... | tri< a ¬b ¬c = {!!} where
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194 zc1 : ZChain A (& A) _<_
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195 zc1 = prev (& A) a
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196 ... | tri≈ ¬a b ¬c = {!!} where
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197 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
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198 ... | no ¬Ax = {!!} where
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199 ... | yes ax with is-o∅ (& (Gtx ax))
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200 ... | 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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201 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
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202 x-is-maximal m am = ¬x<m where
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203 ¬x<m : ¬ (* x < * m)
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204 ¬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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205 ... | no not = {!!} where
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206 zorn00 : Maximal A _<_
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207 zorn00 with is-o∅ ( & HasMaximal )
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208 ... | 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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209 -- yes we have the maximal
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210 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
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211 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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477
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212 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
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213 zorn01 = proj1 hasm
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477
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214 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
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215 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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216 ... | yes ¬Maximal = ⊥-elim {!!} where
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217 -- if we have no maximal, make ZChain, which contradict SUP condition
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218 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_
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219 z x nomx = TransFinite (ind nomx) x
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220
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221 _⊆'_ : ( A B : HOD ) → Set n
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222 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
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223
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224 MaximumSubset : {L P : HOD}
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225 → o∅ o< & L → o∅ o< & P → P ⊆ L
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226 → PartialOrderSet P _⊆'_
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227 → ( (B : HOD) → B ⊆ P → TotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
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228 → Maximal P (_⊆'_)
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229 MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
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