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1 open import Level
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2 open import Ordinals
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3 module OrdUtil {n : Level} (O : Ordinals {n} ) where
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4
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5 open import logic
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6 open import nat
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7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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8 open import Data.Empty
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9 open import Relation.Binary.PropositionalEquality
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10 open import Relation.Nullary
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11 open import Relation.Binary hiding (_⇔_)
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12
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13 open Ordinals.Ordinals O
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14 open Ordinals.IsOrdinals isOrdinal
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15 open Ordinals.IsNext isNext
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16
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17 o<-dom : { x y : Ordinal } → x o< y → Ordinal
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18 o<-dom {x} _ = x
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19
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20 o<-cod : { x y : Ordinal } → x o< y → Ordinal
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21 o<-cod {_} {y} _ = y
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22
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23 o<-subst : {Z X z x : Ordinal } → Z o< X → Z ≡ z → X ≡ x → z o< x
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24 o<-subst df refl refl = df
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25
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26 o<¬≡ : { ox oy : Ordinal } → ox ≡ oy → ox o< oy → ⊥
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27 o<¬≡ {ox} {oy} eq lt with trio< ox oy
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28 o<¬≡ {ox} {oy} eq lt | tri< a ¬b ¬c = ¬b eq
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29 o<¬≡ {ox} {oy} eq lt | tri≈ ¬a b ¬c = ¬a lt
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30 o<¬≡ {ox} {oy} eq lt | tri> ¬a ¬b c = ¬b eq
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31
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32 o<> : {x y : Ordinal } → y o< x → x o< y → ⊥
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33 o<> {ox} {oy} lt tl with trio< ox oy
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34 o<> {ox} {oy} lt tl | tri< a ¬b ¬c = ¬c lt
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35 o<> {ox} {oy} lt tl | tri≈ ¬a b ¬c = ¬a tl
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36 o<> {ox} {oy} lt tl | tri> ¬a ¬b c = ¬a tl
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37
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38 osuc-< : { x y : Ordinal } → y o< osuc x → x o< y → ⊥
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39 osuc-< {x} {y} y<ox x<y with osuc-≡< y<ox
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40 osuc-< {x} {y} y<ox x<y | case1 refl = o<¬≡ refl x<y
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41 osuc-< {x} {y} y<ox x<y | case2 y<x = o<> x<y y<x
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42
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43 osucc : {ox oy : Ordinal } → oy o< ox → osuc oy o< osuc ox
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44 ---- y < osuc y < x < osuc x
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45 ---- y < osuc y = x < osuc x
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46 ---- y < osuc y > x < osuc x -> y = x ∨ x < y → ⊥
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47 osucc {ox} {oy} oy<ox with trio< (osuc oy) ox
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48 osucc {ox} {oy} oy<ox | tri< a ¬b ¬c = ordtrans a <-osuc
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49 osucc {ox} {oy} oy<ox | tri≈ ¬a refl ¬c = <-osuc
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50 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c with osuc-≡< c
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51 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case1 eq = ⊥-elim (o<¬≡ (sym eq) oy<ox)
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52 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case2 lt = ⊥-elim (o<> lt oy<ox)
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53
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54 osucprev : {ox oy : Ordinal } → osuc oy o< osuc ox → oy o< ox
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55 osucprev {ox} {oy} oy<ox with trio< oy ox
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56 osucprev {ox} {oy} oy<ox | tri< a ¬b ¬c = a
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57 osucprev {ox} {oy} oy<ox | tri≈ ¬a b ¬c = ⊥-elim (o<¬≡ (cong (λ k → osuc k) b) oy<ox )
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58 osucprev {ox} {oy} oy<ox | tri> ¬a ¬b c = ⊥-elim (o<> (osucc c) oy<ox )
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59
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60 open _∧_
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61
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62 osuc2 : ( x y : Ordinal ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
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63 proj2 (osuc2 x y) lt = osucc lt
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64 proj1 (osuc2 x y) ox<ooy with osuc-≡< ox<ooy
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65 proj1 (osuc2 x y) ox<ooy | case1 ox=oy = o<-subst <-osuc refl ox=oy
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66 proj1 (osuc2 x y) ox<ooy | case2 ox<oy = ordtrans <-osuc ox<oy
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67
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538
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68 o≡? : (x y : Ordinal) → Dec ( x ≡ y )
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69 o≡? x y with trio< x y
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70 ... | tri< a ¬b ¬c = no ¬b
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71 ... | tri≈ ¬a b ¬c = yes b
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72 ... | tri> ¬a ¬b c = no ¬b
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73
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431
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74 _o≤_ : Ordinal → Ordinal → Set n
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75 a o≤ b = a o< osuc b -- (a ≡ b) ∨ ( a o< b )
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76
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611
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77 -- o<-irr : { x y : Ordinal } → { lt lt1 : x o< y } → lt ≡ lt1
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431
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78
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79 xo<ab : {oa ob : Ordinal } → ( {ox : Ordinal } → ox o< oa → ox o< ob ) → oa o< osuc ob
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80 xo<ab {oa} {ob} a→b with trio< oa ob
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81 xo<ab {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
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82 xo<ab {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
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83 xo<ab {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) )
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84
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85 maxα : Ordinal → Ordinal → Ordinal
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86 maxα x y with trio< x y
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87 maxα x y | tri< a ¬b ¬c = y
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88 maxα x y | tri> ¬a ¬b c = x
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89 maxα x y | tri≈ ¬a refl ¬c = x
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90
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91 omin : Ordinal → Ordinal → Ordinal
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92 omin x y with trio< x y
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93 omin x y | tri< a ¬b ¬c = x
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94 omin x y | tri> ¬a ¬b c = y
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95 omin x y | tri≈ ¬a refl ¬c = x
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96
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97 min1 : {x y z : Ordinal } → z o< x → z o< y → z o< omin x y
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98 min1 {x} {y} {z} z<x z<y with trio< x y
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99 min1 {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x
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100 min1 {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x
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101 min1 {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y
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102
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103 --
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104 -- max ( osuc x , osuc y )
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105 --
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106
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107 omax : ( x y : Ordinal ) → Ordinal
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108 omax x y with trio< x y
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109 omax x y | tri< a ¬b ¬c = osuc y
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110 omax x y | tri> ¬a ¬b c = osuc x
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111 omax x y | tri≈ ¬a refl ¬c = osuc x
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112
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113 omax< : ( x y : Ordinal ) → x o< y → osuc y ≡ omax x y
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114 omax< x y lt with trio< x y
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115 omax< x y lt | tri< a ¬b ¬c = refl
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116 omax< x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
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117 omax< x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
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118
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119 omax≤ : ( x y : Ordinal ) → x o≤ y → osuc y ≡ omax x y
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120 omax≤ x y le with trio< x y
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121 omax≤ x y le | tri< a ¬b ¬c = refl
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122 omax≤ x y le | tri≈ ¬a refl ¬c = refl
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123 omax≤ x y le | tri> ¬a ¬b c with osuc-≡< le
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124 omax≤ x y le | tri> ¬a ¬b c | case1 eq = ⊥-elim (¬b eq)
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125 omax≤ x y le | tri> ¬a ¬b c | case2 x<y = ⊥-elim (¬a x<y)
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126
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127 omax≡ : ( x y : Ordinal ) → x ≡ y → osuc y ≡ omax x y
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128 omax≡ x y eq with trio< x y
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129 omax≡ x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
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130 omax≡ x y eq | tri≈ ¬a refl ¬c = refl
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131 omax≡ x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
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132
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133 omax-x : ( x y : Ordinal ) → x o< omax x y
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134 omax-x x y with trio< x y
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135 omax-x x y | tri< a ¬b ¬c = ordtrans a <-osuc
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136 omax-x x y | tri> ¬a ¬b c = <-osuc
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137 omax-x x y | tri≈ ¬a refl ¬c = <-osuc
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138
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139 omax-y : ( x y : Ordinal ) → y o< omax x y
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140 omax-y x y with trio< x y
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141 omax-y x y | tri< a ¬b ¬c = <-osuc
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142 omax-y x y | tri> ¬a ¬b c = ordtrans c <-osuc
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143 omax-y x y | tri≈ ¬a refl ¬c = <-osuc
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144
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145 omxx : ( x : Ordinal ) → omax x x ≡ osuc x
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146 omxx x with trio< x x
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147 omxx x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
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148 omxx x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
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149 omxx x | tri≈ ¬a refl ¬c = refl
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150
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151 omxxx : ( x : Ordinal ) → omax x (omax x x ) ≡ osuc (osuc x)
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152 omxxx x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
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153
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154 open _∧_
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155
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156 o≤-refl : { i j : Ordinal } → i ≡ j → i o≤ j
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157 o≤-refl {i} {j} eq = subst (λ k → i o< osuc k ) eq <-osuc
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158 OrdTrans : Transitive _o≤_
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159 OrdTrans a≤b b≤c with osuc-≡< a≤b | osuc-≡< b≤c
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160 OrdTrans a≤b b≤c | case1 refl | case1 refl = <-osuc
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161 OrdTrans a≤b b≤c | case1 refl | case2 a≤c = ordtrans a≤c <-osuc
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162 OrdTrans a≤b b≤c | case2 a≤c | case1 refl = ordtrans a≤c <-osuc
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163 OrdTrans a≤b b≤c | case2 a<b | case2 b<c = ordtrans (ordtrans a<b b<c) <-osuc
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164
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165 OrdPreorder : Preorder n n n
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166 OrdPreorder = record { Carrier = Ordinal
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167 ; _≈_ = _≡_
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168 ; _∼_ = _o≤_
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169 ; isPreorder = record {
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170 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
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171 ; reflexive = o≤-refl
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172 ; trans = OrdTrans
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173 }
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174 }
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175
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176 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
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177 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
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178 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
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179 → ¬ p
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180 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
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181
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182 nexto∅ : {x : Ordinal} → o∅ o< next x
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183 nexto∅ {x} with trio< o∅ x
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184 nexto∅ {x} | tri< a ¬b ¬c = ordtrans a x<nx
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185 nexto∅ {x} | tri≈ ¬a b ¬c = subst (λ k → k o< next x) (sym b) x<nx
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186 nexto∅ {x} | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
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187
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188 next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
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189 next< {x} {y} {z} x<nz y<nx with trio< y (next z)
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190 next< {x} {y} {z} x<nz y<nx | tri< a ¬b ¬c = a
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191 next< {x} {y} {z} x<nz y<nx | tri≈ ¬a b ¬c = ⊥-elim (¬nx<nx x<nz (subst (λ k → k o< next x) b y<nx)
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192 (λ w nz=ow → o<¬≡ nz=ow (subst₂ (λ j k → j o< k ) (sym nz=ow) nz=ow (osuc<nx (subst (λ k → w o< k ) (sym nz=ow) <-osuc) ))))
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193 next< {x} {y} {z} x<nz y<nx | tri> ¬a ¬b c = ⊥-elim (¬nx<nx x<nz (ordtrans c y<nx )
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194 (λ w nz=ow → o<¬≡ (sym nz=ow) (osuc<nx (subst (λ k → w o< k ) (sym nz=ow) <-osuc ))))
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195
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196 osuc< : {x y : Ordinal} → osuc x ≡ y → x o< y
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197 osuc< {x} {y} refl = <-osuc
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198
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199 nexto=n : {x y : Ordinal} → x o< next (osuc y) → x o< next y
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200 nexto=n {x} {y} x<noy = next< (osuc<nx x<nx) x<noy
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201
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202 nexto≡ : {x : Ordinal} → next x ≡ next (osuc x)
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203 nexto≡ {x} with trio< (next x) (next (osuc x) )
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204 -- next x o< next (osuc x ) -> osuc x o< next x o< next (osuc x) -> next x ≡ osuc z -> z o o< next x -> osuc z o< next x -> next x o< next x
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205 nexto≡ {x} | tri< a ¬b ¬c = ⊥-elim (¬nx<nx (osuc<nx x<nx ) a
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206 (λ z eq → o<¬≡ (sym eq) (osuc<nx (osuc< (sym eq)))))
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207 nexto≡ {x} | tri≈ ¬a b ¬c = b
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208 -- next (osuc x) o< next x -> osuc x o< next (osuc x) o< next x -> next (osuc x) ≡ osuc z -> z o o< next (osuc x) ...
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209 nexto≡ {x} | tri> ¬a ¬b c = ⊥-elim (¬nx<nx (ordtrans <-osuc x<nx) c
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210 (λ z eq → o<¬≡ (sym eq) (osuc<nx (osuc< (sym eq)))))
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211
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212 next-is-limit : {x y : Ordinal} → ¬ (next x ≡ osuc y)
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213 next-is-limit {x} {y} eq = o<¬≡ (sym eq) (osuc<nx y<nx) where
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214 y<nx : y o< next x
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215 y<nx = osuc< (sym eq)
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216
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217 omax<next : {x y : Ordinal} → x o< y → omax x y o< next y
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218 omax<next {x} {y} x<y = subst (λ k → k o< next y ) (omax< _ _ x<y ) (osuc<nx x<nx)
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219
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220 x<ny→≡next : {x y : Ordinal} → x o< y → y o< next x → next x ≡ next y
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221 x<ny→≡next {x} {y} x<y y<nx with trio< (next x) (next y)
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222 x<ny→≡next {x} {y} x<y y<nx | tri< a ¬b ¬c = -- x < y < next x < next y ∧ next x = osuc z
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223 ⊥-elim ( ¬nx<nx y<nx a (λ z eq → o<¬≡ (sym eq) (osuc<nx (subst (λ k → z o< k ) (sym eq) <-osuc ))))
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224 x<ny→≡next {x} {y} x<y y<nx | tri≈ ¬a b ¬c = b
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225 x<ny→≡next {x} {y} x<y y<nx | tri> ¬a ¬b c = -- x < y < next y < next x
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226 ⊥-elim ( ¬nx<nx (ordtrans x<y x<nx) c (λ z eq → o<¬≡ (sym eq) (osuc<nx (subst (λ k → z o< k ) (sym eq) <-osuc ))))
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227
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228 ≤next : {x y : Ordinal} → x o≤ y → next x o≤ next y
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229 ≤next {x} {y} x≤y with trio< (next x) y
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230 ≤next {x} {y} x≤y | tri< a ¬b ¬c = ordtrans a (ordtrans x<nx <-osuc )
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231 ≤next {x} {y} x≤y | tri≈ ¬a refl ¬c = (ordtrans x<nx <-osuc )
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232 ≤next {x} {y} x≤y | tri> ¬a ¬b c with osuc-≡< x≤y
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233 ≤next {x} {y} x≤y | tri> ¬a ¬b c | case1 refl = o≤-refl refl -- x = y < next x
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234 ≤next {x} {y} x≤y | tri> ¬a ¬b c | case2 x<y = o≤-refl (x<ny→≡next x<y c) -- x ≤ y < next x
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235
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236 x<ny→≤next : {x y : Ordinal} → x o< next y → next x o≤ next y
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237 x<ny→≤next {x} {y} x<ny with trio< x y
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238 x<ny→≤next {x} {y} x<ny | tri< a ¬b ¬c = ≤next (ordtrans a <-osuc )
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239 x<ny→≤next {x} {y} x<ny | tri≈ ¬a refl ¬c = o≤-refl refl
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240 x<ny→≤next {x} {y} x<ny | tri> ¬a ¬b c = o≤-refl (sym ( x<ny→≡next c x<ny ))
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241
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242 omax<nomax : {x y : Ordinal} → omax x y o< next (omax x y )
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243 omax<nomax {x} {y} with trio< x y
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244 omax<nomax {x} {y} | tri< a ¬b ¬c = subst (λ k → osuc y o< k ) nexto≡ (osuc<nx x<nx )
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245 omax<nomax {x} {y} | tri≈ ¬a refl ¬c = subst (λ k → osuc x o< k ) nexto≡ (osuc<nx x<nx )
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246 omax<nomax {x} {y} | tri> ¬a ¬b c = subst (λ k → osuc x o< k ) nexto≡ (osuc<nx x<nx )
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247
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248 omax<nx : {x y z : Ordinal} → x o< next z → y o< next z → omax x y o< next z
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249 omax<nx {x} {y} {z} x<nz y<nz with trio< x y
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250 omax<nx {x} {y} {z} x<nz y<nz | tri< a ¬b ¬c = osuc<nx y<nz
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251 omax<nx {x} {y} {z} x<nz y<nz | tri≈ ¬a refl ¬c = osuc<nx y<nz
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252 omax<nx {x} {y} {z} x<nz y<nz | tri> ¬a ¬b c = osuc<nx x<nz
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253
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254 nn<omax : {x nx ny : Ordinal} → x o< next nx → x o< next ny → x o< next (omax nx ny)
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255 nn<omax {x} {nx} {ny} xnx xny with trio< nx ny
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256 nn<omax {x} {nx} {ny} xnx xny | tri< a ¬b ¬c = subst (λ k → x o< k ) nexto≡ xny
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257 nn<omax {x} {nx} {ny} xnx xny | tri≈ ¬a refl ¬c = subst (λ k → x o< k ) nexto≡ xny
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258 nn<omax {x} {nx} {ny} xnx xny | tri> ¬a ¬b c = subst (λ k → x o< k ) nexto≡ xnx
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259
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260 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
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261 field
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262 os→ : (x : Ordinal) → x o< maxordinal → Ordinal
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263 os← : Ordinal → Ordinal
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264 os←limit : (x : Ordinal) → os← x o< maxordinal
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265 os-iso← : {x : Ordinal} → os→ (os← x) (os←limit x) ≡ x
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266 os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) → os← (os→ x lt) ≡ x
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267
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268 module o≤-Reasoning {n : Level} (O : Ordinals {n} ) where
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269
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270 -- open inOrdinal O
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271
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272 resp-o< : _o<_ Respects₂ _≡_
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273 resp-o< = resp₂ _o<_
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274
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275 trans1 : {i j k : Ordinal} → i o< j → j o< osuc k → i o< k
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276 trans1 {i} {j} {k} i<j j<ok with osuc-≡< j<ok
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277 trans1 {i} {j} {k} i<j j<ok | case1 refl = i<j
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278 trans1 {i} {j} {k} i<j j<ok | case2 j<k = ordtrans i<j j<k
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279
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280 trans2 : {i j k : Ordinal} → i o< osuc j → j o< k → i o< k
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281 trans2 {i} {j} {k} i<oj j<k with osuc-≡< i<oj
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282 trans2 {i} {j} {k} i<oj j<k | case1 refl = j<k
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283 trans2 {i} {j} {k} i<oj j<k | case2 i<j = ordtrans i<j j<k
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284
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285 open import Relation.Binary.Reasoning.Base.Triple
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286 (Preorder.isPreorder OrdPreorder)
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287 ordtrans --<-trans
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288 (resp₂ _o<_) --(resp₂ _<_)
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289 (λ x → ordtrans x <-osuc ) --<⇒≤
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290 trans1 --<-transˡ
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291 trans2 --<-transʳ
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292 public
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293 -- hiding (_≈⟨_⟩_)
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294
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