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1 {-# OPTIONS --allow-unsolved-metas #-}
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2
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3 module fin where
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4
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283
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5 open import Data.Fin hiding (_<_ ; _≤_ ; _>_ ; _+_ )
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284
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6 open import Data.Fin.Properties hiding (≤-trans ; <-trans ; ≤-refl ) renaming ( <-cmp to <-fcmp )
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163
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7 open import Data.Nat
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284
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8 open import Data.Nat.Properties
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163
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9 open import logic
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10 open import nat
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11 open import Relation.Binary.PropositionalEquality
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12
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13
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14 -- toℕ<n
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15 fin<n : {n : ℕ} {f : Fin n} → toℕ f < n
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16 fin<n {_} {zero} = s≤s z≤n
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17 fin<n {suc n} {suc f} = s≤s (fin<n {n} {f})
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18
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19 -- toℕ≤n
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20 fin≤n : {n : ℕ} (f : Fin (suc n)) → toℕ f ≤ n
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21 fin≤n {_} zero = z≤n
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22 fin≤n {suc n} (suc f) = s≤s (fin≤n {n} f)
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23
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24 pred<n : {n : ℕ} {f : Fin (suc n)} → n > 0 → Data.Nat.pred (toℕ f) < n
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25 pred<n {suc n} {zero} (s≤s z≤n) = s≤s z≤n
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26 pred<n {suc n} {suc f} (s≤s z≤n) = fin<n
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27
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28 fin<asa : {n : ℕ} → toℕ (fromℕ< {n} a<sa) ≡ n
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29 fin<asa = toℕ-fromℕ< nat.a<sa
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30
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31 -- fromℕ<-toℕ
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32 toℕ→from : {n : ℕ} {x : Fin (suc n)} → toℕ x ≡ n → fromℕ n ≡ x
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33 toℕ→from {0} {zero} refl = refl
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34 toℕ→from {suc n} {suc x} eq = cong (λ k → suc k ) ( toℕ→from {n} {x} (cong (λ k → Data.Nat.pred k ) eq ))
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35
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36 0≤fmax : {n : ℕ } → (# 0) Data.Fin.≤ fromℕ< {n} a<sa
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37 0≤fmax = subst (λ k → 0 ≤ k ) (sym (toℕ-fromℕ< a<sa)) z≤n
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38
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39 0<fmax : {n : ℕ } → (# 0) Data.Fin.< fromℕ< {suc n} a<sa
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40 0<fmax = subst (λ k → 0 < k ) (sym (toℕ-fromℕ< a<sa)) (s≤s z≤n)
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41
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42 -- toℕ-injective
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43 i=j : {n : ℕ} (i j : Fin n) → toℕ i ≡ toℕ j → i ≡ j
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44 i=j {suc n} zero zero refl = refl
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45 i=j {suc n} (suc i) (suc j) eq = cong ( λ k → suc k ) ( i=j i j (cong ( λ k → Data.Nat.pred k ) eq) )
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46
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47 -- raise 1
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48 fin+1 : { n : ℕ } → Fin n → Fin (suc n)
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49 fin+1 zero = zero
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50 fin+1 (suc x) = suc (fin+1 x)
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51
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52 open import Data.Nat.Properties as NatP hiding ( _≟_ )
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53
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54 fin+1≤ : { i n : ℕ } → (a : i < n) → fin+1 (fromℕ< a) ≡ fromℕ< (<-trans a a<sa)
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55 fin+1≤ {0} {suc i} (s≤s z≤n) = refl
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56 fin+1≤ {suc n} {suc (suc i)} (s≤s (s≤s a)) = cong (λ k → suc k ) ( fin+1≤ {n} {suc i} (s≤s a) )
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57
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58 fin+1-toℕ : { n : ℕ } → { x : Fin n} → toℕ (fin+1 x) ≡ toℕ x
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59 fin+1-toℕ {suc n} {zero} = refl
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60 fin+1-toℕ {suc n} {suc x} = cong (λ k → suc k ) (fin+1-toℕ {n} {x})
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61
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62 open import Relation.Nullary
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63 open import Data.Empty
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64
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65 fin-1 : { n : ℕ } → (x : Fin (suc n)) → ¬ (x ≡ zero ) → Fin n
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66 fin-1 zero ne = ⊥-elim (ne refl )
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67 fin-1 {n} (suc x) ne = x
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68
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69 fin-1-sx : { n : ℕ } → (x : Fin n) → fin-1 (suc x) (λ ()) ≡ x
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70 fin-1-sx zero = refl
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71 fin-1-sx (suc x) = refl
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72
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73 fin-1-xs : { n : ℕ } → (x : Fin (suc n)) → (ne : ¬ (x ≡ zero )) → suc (fin-1 x ne ) ≡ x
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74 fin-1-xs zero ne = ⊥-elim ( ne refl )
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75 fin-1-xs (suc x) ne = refl
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76
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77 -- suc-injective
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78 -- suc-eq : {n : ℕ } {x y : Fin n} → Fin.suc x ≡ Fin.suc y → x ≡ y
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79 -- suc-eq {n} {x} {y} eq = subst₂ (λ j k → j ≡ k ) {!!} {!!} (cong (λ k → Data.Fin.pred k ) eq )
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80
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81 -- this is refl
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82 lemma3 : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt)
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83 lemma3 (s≤s lt) = refl
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84
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85 -- fromℕ<-toℕ
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86 lemma12 : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m
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87 lemma12 {zero} {suc m} (s≤s z≤n) zero refl = refl
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88 lemma12 {suc n} {suc m} (s≤s n<m) (suc f) refl = cong suc ( lemma12 {n} {m} n<m f refl )
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89
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90 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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91
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92 -- <-irrelevant
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93 <-nat=irr : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n
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94 <-nat=irr {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
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95 <-nat=irr {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( <-nat=irr {i} {i} {n} refl )
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96
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97 lemma8 : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n
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98 lemma8 {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
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99 lemma8 {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( lemma8 {i} {i} {n} refl )
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100
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101 -- fromℕ<-irrelevant
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102 lemma10 : {n i j : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → fromℕ< i<n ≡ fromℕ< j<n
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103 lemma10 {n} refl = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8 refl ))
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104
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105 lemma31 : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NatP.<-trans a<b b<c ≡ a<c
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106 lemma31 {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8 refl)
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107
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108 -- toℕ-fromℕ<
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109 lemma11 : {n m : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x
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110 lemma11 {n} {m} {x} n<m = begin
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111 toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m))
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112 ≡⟨ toℕ-fromℕ< _ ⟩
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113 toℕ x
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114 ∎ where
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115 open ≡-Reasoning
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116
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284
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117 x<y→fin-1 : {n : ℕ } → { x y : Fin (suc n)} → toℕ x < toℕ y → Fin n
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118 x<y→fin-1 {n} {x} {y} lt = fromℕ< (≤-trans lt (fin≤n _ ))
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119
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120 x<y→fin-1-eq : {n : ℕ } → { x y : Fin (suc n)} → (lt : toℕ x < toℕ y ) → toℕ x ≡ toℕ (x<y→fin-1 lt )
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121 x<y→fin-1-eq {n} {x} {y} lt = sym ( begin
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122 toℕ (fromℕ< (≤-trans lt (fin≤n y)) ) ≡⟨ toℕ-fromℕ< _ ⟩
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123 toℕ x ∎ ) where open ≡-Reasoning
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124
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283
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125 open import Data.List
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126 open import Relation.Binary.Definitions
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127
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128 fin-phase2 : { n : ℕ } (q : Fin n) (qs : List (Fin n) ) → Bool
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129 fin-phase2 q [] = false
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130 fin-phase2 q (x ∷ qs) with <-fcmp q x
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131 ... | tri< a ¬b ¬c = fin-phase2 q qs
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132 ... | tri≈ ¬a b ¬c = true
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133 ... | tri> ¬a ¬b c = fin-phase2 q qs
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134 fin-phase1 : { n : ℕ } (q : Fin n) (qs : List (Fin n) ) → Bool
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135 fin-phase1 q [] = false
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136 fin-phase1 q (x ∷ qs) with <-fcmp q x
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137 ... | tri< a ¬b ¬c = fin-phase1 q qs
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138 ... | tri≈ ¬a b ¬c = fin-phase2 q qs
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139 ... | tri> ¬a ¬b c = fin-phase1 q qs
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140
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141 fin-dup-in-list : { n : ℕ} (q : Fin n) (qs : List (Fin n) ) → Bool
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142 fin-dup-in-list {n} q qs = fin-phase1 q qs
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143
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144 record FDup-in-list (n : ℕ ) (qs : List (Fin n)) : Set where
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145 field
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146 dup : Fin n
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147 is-dup : fin-dup-in-list dup qs ≡ true
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148
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149 list-less : {n : ℕ } → List (Fin (suc n)) → List (Fin n)
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150 list-less [] = []
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151 list-less {n} (i ∷ ls) with NatP.<-cmp (toℕ i) n
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152 ... | tri< a ¬b ¬c = fromℕ< a ∷ list-less ls
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153 ... | tri≈ ¬a b ¬c = list-less ls
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154 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> (fin≤n i) c )
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155
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285
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156 record NList (n : ℕ) (qs : List (Fin (suc n))) : Set where
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283
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157 field
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158 ls : List (Fin n)
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159 lseq : list-less qs ≡ ls
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160 ls< : (length ls ≡ length qs) ∨ (suc (length ls) ≡ length qs)
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284
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161
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283
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162 fin-dup-in-list>n : {n : ℕ } → (qs : List (Fin n)) → (len> : length qs > n ) → FDup-in-list n qs
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163 fin-dup-in-list>n {zero} [] ()
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164 fin-dup-in-list>n {zero} (() ∷ qs) lt
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165 fin-dup-in-list>n {suc n} qs lt = fdup-phase0 where
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284
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166 open import Level using ( Level )
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283
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167 fdup+1 : (qs : List (Fin (suc n))) (i : Fin n) → fin-dup-in-list i (list-less qs) ≡ true → fin-dup-in-list (fin+1 i) qs ≡ true
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168 fdup+1 qs i p = f1-phase1 qs p where
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169 f1-phase2 : (qs : List (Fin (suc n)) ) → fin-phase2 i (list-less qs) ≡ true → fin-phase2 (fin+1 i) qs ≡ true
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286
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170 f1-phase2 (x ∷ qs) p with NatP.<-cmp (toℕ x) n
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171 f1-phase2 (x ∷ qs) p | tri< a ¬b ¬c with <-fcmp (fin+1 i) x
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172 ... | tri< a₁ ¬b₁ ¬c₁ = f1-phase2 qs {!!}
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173 ... | tri≈ ¬a b ¬c₁ = refl
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174 ... | tri> ¬a ¬b₁ c = f1-phase2 qs {!!}
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175 f1-phase2 (x ∷ qs) p | tri≈ ¬a b ¬c = {!!}
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176 f1-phase2 (x ∷ qs) p | tri> ¬a ¬b c = ⊥-elim ( nat-≤> (fin≤n x) c )
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283
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177 f1-phase1 : (qs : List (Fin (suc n)) ) → fin-phase1 i (list-less qs) ≡ true → fin-phase1 (fin+1 i) qs ≡ true
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178 f1-phase1 [] ()
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179 f1-phase1 (x ∷ qs) p with <-fcmp (fin+1 i) x
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180 ... | tri< a ¬b ¬c = f1-phase1 qs {!!}
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181 ... | tri≈ ¬a b ¬c = f1-phase2 qs {!!}
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182 ... | tri> ¬a ¬b c = f1-phase1 qs {!!}
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285
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183 fdup-phase2 : (qs : List (Fin (suc n)) )
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184 → ( fin-phase2 (fromℕ< a<sa ) qs ≡ true ) ∨ NList n qs
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286
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185 fdup-phase2 [] = case2 record { ls = [] ; lseq = refl ; ls< = case1 refl }
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285
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186 fdup-phase2 (x ∷ qs) with <-fcmp (fromℕ< a<sa) x
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284
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187 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> a (subst (λ k → toℕ x < suc k) (sym fin<asa) fin<n ))
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285
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188 fdup-phase2 (x ∷ qs) | tri≈ ¬a b ¬c = case1 refl
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189 fdup-phase2 (x ∷ qs) | tri> ¬a ¬b c with fdup-phase2 qs
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283
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190 ... | case1 p = case1 p
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286
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191 ... | case2 nlist = case2 record { ls = x<y→fin-1 c ∷ NList.ls nlist ; lseq = fdup01 ; ls< = case1 {!!} } where
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285
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192 fdup01 : list-less (x ∷ qs) ≡ x<y→fin-1 c ∷ NList.ls nlist
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193 fdup01 with NatP.<-cmp (toℕ x) n
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194 ... | tri< a ¬b ¬c = begin
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195 fromℕ< a ∷ list-less qs ≡⟨ cong₂ (λ j k → j ∷ k ) (lemma10 refl) (NList.lseq nlist) ⟩
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196 fromℕ< (≤-trans c (fin≤n (fromℕ< a<sa))) ∷ NList.ls nlist ∎ where open ≡-Reasoning
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197 ... | tri≈ ¬a b ¬c = ⊥-elim ( nat-≡< b (subst (λ k → toℕ x < k ) fin<asa c ))
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198 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> (fin≤n x) c )
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199 fdup-phase1 : (qs : List (Fin (suc n)) ) → (fin-phase1 (fromℕ< a<sa) qs ≡ true) ∨ NList n qs
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286
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200 fdup-phase1 [] = case2 record { ls = [] ; lseq = refl ; ls< = case1 refl }
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285
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201 fdup-phase1 (x ∷ qs) with <-fcmp (fromℕ< a<sa) x
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202 fdup-phase1 (x ∷ qs) | tri< a ¬b ¬c = ⊥-elim ( nat-≤> a (subst (λ k → toℕ x < suc k) (sym fin<asa) fin<n ))
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203 fdup-phase1 (x ∷ qs) | tri≈ ¬a b ¬c with fdup-phase2 qs
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283
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204 ... | case1 p = case1 p
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286
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205 ... | case2 nlist = case2 record { ls = NList.ls nlist ; lseq = {!!} ; ls< = case2 {!!} } where
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285
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206 fdup03 : list-less (x ∷ qs) ≡ NList.ls nlist
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207 fdup03 = {!!}
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286
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208 fdup06 : suc (length (NList.ls nlist)) ≡ length (x ∷ qs)
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209 fdup06 = {!!}
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285
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210 fdup-phase1 (x ∷ qs) | tri> ¬a ¬b c with fdup-phase1 qs
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283
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211 ... | case1 p = case1 p
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286
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212 ... | case2 nlist = case2 record { ls = x<y→fin-1 c ∷ NList.ls nlist ; lseq = {!!} ; ls< = case1 fdup5 } where
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213 fdup5 : length (x<y→fin-1 c ∷ NList.ls nlist) ≡ length (x ∷ qs)
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214 fdup5 = {!!}
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283
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215 fdup-phase0 : FDup-in-list (suc n) qs
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285
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216 fdup-phase0 with fdup-phase1 qs
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283
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217 ... | case1 dup = record { dup = fromℕ< a<sa ; is-dup = dup }
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218 ... | case2 nlist = record { dup = fin+1 (FDup-in-list.dup fdup)
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219 ; is-dup = fdup+1 qs (FDup-in-list.dup fdup) (FDup-in-list.is-dup fdup) } where
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286
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220 fdup04 : (length (NList.ls nlist) ≡ length qs) ∨ (suc (length (NList.ls nlist)) ≡ length qs) → length (list-less qs) > n
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221 fdup04 (case1 eq) = px≤py ( begin
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222 suc (suc n) ≤⟨ lt ⟩
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223 length qs ≡⟨ sym eq ⟩
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224 length (NList.ls nlist) ≡⟨ cong (λ k → length k) (sym (NList.lseq nlist )) ⟩
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225 length (list-less qs) ≤⟨ refl-≤s ⟩
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226 suc (length (list-less qs)) ∎ ) where open ≤-Reasoning
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227 fdup04 (case2 eq) = px≤py ( begin
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228 suc (suc n) ≤⟨ lt ⟩
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229 length qs ≡⟨ sym eq ⟩
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230 suc (length (NList.ls nlist)) ≡⟨ cong (λ k → suc (length k)) (sym (NList.lseq nlist )) ⟩
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231 suc (length (list-less qs)) ∎ ) where open ≤-Reasoning
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283
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232 fdup : FDup-in-list n (list-less qs)
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286
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233 fdup = fin-dup-in-list>n (list-less qs) ( fdup04 (NList.ls< nlist) )
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