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1 module Symmetric where
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3 open import Level hiding ( suc ; zero )
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4 open import Algebra
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5 open import Algebra.Structures
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6 open import Data.Fin hiding ( _<_ ; _≤_ ; _-_ ; _+_ )
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7 open import Data.Fin.Properties hiding ( <-cmp ; <-trans ; ≤-trans )
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8 open import Data.Product
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9 open import Data.Fin.Permutation
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10 open import Function hiding (id ; flip)
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11 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
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12 open import Function.LeftInverse using ( _LeftInverseOf_ )
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13 open import Function.Equality using (Π)
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14 open import Data.Nat -- using (ℕ; suc; zero; s≤s ; z≤n )
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15 open import Data.Nat.Properties -- using (<-trans)
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16 open import Relation.Binary.PropositionalEquality
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17 open import Data.List using (List; []; _∷_ ; length)
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18 open import nat
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19
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20 fid : {p : ℕ } → Fin p → Fin p
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21 fid x = x
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22
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23 -- Data.Fin.Permutation.id
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24 pid : {p : ℕ } → Permutation p p
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25 pid = permutation fid fid record { left-inverse-of = λ x → refl ; right-inverse-of = λ x → refl }
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26
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27 -- Data.Fin.Permutation.flip
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28 pinv : {p : ℕ } → Permutation p p → Permutation p p
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29 pinv {p} P = permutation (_⟨$⟩ˡ_ P) (_⟨$⟩ʳ_ P ) record { left-inverse-of = λ x → inverseʳ P ; right-inverse-of = λ x → inverseˡ P }
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31 record _=p=_ {p : ℕ } ( x y : Permutation p p ) : Set where
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32 field
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33 peq : ( q : Fin p ) → x ⟨$⟩ʳ q ≡ y ⟨$⟩ʳ q
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35 open _=p=_
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36
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37 prefl : {p : ℕ } { x : Permutation p p } → x =p= x
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38 peq (prefl {p} {x}) q = refl
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39
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40 psym : {p : ℕ } { x y : Permutation p p } → x =p= y → y =p= x
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41 peq (psym {p} {x} {y} eq ) q = sym (peq eq q)
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42
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43 ptrans : {p : ℕ } { x y z : Permutation p p } → x =p= y → y =p= z → x =p= z
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44 peq (ptrans {p} {x} {y} x=y y=z ) q = trans (peq x=y q) (peq y=z q)
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45
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46 Symmetric : ℕ → Group Level.zero Level.zero
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47 Symmetric p = record {
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48 Carrier = Permutation p p
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49 ; _≈_ = _=p=_
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50 ; _∙_ = _∘ₚ_
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51 ; ε = pid
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52 ; _⁻¹ = pinv
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53 ; isGroup = record { isMonoid = record { isSemigroup = record { isMagma = record {
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54 isEquivalence = record {refl = prefl ; trans = ptrans ; sym = psym }
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55 ; ∙-cong = presp }
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56 ; assoc = passoc }
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57 ; identity = ( (λ q → record { peq = λ q → refl } ) , (λ q → record { peq = λ q → refl } )) }
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58 ; inverse = ( (λ x → record { peq = λ q → inverseʳ x} ) , (λ x → record { peq = λ q → inverseˡ x} ))
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59 ; ⁻¹-cong = λ i=j → record { peq = λ q → p-inv i=j q }
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60 }
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61 } where
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62 presp : {x y u v : Permutation p p } → x =p= y → u =p= v → (x ∘ₚ u) =p= (y ∘ₚ v)
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63 presp {x} {y} {u} {v} x=y u=v = record { peq = λ q → lemma4 q } where
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64 lemma4 : (q : Fin p) → ((x ∘ₚ u) ⟨$⟩ʳ q) ≡ ((y ∘ₚ v) ⟨$⟩ʳ q)
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65 lemma4 q = trans (cong (λ k → Inverse.to u Π.⟨$⟩ k) (peq x=y q) ) (peq u=v _ )
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66 passoc : (x y z : Permutation p p) → ((x ∘ₚ y) ∘ₚ z) =p= (x ∘ₚ (y ∘ₚ z))
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67 passoc x y z = record { peq = λ q → refl }
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68 p-inv : {i j : Permutation p p} → i =p= j → (q : Fin p) → pinv i ⟨$⟩ʳ q ≡ pinv j ⟨$⟩ʳ q
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69 p-inv {i} {j} i=j q = begin
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70 i ⟨$⟩ˡ q ≡⟨ cong (λ k → i ⟨$⟩ˡ k) (sym (inverseʳ j) ) ⟩
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71 i ⟨$⟩ˡ ( j ⟨$⟩ʳ ( j ⟨$⟩ˡ q )) ≡⟨ cong (λ k → i ⟨$⟩ˡ k) (sym (peq i=j _ )) ⟩
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72 i ⟨$⟩ˡ ( i ⟨$⟩ʳ ( j ⟨$⟩ˡ q )) ≡⟨ inverseˡ i ⟩
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73 j ⟨$⟩ˡ q
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74 ∎ where open ≡-Reasoning
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76 open import Relation.Nullary
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77 open import Data.Empty
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78 open import Relation.Binary.Core
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79 open import fin
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80
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81 -- An inductive construction of permutation
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82
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83 pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n)
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84 pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where
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85 p→ : Fin (suc n) → Fin (suc n)
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86 p→ zero = zero
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87 p→ (suc x) = suc ( perm ⟨$⟩ˡ x)
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88
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89 p← : Fin (suc n) → Fin (suc n)
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90 p← zero = zero
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91 p← (suc x) = suc ( perm ⟨$⟩ʳ x)
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92
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93 piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x
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94 piso← zero = refl
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95 piso← (suc x) = cong (λ k → suc k ) (inverseˡ perm)
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96
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97 piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x
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98 piso→ zero = refl
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99 piso→ (suc x) = cong (λ k → suc k ) (inverseʳ perm)
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100
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101 pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n ))
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102 pswap {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where
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103 p→ : Fin (suc (suc n)) → Fin (suc (suc n))
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104 p→ zero = suc zero
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105 p→ (suc zero) = zero
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106 p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) )
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107
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108 p← : Fin (suc (suc n)) → Fin (suc (suc n))
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109 p← zero = suc zero
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110 p← (suc zero) = zero
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111 p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) )
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112
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113 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x
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114 piso← zero = refl
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115 piso← (suc zero) = refl
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116 piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm)
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117
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118 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x
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119 piso→ zero = refl
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120 piso→ (suc zero) = refl
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121 piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm)
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122
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123 -- enumeration
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124
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125 psawpn : {n m : ℕ} → suc m < n → Permutation n n
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126 psawpn {suc zero} {m} (s≤s ())
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127 psawpn {suc n} {m} (s≤s (s≤s x)) = pswap pid
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128
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129 pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n
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130 pfill {n} {m} m≤n perm = pfill1 (n - m) (n-m<n n m ) (subst (λ k → Permutation k k ) (n-n-m=m m≤n ) perm) where
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131 pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n
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132 pfill1 0 _ perm = perm
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133 pfill1 (suc i) i<n perm = pfill1 i (≤to< i<n) (subst (λ k → Permutation k k ) (si-sn=i-n i<n ) ( pprep perm ) )
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134
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135 eperm : {n m : ℕ} → m < n → Permutation n n → Permutation (suc n) (suc n)
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136 eperm {zero} ()
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137 eperm {n} {0} (s≤s z≤n) perm = pprep perm
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138 eperm {n} {suc m} (s≤s m<n) perm = eperm1 m 2 lemm3 (pswap {0} pid ) (pprep perm) where
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139 lemm3 : 2 + m ≤ suc n
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140 lemm3 = ≤-trans (s≤s m<n) refl-≤s
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141 eperm1 : (m i : ℕ ) → i + m ≤ suc n → Permutation i i → Permutation (suc n)(suc n)→ Permutation (suc n)(suc n)
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142 eperm1 zero i i<ssm sw perm = perm ∘ₚ ( pfill (subst (λ k → k ≤ suc n) (+-comm i _) i<ssm) sw ) -- i + zero ≤ suc (suc n) → i ≤ suc (suc n)
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143 eperm1 (suc m) i i<ssm sw perm = eperm1 m (suc i) (lemm4 i<ssm ) (pprep sw) perm where
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144 lemm4 : i + suc m ≤ suc n → suc i + m ≤ suc n
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145 lemm4 lt = begin
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146 suc i + m ≡⟨ cong (λ k → suc k ) ( +-comm i _ ) ⟩
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147 suc m + i ≡⟨ +-comm (suc m) _ ⟩
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148 i + suc m ≤⟨ lt ⟩
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149 suc n
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150 ∎ where open ≤-Reasoning
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152
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153 finpid : (n i : ℕ ) → i < n → List (Fin n)
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154 finpid (suc n) zero _ = fromℕ≤ {zero} (s≤s z≤n) ∷ []
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155 finpid (suc n) (suc i) (s≤s lt) = fromℕ≤ (s≤s lt) ∷ finpid (suc n) i (<-trans lt a<sa)
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156
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157 fpid : (n : ℕ ) → List (Fin n)
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158 fpid 0 = []
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159 fpid (suc j) = finpid (suc j) j a<sa where
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160
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161 plist : {n : ℕ} → Permutation n n → List ℕ
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162 plist {0} perm = []
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163 plist {suc j} perm = plist1 j a<sa where
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164 n = suc j
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165 plist1 : (i : ℕ ) → i < n → List ℕ
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166 plist1 zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ≤ {zero} (s≤s z≤n))) ∷ []
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167 plist1 (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ≤ (s≤s lt))) ∷ plist1 i (<-trans lt a<sa)
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168
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169 test = eperm {3} ( s≤s ( s≤s z≤n )) ( eperm (s≤s z≤n) pid )
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170
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171 NL : (n : ℕ ) → Set
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172 NL 0 = ℕ
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173 NL (suc n) = List ( NL n )
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174
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175 pls : (n : ℕ ) → List (List ℕ )
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176 pls n = pls1 n n lem0 where
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177 lem0 : {n : ℕ } → n ≤ n
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178 lem0 {zero} = z≤n
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179 lem0 {suc n} = s≤s lem0
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180 lem1 : {i n : ℕ } → i ≤ n → i < suc n
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181 lem1 z≤n = s≤s z≤n
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182 lem1 (s≤s lt) = s≤s (lem1 lt)
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183 lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n
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184 lem2 i≤n = ≤-trans i≤n ( refl-≤s )
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185 pls2 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation (suc n) (suc n)
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186 pls2 0 n i≤<n = pid
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187 pls2 (suc i) (suc n) (s≤s i≤n) = eperm {suc n} {i} (lem1 i≤n) ( pls2 i n i≤n)
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188 pls1 : ( i n : ℕ ) → i ≤ n → List (List ℕ)
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189 pls1 0 n _ = []
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190 pls1 (suc i) n (s≤s i≤n) = plist ( pls2 i n (lem2 i≤n)) ∷ pls1 i n (lem2 i≤n)
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