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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 )
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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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34
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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 ⟨$⟩ˡ q
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74 ∎ where open ≡-Reasoning
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75
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76 perm0 : Permutation zero zero
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77 perm0 = permutation fid fid record { left-inverse-of = λ x → refl ; right-inverse-of = λ x → refl }
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78
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79 open import Relation.Nullary
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80 open import Data.Empty
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81 open import Relation.Binary.Core
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82 open import fin
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83
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84 fperm : {n m : ℕ} → m < n → Permutation n n → Permutation (suc n) (suc n)
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85 fperm {zero} ()
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86 fperm {suc n} {m} (s≤s m<n) perm = permutation p→ p← record { left-inverse-of = piso← ; right-inverse-of = piso→ } where
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87 p→ : Fin (suc (suc n)) → Fin (suc (suc n))
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88 p→ x with <-cmp (toℕ x) m
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89 p→ x | tri< a ¬b ¬c = fin+1 (perm ⟨$⟩ʳ (fromℕ≤ x<sn)) where
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90 x<sn : toℕ x < suc n
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91 x<sn = <-trans a (s≤s m<n)
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92 p→ x | tri≈ ¬a b ¬c = fromℕ≤ a<sa
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93 p→ x | tri> ¬a ¬b c = fin+1 (perm ⟨$⟩ʳ (fromℕ≤ (pred<n {_} {x} 0<s )))
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94
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95 p← : Fin (suc (suc n)) → Fin (suc (suc n))
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96 p← x = lemma (suc n) refl x where
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97 lemma : (i : ℕ ) → i ≡ suc n → (x : Fin (suc (suc n))) → Fin (suc (suc n))
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98 lemma i refl x with <-cmp (toℕ x) i
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99 lemma i refl x | tri< a ¬b ¬c = fin+1 (perm ⟨$⟩ˡ (fromℕ≤ x<sn)) where
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100 x<sn : toℕ x < suc n
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101 x<sn = a
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102 lemma i refl x | tri≈ ¬a b ¬c = fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )
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103 lemma i refl x | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c fin<n )
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104 lem8 : {x : Fin (suc (suc n)) } → toℕ ( fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )) ≡ m
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105 lem8 {x} = toℕ-fromℕ≤ (<-trans (s≤s m<n ) a<sa )
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106 piso← : (x : Fin (suc (suc n))) → p← ( p→ x ) ≡ x
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107 piso← x with <-cmp (toℕ x) m
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108 piso← x | tri< a ¬b ¬c = begin
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109 p← ( fin+1 (perm ⟨$⟩ʳ (fromℕ≤ x<sn)) )
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110 ≡⟨ {!!} ⟩
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111 fin+1 (perm ⟨$⟩ˡ (perm ⟨$⟩ʳ fromℕ≤ (<-trans a (s≤s m<n))))
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112 ≡⟨ cong (λ k → fin+1 k ) (inverseˡ perm) ⟩
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113 fin+1 (fromℕ≤ x<sn)
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114 ≡⟨ {!!} ⟩
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115 x
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116 ∎ where
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117 open ≡-Reasoning
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118 k = inverseˡ perm
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119 x<sn : toℕ x < suc n
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120 x<sn = <-trans a (s≤s m<n)
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121 piso← x | tri> ¬a ¬b c = begin
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122 p← ( fin+1 (perm ⟨$⟩ʳ (fromℕ≤ (pred<n {_} {x} 0<s ))))
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123 ≡⟨ {!!} ⟩
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124 x
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125 ∎ where
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126 open ≡-Reasoning
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127 piso← x | tri≈ ¬a refl ¬c = begin
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128 p← ( fromℕ≤ a<sa )
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129 ≡⟨ lem4 refl ⟩
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130 fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )
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131 ≡⟨ {!!} ⟩
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132 x
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133 ∎ where
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134 open ≡-Reasoning
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135 lem4 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ {suc n} a<sa → p← x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )
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136 lem4 {x} refl with <-cmp (toℕ x) (suc n)
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137 lem4 refl | tri< a ¬b ¬c = ⊥-elim ( ¬b {!!} )
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138 lem4 refl | tri≈ ¬a b ¬c = refl
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139 lem4 refl | tri> ¬a ¬b c = ⊥-elim ( ¬b {!!} )
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140 piso→ : (x : Fin (suc (suc n))) → p→ ( p← x ) ≡ x
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141 piso→ x = lemma2 (suc n) refl x where
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142 lemma2 : (i : ℕ ) → i ≡ suc n → (x : Fin (suc (suc n))) → p→ ( p← x ) ≡ x
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143 lemma2 i refl x with <-cmp (toℕ x) i
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144 lemma2 i refl x | tri< a ¬b ¬c = begin
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145 p→ ( fin+1 (perm ⟨$⟩ˡ (fromℕ≤ x<sn)))
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146 ≡⟨ {!!} ⟩
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147 x
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148 ∎ where
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149 open ≡-Reasoning
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150 x<sn : toℕ x < suc n
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151 x<sn = a
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152 lemma2 i refl x | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c fin<n )
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153 lemma2 i refl x | tri≈ ¬a b ¬c = begin
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154 p→ (fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ))
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155 ≡⟨ lem5 (fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )) {!!} ⟩
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156 fromℕ≤ a<sa
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157 ≡⟨ {!!} ⟩
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158 x
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159 ∎ where
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160 open ≡-Reasoning
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161 lem7 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ (s≤s (s≤s m<n)) → toℕ x ≡ m
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162 lem7 refl = trans (toℕ-fromℕ≤ _) {!!}
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163 lem6 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ (s≤s (s≤s m<n)) → toℕ x ≡ (suc m)
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164 lem6 refl = toℕ-fromℕ≤ _
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165 -- lem5 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) → p→ x ≡ fromℕ≤ a<sa
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166 lem5 : (x : Fin (suc (suc n)) ) → x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) → p→ x ≡ fromℕ≤ a<sa
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167 lem5 x eq with <-cmp (toℕ x) m
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168 lem5 x eq | tri< a ¬b ¬c = {!!}
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169 lem5 x eq | tri≈ ¬a refl ¬c = refl
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170 lem5 x eq | tri> ¬a ¬b c = {!!}
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