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1 module regular-language where
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2
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3 open import Level renaming ( suc to Suc ; zero to Zero )
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4 open import Data.List
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5 open import Data.Nat hiding ( _≟_ )
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6 open import Data.Fin hiding ( _+_ )
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7 open import Data.Empty
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8 open import Data.Product
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9 -- open import Data.Maybe
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10 open import Relation.Nullary
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11 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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12 open import logic
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13 open import nat
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14 open import automaton
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15 open import finiteSet
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16
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17 language : { Σ : Set } → Set
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18 language {Σ} = List Σ → Bool
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19
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20 language-L : { Σ : Set } → Set
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21 language-L {Σ} = List (List Σ)
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22
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23 open Automaton
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24
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25 record RegularLanguage ( Σ : Set ) : Set (Suc Zero) where
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26 field
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27 states : Set
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28 astart : states
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29 aℕ : ℕ
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30 afin : FiniteSet states {aℕ}
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31 automaton : Automaton states Σ
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32 contain : List Σ → Bool
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33 contain x = accept automaton astart x
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34
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35 Union : {Σ : Set} → ( A B : language {Σ} ) → language {Σ}
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36 Union {Σ} A B x = (A x ) \/ (B x)
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37
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38 split : {Σ : Set} → (List Σ → Bool)
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39 → ( List Σ → Bool) → List Σ → Bool
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40 split x y [] = x [] /\ y []
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41 split x y (h ∷ t) = (x [] /\ y (h ∷ t)) \/
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42 split (λ t1 → x ( h ∷ t1 )) (λ t2 → y t2 ) t
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43
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44 Concat : {Σ : Set} → ( A B : language {Σ} ) → language {Σ}
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45 Concat {Σ} A B = split A B
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46
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47 {-# TERMINATING #-}
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48 Star : {Σ : Set} → ( A : language {Σ} ) → language {Σ}
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49 Star {Σ} A = split A ( Star {Σ} A )
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50
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51 test-split : {Σ : Set} → {A B : List In2 → Bool} → split A B ( i0 ∷ i1 ∷ i0 ∷ [] ) ≡ (
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52 ( A [] /\ B ( i0 ∷ i1 ∷ i0 ∷ [] ) ) \/
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53 ( A ( i0 ∷ [] ) /\ B ( i1 ∷ i0 ∷ [] ) ) \/
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54 ( A ( i0 ∷ i1 ∷ [] ) /\ B ( i0 ∷ [] ) ) \/
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55 ( A ( i0 ∷ i1 ∷ i0 ∷ [] ) /\ B [] )
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56 )
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57 test-split {_} {A} {B} = refl
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58
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59 open RegularLanguage
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60 isRegular : {Σ : Set} → (A : language {Σ} ) → ( x : List Σ ) → (r : RegularLanguage Σ ) → Set
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61 isRegular A x r = A x ≡ contain r x
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62
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63 postulate
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64 fin-× : {A B : Set} → { a b : ℕ } → FiniteSet A {a} → FiniteSet B {b} → FiniteSet (A × B) {a * b}
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66 M-Union : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
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67 M-Union {Σ} A B = record {
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68 states = states A × states B
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69 ; astart = ( astart A , astart B )
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70 ; aℕ = aℕ A * aℕ B
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71 ; afin = fin-× (afin A) (afin B)
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72 ; automaton = record {
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73 δ = λ q x → ( δ (automaton A) (proj₁ q) x , δ (automaton B) (proj₂ q) x )
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74 ; aend = λ q → ( aend (automaton A) (proj₁ q) \/ aend (automaton B) (proj₂ q) )
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75 }
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76 }
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77
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78 closed-in-union : {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Union (contain A) (contain B)) x ( M-Union A B )
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79 closed-in-union A B [] = lemma where
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80 lemma : aend (automaton A) (astart A) \/ aend (automaton B) (astart B) ≡
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81 aend (automaton A) (astart A) \/ aend (automaton B) (astart B)
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82 lemma = refl
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83 closed-in-union {Σ} A B ( h ∷ t ) = lemma1 t ((δ (automaton A) (astart A) h)) ((δ (automaton B) (astart B) h)) where
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84 lemma1 : (t : List Σ) → (qa : states A ) → (qb : states B ) →
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85 accept (automaton A) qa t \/ accept (automaton B) qb t
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86 ≡ accept (automaton (M-Union A B)) (qa , qb) t
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87 lemma1 [] qa qb = refl
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88 lemma1 (h ∷ t ) qa qb = lemma1 t ((δ (automaton A) qa h)) ((δ (automaton B) qb h))
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89
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90 -- M-Concat : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
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91 -- M-Concat {Σ} A B = record {
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92 -- states = states A ∨ states B
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93 -- ; astart = case1 (astart A )
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94 -- ; automaton = record {
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95 -- δ = {!!}
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96 -- ; aend = {!!}
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97 -- }
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98 -- }
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99 --
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100 -- closed-in-concat : {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Concat (contain A) (contain B)) x ( M-Concat A B )
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101 -- closed-in-concat = {!!}
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102
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103 open import nfa
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104 open import sbconst2
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105 open FiniteSet
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106 open import Data.Nat.Properties hiding ( _≟_ )
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107 open import Relation.Binary as B hiding (Decidable)
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108
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109 postulate
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110 fin-∨ : {A B : Set} → { a b : ℕ } → FiniteSet A {a} → FiniteSet B {b} → FiniteSet (A ∨ B) {a + b}
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111 fin→ : {A : Set} → { a : ℕ } → FiniteSet A {a} → FiniteSet (A → Bool ) {exp 2 a}
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112
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113 Concat-NFA : {Σ : Set} → (A B : RegularLanguage Σ ) → NAutomaton (states A ∨ states B) Σ
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114 Concat-NFA {Σ} A B = record { Nδ = δnfa ; Nend = nend }
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115 module Concat-NFA where
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116 δnfa : states A ∨ states B → Σ → states A ∨ states B → Bool
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117 δnfa (case1 q) i (case1 q₁) = equal? (afin A) (δ (automaton A) q i) q₁
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118 δnfa (case1 qa) i (case2 qb) = (aend (automaton A) qa ) /\ (equal? (afin B) qb (astart B) )
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119 δnfa (case2 q) i (case2 q₁) = equal? (afin B) (δ (automaton B) q i) q₁
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120 δnfa _ i _ = false
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121 nend : states A ∨ states B → Bool
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122 nend (case2 q) = aend (automaton B) q
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123 nend _ = false
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124
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125 -- Concat-NFA-start : {Σ : Set} → (A B : RegularLanguage Σ ) → states A ∨ states B → Bool
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126 -- Concat-NFA-start A B (case1 q) = equal? (afin A) q (astart A)
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127 -- Concat-NFA-start _ _ _ = false
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128
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129 Concat-NFA-start : {Σ : Set} → (A B : RegularLanguage Σ ) → states A ∨ states B → Bool
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130 Concat-NFA-start A B q = equal? (fin-∨ (afin A) (afin B)) (case1 (astart A)) q
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131
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132 M-Concat : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
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133 M-Concat {Σ} A B = record {
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134 states = states A ∨ states B → Bool
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135 ; astart = Concat-NFA-start A B
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136 ; aℕ = finℕ finf
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137 ; afin = finf
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138 ; automaton = subset-construction fin (Concat-NFA A B) (case1 (astart A))
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139 } where
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140 fin : FiniteSet (states A ∨ states B ) {aℕ A + aℕ B}
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141 fin = fin-∨ (afin A) (afin B)
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142 finf : FiniteSet (states A ∨ states B → Bool )
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143 finf = fin→ fin
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144
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145 record Split {Σ : Set} (A : List Σ → Bool ) ( B : List Σ → Bool ) (x : List Σ ) : Set where
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146 field
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147 sp0 : List Σ
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148 sp1 : List Σ
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149 sp-concat : sp0 ++ sp1 ≡ x
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150 prop0 : A sp0 ≡ true
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151 prop1 : B sp1 ≡ true
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152
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153 open Split
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154
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155 list-empty++ : {Σ : Set} → (x y : List Σ) → x ++ y ≡ [] → (x ≡ [] ) ∧ (y ≡ [] )
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156 list-empty++ [] [] refl = record { proj1 = refl ; proj2 = refl }
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157 list-empty++ [] (x ∷ y) ()
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158 list-empty++ (x ∷ x₁) y ()
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159
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160 open _∧_
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161
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162 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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163
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164 c-split-lemma : {Σ : Set} → (A B : List Σ → Bool ) → (h : Σ) → ( t : List Σ ) → split A B (h ∷ t ) ≡ true
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165 → ( ¬ (A [] ≡ true )) ∨ ( ¬ (B ( h ∷ t ) ≡ true) )
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166 → split (λ t1 → A (h ∷ t1)) B t ≡ true
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167 c-split-lemma {Σ} A B h t eq (case1 ¬p ) = sym ( begin
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168 true
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169 ≡⟨ sym eq ⟩
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170 split A B (h ∷ t )
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171 ≡⟨⟩
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172 A [] /\ B (h ∷ t) \/ split (λ t1 → A (h ∷ t1)) B t
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173 ≡⟨ cong ( λ k → k \/ split (λ t1 → A (h ∷ t1)) B t ) (bool-and-1 (¬-bool-t ¬p)) ⟩
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174 false \/ split (λ t1 → A (h ∷ t1)) B t
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175 ≡⟨ bool-or-1 refl ⟩
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176 split (λ t1 → A (h ∷ t1)) B t
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177 ∎ ) where open ≡-Reasoning
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178 c-split-lemma {Σ} A B h t eq (case2 ¬p ) = sym ( begin
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179 true
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180 ≡⟨ sym eq ⟩
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181 split A B (h ∷ t )
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182 ≡⟨⟩
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183 A [] /\ B (h ∷ t) \/ split (λ t1 → A (h ∷ t1)) B t
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184 ≡⟨ cong ( λ k → k \/ split (λ t1 → A (h ∷ t1)) B t ) (bool-and-2 (¬-bool-t ¬p)) ⟩
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185 false \/ split (λ t1 → A (h ∷ t1)) B t
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186 ≡⟨ bool-or-1 refl ⟩
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187 split (λ t1 → A (h ∷ t1)) B t
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188 ∎ ) where open ≡-Reasoning
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189
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190 c-split : {Σ : Set} → (A B : List Σ → Bool ) → ( x : List Σ ) → split A B x ≡ true → Split A B x
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191 c-split {Σ} A B [] eq with bool-≡-? (A []) true | bool-≡-? (B []) true
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192 c-split {Σ} A B [] eq | yes eqa | yes eqb =
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193 record { sp0 = [] ; sp1 = [] ; sp-concat = refl ; prop0 = eqa ; prop1 = eqb }
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194 c-split {Σ} A B [] eq | yes p | no ¬p = ⊥-elim (lemma-∧-1 eq (¬-bool-t ¬p ))
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195 c-split {Σ} A B [] eq | no ¬p | t = ⊥-elim (lemma-∧-0 eq (¬-bool-t ¬p ))
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196 c-split {Σ} A B (h ∷ t ) eq with bool-≡-? (A []) true | bool-≡-? (B (h ∷ t )) true
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197 ... | yes px | yes py = record { sp0 = [] ; sp1 = h ∷ t ; sp-concat = refl ; prop0 = px ; prop1 = py }
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198 ... | no px | _ with c-split (λ t1 → A ( h ∷ t1 )) B t (c-split-lemma A B h t eq (case1 px) )
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199 ... | S = record { sp0 = h ∷ sp0 S ; sp1 = sp1 S ; sp-concat = cong ( λ k → h ∷ k) (sp-concat S) ; prop0 = prop0 S ; prop1 = prop1 S }
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200 c-split {Σ} A B (h ∷ t ) eq | _ | no px with c-split (λ t1 → A ( h ∷ t1 )) B t (c-split-lemma A B h t eq (case2 px) )
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201 ... | S = record { sp0 = h ∷ sp0 S ; sp1 = sp1 S ; sp-concat = cong ( λ k → h ∷ k) (sp-concat S) ; prop0 = prop0 S ; prop1 = prop1 S }
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202
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203 split++ : {Σ : Set} → (A B : List Σ → Bool ) → ( x y : List Σ ) → A x ≡ true → B y ≡ true → split A B (x ++ y ) ≡ true
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204 split++ {Σ} A B [] [] eqa eqb = begin
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205 split A B []
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206 ≡⟨⟩
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207 A [] /\ B []
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208 ≡⟨ cong₂ (λ j k → j /\ k ) eqa eqb ⟩
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209 true
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210 ∎ where open ≡-Reasoning
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211 split++ {Σ} A B [] (h ∷ y ) eqa eqb = begin
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212 split A B (h ∷ y )
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213 ≡⟨⟩
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214 A [] /\ B (h ∷ y) \/ split (λ t1 → A (h ∷ t1)) B y
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215 ≡⟨ cong₂ (λ j k → j /\ k \/ split (λ t1 → A (h ∷ t1)) B y ) eqa eqb ⟩
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216 true /\ true \/ split (λ t1 → A (h ∷ t1)) B y
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217 ≡⟨⟩
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218 true \/ split (λ t1 → A (h ∷ t1)) B y
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219 ≡⟨⟩
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220 true
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221 ∎ where open ≡-Reasoning
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222 split++ {Σ} A B (h ∷ t) y eqa eqb = begin
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223 split A B ((h ∷ t) ++ y)
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224 ≡⟨⟩
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225 A [] /\ B (h ∷ t ++ y) \/ split (λ t1 → A (h ∷ t1)) B (t ++ y)
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226 ≡⟨ cong ( λ k → A [] /\ B (h ∷ t ++ y) \/ k ) ( begin
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227 split (λ t1 → A (h ∷ t1)) B (t ++ y)
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228 ≡⟨ split++ {Σ} (λ t1 → A (h ∷ t1)) B t y eqa eqb ⟩
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229 true
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230 ∎ ) ⟩
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231 A [] /\ B (h ∷ t ++ y) \/ true
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232 ≡⟨ bool-or-3 ⟩
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233 true
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234 ∎ where open ≡-Reasoning
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235
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236 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n n -- (Level.suc n) already in finiteSet
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237
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238 open NAutomaton
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239
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240 closed-in-concat : {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Concat (contain A) (contain B)) x ( M-Concat A B )
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241 closed-in-concat {Σ} A B x = ≡-Bool-func lemma3 lemma4 where
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242 finav = (fin-∨ (afin A) (afin B))
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243 NFA = (Concat-NFA A B)
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244 abmove : (q : states A ∨ states B) → (h : Σ ) → states A ∨ states B
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245 abmove (case1 q) h = case1 (δ (automaton A) q h)
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246 abmove (case2 q) h = case2 (δ (automaton B) q h)
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247 nmove : (q : states A ∨ states B) (nq : states A ∨ states B → Bool ) → (nq q ≡ true) → ( h : Σ ) →
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248 exists finav (λ qn → nq qn /\ Nδ NFA qn h (abmove q h)) ≡ true
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249 nmove (case1 q) nq nqt h = found finav {_} {(case1 q)} ( bool-and-tt nqt lemma-nmove-a ) where
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250 lemma-nmove-a : Nδ NFA (case1 q) h (abmove (case1 q) h) ≡ true
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251 lemma-nmove-a with F←Q (afin A) (δ (automaton A) q h) ≟ F←Q (afin A) (δ (automaton A) q h)
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252 lemma-nmove-a | yes refl = refl
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253 lemma-nmove-a | no ne = ⊥-elim (ne refl)
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254 nmove (case2 q) nq nqt h = found finav {_} {(case2 q)} ( bool-and-tt nqt lemma-nmove ) where
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255 lemma-nmove : Nδ NFA (case2 q) h (abmove (case2 q) h) ≡ true
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256 lemma-nmove with F←Q (afin B) (δ (automaton B) q h) ≟ F←Q (afin B) (δ (automaton B) q h)
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257 lemma-nmove | yes refl = refl
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258 lemma-nmove | no ne = ⊥-elim (ne refl)
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259 lemma6 : (z : List Σ) → (q : states B) → (nq : states A ∨ states B → Bool ) → (nq (case2 q) ≡ true) → ( accept (automaton B) q z ≡ true )
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260 → Naccept NFA finav nq z ≡ true
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261 lemma6 [] q nq nqt fb = lemma8 where
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262 lemma8 : exists finav ( λ q → nq q /\ Nend NFA q ) ≡ true
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263 lemma8 = found finav {_} {case2 q} ( bool-and-tt nqt fb )
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264 lemma6 (h ∷ t ) q nq nq=q fb = lemma6 t (δ (automaton B) q h) (Nmoves NFA finav nq h) (nmove (case2 q) nq nq=q h) fb
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265 lemma7 : (y z : List Σ) → (q : states A) → (nq : states A ∨ states B → Bool ) → (nq (case1 q) ≡ true)
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266 → ( accept (automaton A) q y ≡ true ) → ( accept (automaton B) (astart B) z ≡ true )
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267 → Naccept NFA finav nq (y ++ z) ≡ true
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268 lemma7 [] z q nq nq=q fa fb = lemma6 z (astart B) nq lemma71 fb where
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269 lemma71 : nq (case2 (astart B)) ≡ true
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270 lemma71 = {!!}
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271 lemma-nq=q : (nq (case1 q) ≡ true)
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272 lemma-nq=q = nq=q
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273 lemma7 (h ∷ t) z q nq nq=q fa fb = lemma7 t z (δ (automaton A) q h) (Nmoves NFA finav nq h) (nmove (case1 q) nq nq=q h) fa fb where
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274 lemma9 : equal? finav (case1 (astart A)) (case1 (astart A)) ≡ true
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275 lemma9 with Data.Fin._≟_ (F←Q finav (case1 (astart A))) ( F←Q finav (case1 (astart A)) )
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276 lemma9 | yes refl = refl
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277 lemma9 | no ¬p = ⊥-elim ( ¬p refl )
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75
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278 lemma5 : Split (contain A) (contain B) x
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76
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279 → Naccept NFA finav (equal? finav (case1 (astart A))) x ≡ true
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280 lemma5 S = subst ( λ k → Naccept NFA finav (equal? finav (case1 (astart A))) k ≡ true ) ( sp-concat S )
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281 (lemma7 (sp0 S) (sp1 S) (astart A) (equal? finav (case1 (astart A))) lemma9 (prop0 S) (prop1 S) )
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71
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282 lemma3 : Concat (contain A) (contain B) x ≡ true → contain (M-Concat A B) x ≡ true
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74
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283 lemma3 concat with c-split (contain A) (contain B) x concat
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75
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284 ... | S = begin
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76
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285 accept (subset-construction finav NFA (case1 (astart A))) (Concat-NFA-start A B ) x
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286 ≡⟨ ≡-Bool-func (subset-construction-lemma← finav NFA (case1 (astart A)) x )
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287 (subset-construction-lemma→ finav NFA (case1 (astart A)) x ) ⟩
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288 Naccept NFA finav (equal? finav (case1 (astart A))) x
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75
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289 ≡⟨ lemma5 S ⟩
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290 true
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291 ∎ where open ≡-Reasoning
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71
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292 lemma4 : contain (M-Concat A B) x ≡ true → Concat (contain A) (contain B) x ≡ true
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74
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293 lemma4 C = {!!} -- split++ (contain A) (contain B) x y (accept ?) (accept ?)
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71
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294
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295
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