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1 open import Level renaming (suc to Suc ; zero to Zero )
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2 module WhileTest where
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3
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4 open import Relation.Binary.PropositionalEquality
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5 open import Relation.Binary.Core
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6 open import Data.Nat hiding (compare)
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7 open import Data.Maybe
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8 open import Data.List
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9 open import Function
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10 open import logic
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11
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13
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12 open import Data.Bool hiding ( _∧_ ; _≟_ ) -- ; _∧_ ; _≤_ ; _<_)
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13 open import Data.Product
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11
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14 open import Agda.Builtin.Unit
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13
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15 open import Relation.Nullary using (¬_; Dec; yes; no)
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16
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17 open import Data.Empty
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18 open import Data.Nat.Properties
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19
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20 -- logicへ
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21 +zero : { y : ℕ } → y + zero ≡ y
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22 +zero {zero} = refl
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23 +zero {suc y} = cong ( λ x → suc x ) ( +zero {y} )
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24
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25 -- utilへ
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26 _-_ : ℕ → ℕ → ℕ
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27 x - zero = x
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28 zero - _ = zero
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29 (suc x) - (suc y) = x - y
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30
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31 record Env : Set (Suc Zero) where
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32 field
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33 c10 : ℕ
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34 varn : ℕ
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35 vari : ℕ
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36 open Env
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37
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38 data _implies_ (A B : Set ) : Set (Suc Zero) where
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39 proof : ( A → B ) → A implies B
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40
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41 implies2p : {A B : Set } → A implies B → A → B
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42 implies2p (proof x) = x
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43
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44 data whileTestState : Set where
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45 s1 : whileTestState
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46 s2 : whileTestState
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47 sf : whileTestState
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48
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49 whileTestStateP : whileTestState → Env → Set
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50 whileTestStateP s1 env = (vari env ≡ 0) ∧ (varn env ≡ c10 env)
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51 whileTestStateP s2 env = (varn env + vari env ≡ c10 env)
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52 whileTestStateP sf env = (vari env ≡ c10 env)
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53
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54 record WhileTest {m : Level } {t : Set m } : Set (Suc m) where
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55 field
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56 env : Env
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57 whileInit : {m : Level } {t : Set m } → (c10 : ℕ) → (Env → t) → t
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58 whileInit c10 next = next (record {c10 = c10 ; varn = c10 ; vari = 0 } )
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59 whileLoop : Env → (Code : Env → t) → t
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60 whileLoop env next = whileLoop1 (varn env) env where
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61 whileLoop1 : ℕ → Env → t
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62 whileLoop1 zero env = next env
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63 whileLoop1 (suc t ) env = whileLoop1 t (record env {varn = t ; vari = (vari env) + 1})
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64 whileTest : (c10 : ℕ) → (Env → t) → t
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65 whileTest c10 next = whileInit c10 $ λ env → whileLoop env next
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66
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67 loopPP : (n : ℕ) → (input : Env ) → (n ≡ varn input) → Env
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68 loopPP zero input refl = input
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69 loopPP (suc n) input refl = loopPP n (record input { varn = pred (varn input) ; vari = suc (vari input)}) refl
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70
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71 -- init
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72 whileInit-impl : (c10 : ℕ) → whileInit c10 (λ env → ⊤ implies (whileTestStateP s1 env) )
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73 whileInit-impl c = proof ( λ _ → record { proj1 = refl ; proj2 = refl } )
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74 whileTestPSemSound : (c : ℕ ) (output : Env ) → output ≡ whileInit c (λ e → e) → ⊤ implies ((vari output ≡ 0) ∧ (varn output ≡ c))
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75 whileTestPSemSound c output refl = whileInit-impl c
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76 -- init → loop
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77 whileConvPSemSound : {l : Level} → (input : Env) → (whileTestStateP s1 input ) implies (whileTestStateP s2 input)
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78 whileConvPSemSound input = proof λ x → (conv input x) where
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79 conv : (env : Env ) → (vari env ≡ 0) ∧ (varn env ≡ c10 env ) → varn env + vari env ≡ c10 env
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80 conv e record { proj1 = refl ; proj2 = refl } = +zero
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81 -- loop → loop
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82 whileLoopPSem : {l : Level} {t : Set l} → (input : Env ) → whileTestStateP s2 input
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83 → (next : (output : Env ) → (whileTestStateP s2 input ) implies (whileTestStateP s2 output) → t)
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84 → (exit : (output : Env ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output) → t) → t
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85 whileLoopPSem env s next exit with varn env | s
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86 ... | zero | _ = exit env (proof (λ z → z))
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87 ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof λ x → +-suc varn (vari env) )
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88 -- loop → fin
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89
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90 loopPPSem : (input output : Env ) → output ≡ loopPP (varn input) input refl
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91 → (whileTestStateP s2 input ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output)
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92 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p
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93 where
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94 lem : (n : ℕ) → (env : Env) → n + suc (vari env) ≡ suc (n + vari env)
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95 lem n env = +-suc (n) (vari env)
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96 loopPPSemInduct : (n : ℕ) → (current : Env) → (eq : n ≡ varn current) → (loopeq : output ≡ loopPP n current eq)
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97 → (whileTestStateP s2 current ) → (whileTestStateP s2 current ) implies (whileTestStateP sf output)
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98 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (λ x → refl)
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99 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
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100 whileLoopPSem current refl
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101 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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102 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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103
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104
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105 whileLoopPSemSound : {l : Level} → (input output : Env )
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106 → whileTestStateP s2 input
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107 → output ≡ loopPP (varn input) input refl
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108 → (whileTestStateP s2 input ) implies ( whileTestStateP sf output )
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109 whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre
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110
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111 whileTestSound : {l : Level} (c : ℕ) → (output : Env) → ⊤ → whileTestStateP sf output
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112 whileTestSound {l} c record { c10 = c10 ; varn = varn ; vari = vari } top =
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113 implies2p (whileLoopPSemSound {l} (record { c10 = c ; varn = c ; vari = zero }) (record { c10 = c10 ; varn = c ; vari = vari}) (+zero) {!!}) (implies2p (whileConvPSemSound {l} (record { c10 = c ; varn = c ; vari = zero })) (implies2p (whileTestPSemSound c (whileInit c (λ e → e)) refl) top))
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114
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115 open WhileTest
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116
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117 createWhileTest : {m : Level} {t : Set m } → WhileTest {m} {t}
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118 createWhileTest = record { env = record { c10 = 0; varn = 0; vari = 0 } }
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119
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120 test2 : ℕ
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121 test2 = whileTest createWhileTest 10 $ λ e → vari e
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122
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123 ---
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124
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125 {-
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126 whileTestStateP : whileTestState → Envc → Set
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127 whileTestStateP s1 env = (vari env ≡ 0) /\ (varn env ≡ c10 env)
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128 whileTestStateP s2 env = (varn env + vari env ≡ c10 env)
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129 whileTestStateP sf env = (vari env ≡ c10 env)
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130
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131 whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Envc → t) → t
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132 whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } )
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133
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134 whileTestPSem : (c : ℕ) → whileTestP c ( λ env → ⊤ implies (whileTestStateP s1 env) )
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135 whileTestPSem c = proof ( λ _ → record { pi1 = refl ; pi2 = refl } )
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136 -}
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