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1 module whileTestGears1 where
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3 open import Function
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4 open import Data.Nat
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5 open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_ )
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6 open import Level renaming ( suc to succ ; zero to Zero )
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7 open import Relation.Nullary using (¬_; Dec; yes; no)
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8 open import Relation.Binary.PropositionalEquality
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9
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10 open import utilities
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11 open _/\_
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12
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13 record Env : Set where
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14 field
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15 varn : ℕ
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16 vari : ℕ
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17 open Env
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18
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19 whileTestS : { m : Level} → (c10 : ℕ) → (Code : Env → Set m) → Set m
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20 whileTestS c10 next = next (record {varn = c10 ; vari = 0} )
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21
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22 whileTestS1 : (c10 : ℕ) → whileTestS c10 (λ e → ((varn e ≡ c10) /\ (vari e ≡ 0 )) )
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23 whileTestS1 c10 = record { pi1 = refl ; pi2 = refl }
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24
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25
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26 whileTest : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Env → t) → t
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27 whileTest c10 next = next (record {varn = c10 ; vari = 0} )
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28
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29 {-# TERMINATING #-}
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30 whileLoop : {l : Level} {t : Set l} → Env → (Code : Env → t) → t
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31 whileLoop env next with lt 0 (varn env)
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32 whileLoop env next | false = next env
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33 whileLoop env next | true =
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34 whileLoop (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
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35
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36 test1 : Env
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37 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 ))
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38
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39
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40 proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 ))
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41 proof1 = refl
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42
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43 -- ↓PostCondition
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44 whileTest' : {l : Level} {t : Set l} → {c10 : ℕ } → (Code : (env : Env) → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t
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45 whileTest' {_} {_} {c10} next = next env proof2
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46 where
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47 env : Env
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48 env = record {vari = 0 ; varn = c10}
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49 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition
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50 proof2 = record {pi1 = refl ; pi2 = refl}
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51
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52 open import Data.Empty
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53 open import Data.Nat.Properties
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54
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55 lemma1 : {i : ℕ} → ¬ 1 ≤ i → i ≡ 0
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56 lemma1 {zero} not = refl
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57 lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) )
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58
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59 {-# TERMINATING #-} -- ↓PreCondition(Invaliant)
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60 whileLoop' : {l : Level} {t : Set l} → (env : Env) → {c10 : ℕ } → ((varn env) + (vari env) ≡ c10)
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61 → (Code : (e1 : Env )→ vari e1 ≡ c10 → t) → t
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62 whileLoop' env proof next with ( suc zero ≤? (varn env) )
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63 whileLoop' env {c10} proof next | no p = next env ( begin
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64 vari env ≡⟨ refl ⟩
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65 0 + vari env ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
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66 varn env + vari env ≡⟨ proof ⟩
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67 c10 ∎ ) where open ≡-Reasoning
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68 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next where
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69 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
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70 1<0 : 1 ≤ zero → ⊥
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71 1<0 ()
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72 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10
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73 proof3 (s≤s lt) with varn env
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74 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
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75 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in begin
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76 n' + (vari env + 1) ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
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77 n' + (1 + vari env ) ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
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78 (n' + 1) + vari env ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
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79 (suc n' ) + vari env ≡⟨⟩
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80 varn env + vari env ≡⟨ proof ⟩
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81 c10
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82 ∎
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83
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84 -- Condition to Invaliant
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85 conversion1 : {l : Level} {t : Set l } → (env : Env) → {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
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86 → (Code : (env1 : Env) → (varn env1 + vari env1 ≡ c10) → t) → t
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87 conversion1 env {c10} p1 next = next env proof4 where
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88 proof4 : varn env + vari env ≡ c10
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89 proof4 = let open ≡-Reasoning in begin
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90 varn env + vari env ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
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91 c10 + vari env ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
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92 c10 + 0 ≡⟨ +-sym {c10} {0} ⟩
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93 c10
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94 ∎
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95
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96 open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_)
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97
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98 whileTestSpec1 : (c10 : ℕ) → (e1 : Env ) → vari e1 ≡ c10 → ⊤
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99 whileTestSpec1 _ _ x = tt
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100
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101 proofGears : {c10 : ℕ } → ⊤
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102 proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 p3 → whileTestSpec1 c10 n2 p3 )))
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103
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104 -- ↓PreCondition(Invaliant)
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105 whileLoopSeg : {l : Level} {t : Set l} → {c10 : ℕ } → (env : Env) → ((varn env) + (vari env) ≡ c10)
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106 → (next : (e1 : Env )→ varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t)
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107 → (exit : (e1 : Env )→ vari e1 ≡ c10 → t) → t
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108 whileLoopSeg env proof next exit with ( suc zero ≤? (varn env) )
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109 whileLoopSeg {_} {_} {c10} env proof next exit | no p = exit env ( begin
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110 vari env ≡⟨ refl ⟩
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111 0 + vari env ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
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112 varn env + vari env ≡⟨ proof ⟩
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113 c10 ∎ ) where open ≡-Reasoning
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114 whileLoopSeg {_} {_} {c10} env proof next exit | yes p = next env1 (proof3 p ) proof4 where
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115 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
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116 1<0 : 1 ≤ zero → ⊥
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117 1<0 ()
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118 proof4 : varn env1 < varn env
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119 proof4 = {!!}
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120 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10
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121 proof3 (s≤s lt) with varn env
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122 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
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123 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in begin
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124 n' + (vari env + 1) ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
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125 n' + (1 + vari env ) ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
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126 (n' + 1) + vari env ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
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127 (suc n' ) + vari env ≡⟨⟩
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128 varn env + vari env ≡⟨ proof ⟩
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129 c10
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130 ∎
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131
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132 open import Relation.Binary.Definitions
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133
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134 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
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135 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x
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136
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137 TerminatingLoop : {l : Level} {t : Set l} {c10 : ℕ } → (i : ℕ) → (env : Env) → i ≡ varn env
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138 → varn env + vari env ≡ c10
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139 → (exit : (e1 : Env )→ vari e1 ≡ c10 → t) → t
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140 TerminatingLoop {_} {t} {c10} i env refl p exit with <-cmp 0 i
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141 ... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} env p (λ e1 eq lt → ⊥-elim (lemma3 e1 b lt) ) exit where
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142 lemma3 : (e1 : Env) → 0 ≡ varn env → varn e1 < varn env → ⊥
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143 lemma3 e refl ()
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144 ... | tri< a ¬b ¬c = whileLoopSeg {_} {t} {c10} env p (TerminatingLoop1 i) exit where
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145 TerminatingLoop1 : (j : ℕ) → (e1 : Env) → varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t
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146 TerminatingLoop1 zero e1 eq lt = whileLoopSeg {_} {t} {c10} env p {!!} exit
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147 TerminatingLoop1 (suc j) e1 eq lt with <-cmp j (varn e1)
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148 ... | tri< (s≤s a) ¬b ¬c = TerminatingLoop1 j e1 {!!} {!!}
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149 ... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} e1 {!!} lemma4 exit where
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150 lemma4 : (e2 : Env) → varn e2 + vari e2 ≡ c10 → varn e2 < varn e1 → t
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151 lemma4 e2 eq lt = TerminatingLoop1 j {!!} {!!} {!!}
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152 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> lt {!!} )
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153
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