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1 module whileTestGears1 where
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2
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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
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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 whileTest : {l : Level} {t : Set l} -> (c10 : ℕ) → (Code : Env -> t) -> t
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20 whileTest c10 next = next (record {varn = c10 ; vari = 0} )
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21
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22 {-# TERMINATING #-}
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23 whileLoop : {l : Level} {t : Set l} -> Env -> (Code : Env -> t) -> t
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24 whileLoop env next with lt 0 (varn env)
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25 whileLoop env next | false = next env
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26 whileLoop env next | true =
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27 whileLoop (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
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28
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29 test1 : Env
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30 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 ))
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31
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32
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33 proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 ))
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34 proof1 = refl
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35
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36 -- ↓PostCondition
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37 whileTest' : {l : Level} {t : Set l} -> {c10 : ℕ } → (Code : (env : Env) -> ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -> t) -> t
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38 whileTest' {_} {_} {c10} next = next env proof2
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39 where
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40 env : Env
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41 env = record {vari = 0 ; varn = c10}
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42 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition
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43 proof2 = record {pi1 = refl ; pi2 = refl}
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44
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45 open import Data.Empty
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46 open import Data.Nat.Properties
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47
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48
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49 {-# TERMINATING #-} -- ↓PreCondition(Invaliant)
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50 whileLoop' : {l : Level} {t : Set l} -> (env : Env) -> {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) -> (Code : Env -> t) -> t
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51 whileLoop' env proof next with ( suc zero ≤? (varn env) )
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52 whileLoop' env proof next | no p = next env
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53 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next
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54 where
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55 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
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56 1<0 : 1 ≤ zero → ⊥
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57 1<0 ()
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58 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10
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59 proof3 (s≤s lt) with varn env
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60 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
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61 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
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62 begin
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63 n' + (vari env + 1)
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64 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
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65 n' + (1 + vari env )
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66 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
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67 (n' + 1) + vari env
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68 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
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69 (suc n' ) + vari env
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70 ≡⟨⟩
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71 varn env + vari env
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72 ≡⟨ proof ⟩
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73 c10
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74 ∎
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75
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76 -- Condition to Invaliant
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77 conversion1 : {l : Level} {t : Set l } → (env : Env) -> {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
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78 -> (Code : (env1 : Env) -> (varn env1 + vari env1 ≡ c10) -> t) -> t
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79 conversion1 env {c10} p1 next = next env proof4
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80 where
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81 proof4 : varn env + vari env ≡ c10
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82 proof4 = let open ≡-Reasoning in
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83 begin
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84 varn env + vari env
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85 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
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86 c10 + vari env
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87 ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
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88 c10 + 0
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89 ≡⟨ +-sym {c10} {0} ⟩
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90 c10
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91 ∎
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92
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93
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94 proofGears : {c10 : ℕ } → Set
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95 proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 ))))
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96
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97 proofGearsMeta : {c10 : ℕ } → whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 ))))
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98 proofGearsMeta {c10} = {!!}
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99
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100
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101
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102 whileTest0 : {l : Level} {t m : Set l} -> (c10 : ℕ) → (Code : m -> Env -> t) -> t
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103 whileTest0 c10 next = next {!!} (record {varn = c10 ; vari = 0} )
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104
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105 {-# TERMINATING #-}
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106 whileLoop0 : {l : Level} {t m : Set l} -> m -> Env -> (Code : m -> Env -> t) -> t
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107 whileLoop0 m env next with lt 0 (varn env)
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108 whileLoop0 m env next | false = next m env
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109 whileLoop0 m env next | true =
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110 whileLoop0 m (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
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