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1 module whileTestGears 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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10
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5 open import Data.Bool hiding ( _≟_ )
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4
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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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4
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12
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6
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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} -> (Code : Env -> t) -> t
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20 whileTest next = next (record {varn = 10 ; 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 (λ env → whileLoop env (λ env1 → env1 ))
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31
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32
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33 proof1 : whileTest (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 whileTest' : {l : Level} {t : Set l} -> (Code : (env : Env) -> ((vari env) ≡ 0) /\ ((varn env) ≡ 10) -> t) -> t
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37 whileTest' next = next env proof2
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38 where
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39 env : Env
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40 env = record {vari = 0 ; varn = 10}
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41 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ 10)
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42 proof2 = record {pi1 = refl ; pi2 = refl}
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11
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43
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44 open import Data.Empty
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45 open import Data.Nat.Properties
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46
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47
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48 {-# TERMINATING #-}
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49 whileLoop' : {l : Level} {t : Set l} -> (env : Env) -> ((varn env) + (vari env) ≡ 10) -> (Code : Env -> t) -> t
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50 whileLoop' env proof next with ( suc zero ≤? (varn env) )
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51 whileLoop' env proof next | no p = next env
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52 whileLoop' env proof next | yes p = whileLoop' env1 (proof3 p ) next
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53 where
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54 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
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11
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55 1<0 : 1 ≤ zero → ⊥
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56 1<0 ()
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57 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ 10
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58 proof3 (s≤s lt) with varn env
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59 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
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60 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
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61 begin
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62 n' + (vari env + 1)
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63 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
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64 n' + (1 + vari env )
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65 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
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66 (n' + 1 ) + vari env
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67 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
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68 (suc n' ) + vari env
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69 ≡⟨⟩
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70 varn env + vari env
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71 ≡⟨ proof ⟩
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72 10
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73 ∎
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74
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75
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76 conversion1 : {l : Level} {t : Set l } → (env : Env) -> ((vari env) ≡ 0) /\ ((varn env) ≡ 10)
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77 -> (Code : (env1 : Env) -> (varn env1 + vari env1 ≡ 10) -> t) -> t
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78 conversion1 env p1 next = next env proof4
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79 where
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80 proof4 : varn env + vari env ≡ 10
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81 proof4 = let open ≡-Reasoning in
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82 begin
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83 varn env + vari env
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84 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
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85 10 + vari env
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86 ≡⟨ cong ( λ n → 10 + n ) (pi1 p1 ) ⟩
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87 10 + 0
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88 ≡⟨⟩
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89 10
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90 ∎
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4
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91
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92
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93 proofGears : Set
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94 proofGears = whileTest' (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ 10 ))))
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9
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95
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96 proofGearsMeta : whileTest' (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ 10 ))))
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97 proofGearsMeta = refl
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