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annotate whileTestPrim.agda @ 3:6be8ee856666

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add simple Hoare logic example
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 14 Dec 2018 16:26:49 +0900
parents
children 64bd5c236002
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3
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 module whileTestPrim where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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2
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 open import Function
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4 open import Data.Nat
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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5 open import Data.Bool hiding ( _≟_ )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6 open import Level renaming ( suc to succ ; zero to Zero )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 open import Relation.Nullary using (¬_; Dec; yes; no)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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8 open import Relation.Binary.Core
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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9
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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10
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12 record Env : Set where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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13 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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14 varn : ℕ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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15 vari : ℕ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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16 open Env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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17
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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18 PrimComm : Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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19 PrimComm = Env → Env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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20
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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21 Cond : Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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22 Cond = (Env → Bool)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 data Comm : Set where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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25 Skip : Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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26 Abort : Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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27 PComm : PrimComm -> Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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28 Seq : Comm -> Comm -> Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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29 If : Cond -> Comm -> Comm -> Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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30 While : Cond -> Comm -> Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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31
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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32 _-_ : ℕ -> ℕ -> ℕ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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33 x - zero = x
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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34 zero - _ = zero
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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35 (suc x) - (suc y) = x - y
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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36
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 lt : ℕ -> ℕ -> Bool
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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38 lt x y with (suc x ) ≤? y
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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39 lt x y | yes p = true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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40 lt x y | no ¬p = false
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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41
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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42 Equal : ℕ -> ℕ -> Bool
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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43 Equal x y with x ≟ y
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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44 Equal x y | yes p = true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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45 Equal x y | no ¬p = false
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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47 program : Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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48 program =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 Seq ( PComm (λ env → record env {varn = 10}))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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50 $ Seq ( PComm (λ env → record env {vari = 0}))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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51 $ While (λ env → lt (varn env ) zero )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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52 (Seq (PComm (λ env → record env {vari = ((vari env) + 1)} ))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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53 $ PComm (λ env → record env {varn = ((varn env) - 1)} ))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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54
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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55 simple : Comm
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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56 simple =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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57 Seq ( PComm (λ env → record env {varn = 10}))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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58 $ PComm (λ env → record env {vari = 0})
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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59
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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60 {-# TERMINATING #-}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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61 interpret : Env → Comm → Env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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62 interpret env Skip = env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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63 interpret env Abort = env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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64 interpret env (PComm x) = x env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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65 interpret env (Seq comm comm1) = interpret (interpret env comm) comm1
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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66 interpret env (If x then else) with x env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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67 ... | true = interpret env then
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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68 ... | false = interpret env else
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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69 interpret env (While x comm) with x env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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70 ... | true = interpret (interpret env comm) (While x comm)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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71 ... | false = env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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72
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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73 test1 : Env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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74 test1 = interpret ( record { vari = 0 ; varn = 0 } ) program
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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75
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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76
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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77 empty-case : (env : Env) → (( λ e → true ) env ) ≡ true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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78 empty-case _ = refl
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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79
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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80 implies : Bool → Bool → Bool
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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81 implies false _ = true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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82 implies true true = true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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83 implies true false = false
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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84
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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85 Axiom : Cond -> PrimComm -> Cond -> Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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86 Axiom pre comm post = ∀ (env : Env) → implies (pre env) ( post (comm env)) ≡ true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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87
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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88 Tautology : Cond -> Cond -> Set
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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89 Tautology pre post = ∀ (env : Env) → implies (pre env) (post env) ≡ true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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90
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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91 _/\_ : Cond -> Cond -> Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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92 x /\ y = λ env → x env ∧ y env
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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93
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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94 neg : Cond -> Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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95 neg x = λ env → not ( x env )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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96
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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97 data HTProof : Cond -> Comm -> Cond -> Set where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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98 PrimRule : {bPre : Cond} -> {pcm : PrimComm} -> {bPost : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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99 (pr : Axiom bPre pcm bPost) ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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100 HTProof bPre (PComm pcm) bPost
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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101 SkipRule : (b : Cond) -> HTProof b Skip b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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102 AbortRule : (bPre : Cond) -> (bPost : Cond) ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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103 HTProof bPre Abort bPost
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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104 WeakeningRule : {bPre : Cond} -> {bPre' : Cond} -> {cm : Comm} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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105 {bPost' : Cond} -> {bPost : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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106 Tautology bPre bPre' ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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107 HTProof bPre' cm bPost' ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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108 Tautology bPost' bPost ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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109 HTProof bPre cm bPost
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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110 SeqRule : {bPre : Cond} -> {cm1 : Comm} -> {bMid : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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111 {cm2 : Comm} -> {bPost : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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112 HTProof bPre cm1 bMid ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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113 HTProof bMid cm2 bPost ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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114 HTProof bPre (Seq cm1 cm2) bPost
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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115 IfRule : {cmThen : Comm} -> {cmElse : Comm} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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116 {bPre : Cond} -> {bPost : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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117 {b : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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118 HTProof (bPre /\ b) cmThen bPost ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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119 HTProof (bPre /\ neg b) cmElse bPost ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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120 HTProof bPre (If b cmThen cmElse) bPost
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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121 WhileRule : {cm : Comm} -> {bInv : Cond} -> {b : Cond} ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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122 HTProof (bInv /\ b) cm bInv ->
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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123 HTProof bInv (While b cm) (bInv /\ neg b)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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124
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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125 initCond : Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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126 initCond env = true
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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127
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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128 stmt1Cond : Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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129 stmt1Cond env = Equal (varn env) 10
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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130
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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131 stmt2Cond : Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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132 stmt2Cond env = (Equal (varn env) 10) ∧ (Equal (vari env) 0)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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133
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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134 whileInv : Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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135 whileInv env = Equal ((varn env) + (vari env)) 10
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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136
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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137 whileInv' : Cond
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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138 whileInv' env = Equal ((varn env) + (vari env)) 11
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139
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140 termCond : Cond
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141 termCond env = Equal (vari env) 10
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142
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143 eqlemma : { x y : ℕ } → Equal x y ≡ true → x ≡ y
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144 eqlemma {x} {y} eq with x ≟ y
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145 eqlemma {x} {y} refl | yes refl = refl
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146 eqlemma {x} {y} () | no ¬p
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147
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148 proofs : HTProof initCond simple stmt2Cond
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149 proofs =
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150 SeqRule {initCond} ( PrimRule empty-case )
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151 $ PrimRule {stmt1Cond} {_} {stmt2Cond} lemma
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152 where
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153 lemma : Axiom stmt1Cond (λ env → record { varn = varn env ; vari = 0 }) stmt2Cond
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154 lemma env with stmt1Cond env
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155 lemma env | false = refl
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156 lemma env | true = refl
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157
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158
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159 proof1 : HTProof initCond program termCond
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160 proof1 =
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161 SeqRule {λ e → true} ( PrimRule empty-case )
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162 $ SeqRule {λ e → Equal (varn e) 10} ( PrimRule lemma1 )
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163 $ WeakeningRule {λ e → (Equal (varn e) 10) ∧ (Equal (vari e) 0)} lemma2 (
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164 WhileRule {_} {λ e → Equal ((varn e) + (vari e)) 10}
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165 $ SeqRule (PrimRule {λ e → whileInv e ∧ lt (varn e) zero } lemma3)
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166 $ PrimRule {whileInv'} {_} {whileInv} lemma4 ) lemma5
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167 where
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168 lemma1 : Axiom stmt1Cond (λ env → record { varn = varn env ; vari = 0 }) stmt2Cond
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169 lemma1 env with stmt1Cond env
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170 lemma1 env | false = refl
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171 lemma1 env | true = refl
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172 lemma2 : Tautology stmt2Cond whileInv
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173 lemma2 env with stmt2Cond env | Equal (varn env + vari env) 10
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174 lemma2 env | false | false = refl
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175 lemma2 env | false | true = refl
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176 lemma2 env | true | true = refl
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177 lemma2 env | true | false = {!!}
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178 lemma3 : Axiom (whileInv /\ (λ env → lt (varn env) zero)) (λ env → record { varn = varn env ; vari = vari env + 1 }) whileInv'
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179 lemma3 = {!!}
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180 lemma4 : Axiom whileInv' (λ env → record { varn = varn env - 1 ; vari = vari env }) whileInv
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181 lemma4 = {!!}
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182 lemma5 : Tautology (whileInv /\ neg (λ z → lt (varn z) zero)) termCond
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183 lemma5 = {!!}
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184
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185