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annotate whileTestPrim.agda @ 17:b95a3cf9727c

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add Gears1
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 16 Dec 2018 22:01:40 +0900
parents 8d546766a9a8
children 6417f6d821e6
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 module whileTestPrim where
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2
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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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8 open import Relation.Binary.PropositionalEquality
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9
10
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10 open import utilities
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11
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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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14 varn : ℕ
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15 vari : ℕ
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16 open Env
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17
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18 PrimComm : Set
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19 PrimComm = Env → Env
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20
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21 Cond : Set
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22 Cond = (Env → Bool)
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23
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24 data Comm : Set where
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25 Skip : Comm
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26 Abort : Comm
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27 PComm : PrimComm -> Comm
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28 Seq : Comm -> Comm -> Comm
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29 If : Cond -> Comm -> Comm -> Comm
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30 While : Cond -> Comm -> Comm
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32 ---------------------------
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33
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34 program : ℕ → Comm
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35 program c10 =
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36 Seq ( PComm (λ env → record env {varn = c10}))
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37 $ Seq ( PComm (λ env → record env {vari = 0}))
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38 $ While (λ env → lt zero (varn env ) )
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39 (Seq (PComm (λ env → record env {vari = ((vari env) + 1)} ))
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40 $ PComm (λ env → record env {varn = ((varn env) - 1)} ))
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41
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42 simple : ℕ → Comm
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43 simple c10 =
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44 Seq ( PComm (λ env → record env {varn = c10}))
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45 $ PComm (λ env → record env {vari = 0})
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46
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47 {-# TERMINATING #-}
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48 interpret : Env → Comm → Env
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49 interpret env Skip = env
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50 interpret env Abort = env
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51 interpret env (PComm x) = x env
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52 interpret env (Seq comm comm1) = interpret (interpret env comm) comm1
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53 interpret env (If x then else) with x env
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54 ... | true = interpret env then
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55 ... | false = interpret env else
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56 interpret env (While x comm) with x env
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57 ... | true = interpret (interpret env comm) (While x comm)
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58 ... | false = env
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59
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60 test1 : Env
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61 test1 = interpret ( record { vari = 0 ; varn = 0 } ) (program 10)
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62
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63 eval-proof : vari test1 ≡ 10
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64 eval-proof = refl
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65
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66 tests : Env
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67 tests = interpret ( record { vari = 0 ; varn = 0 } ) (simple 10)
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68
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69
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70 empty-case : (env : Env) → (( λ e → true ) env ) ≡ true
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71 empty-case _ = refl
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72
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73
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74 Axiom : Cond -> PrimComm -> Cond -> Set
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75 Axiom pre comm post = ∀ (env : Env) → (pre env) ⇒ ( post (comm env)) ≡ true
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76
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77 Tautology : Cond -> Cond -> Set
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78 Tautology pre post = ∀ (env : Env) → (pre env) ⇒ (post env) ≡ true
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79
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80 _and_ : Cond -> Cond -> Cond
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81 x and y = λ env → x env ∧ y env
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82
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83 neg : Cond -> Cond
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84 neg x = λ env → not ( x env )
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85
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86 data HTProof : Cond -> Comm -> Cond -> Set where
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87 PrimRule : {bPre : Cond} -> {pcm : PrimComm} -> {bPost : Cond} ->
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88 (pr : Axiom bPre pcm bPost) ->
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89 HTProof bPre (PComm pcm) bPost
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90 SkipRule : (b : Cond) -> HTProof b Skip b
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91 AbortRule : (bPre : Cond) -> (bPost : Cond) ->
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92 HTProof bPre Abort bPost
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93 WeakeningRule : {bPre : Cond} -> {bPre' : Cond} -> {cm : Comm} ->
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94 {bPost' : Cond} -> {bPost : Cond} ->
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95 Tautology bPre bPre' ->
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96 HTProof bPre' cm bPost' ->
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97 Tautology bPost' bPost ->
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98 HTProof bPre cm bPost
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99 SeqRule : {bPre : Cond} -> {cm1 : Comm} -> {bMid : Cond} ->
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100 {cm2 : Comm} -> {bPost : Cond} ->
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101 HTProof bPre cm1 bMid ->
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102 HTProof bMid cm2 bPost ->
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103 HTProof bPre (Seq cm1 cm2) bPost
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104 IfRule : {cmThen : Comm} -> {cmElse : Comm} ->
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105 {bPre : Cond} -> {bPost : Cond} ->
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106 {b : Cond} ->
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107 HTProof (bPre and b) cmThen bPost ->
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108 HTProof (bPre and neg b) cmElse bPost ->
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109 HTProof bPre (If b cmThen cmElse) bPost
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110 WhileRule : {cm : Comm} -> {bInv : Cond} -> {b : Cond} ->
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111 HTProof (bInv and b) cm bInv ->
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112 HTProof bInv (While b cm) (bInv and neg b)
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113
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114 initCond : Cond
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115 initCond env = true
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116
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117 stmt1Cond : {c10 : ℕ} → Cond
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118 stmt1Cond {c10} env = Equal (varn env) c10
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119
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120 init-case : {c10 : ℕ} → (env : Env) → (( λ e → true ⇒ stmt1Cond {c10} e ) (record { varn = c10 ; vari = vari env }) ) ≡ true
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121 init-case {c10} _ = impl⇒ ( λ cond → ≡→Equal refl )
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122
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123 init-type : {c10 : ℕ} → Axiom (λ env → true) (λ env → record { varn = c10 ; vari = vari env }) (stmt1Cond {c10})
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124 init-type {c10} env = init-case env
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125
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126 stmt2Cond : {c10 : ℕ} → Cond
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127 stmt2Cond {c10} env = (Equal (varn env) c10) ∧ (Equal (vari env) 0)
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128
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129 whileInv : {c10 : ℕ} → Cond
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130 whileInv {c10} env = Equal ((varn env) + (vari env)) c10
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131
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132 whileInv' : {c10 : ℕ} → Cond
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133 whileInv'{c10} env = Equal ((varn env) + (vari env)) (suc c10) ∧ lt zero (varn env)
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134
14
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135 termCond : {c10 : ℕ} → Cond
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136 termCond {c10} env = Equal (vari env) c10
3
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137
15
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138 lemma1 : {c10 : ℕ} → Axiom (stmt1Cond {c10}) (λ env → record { varn = varn env ; vari = 0 }) (stmt2Cond {c10})
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parents: 14
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139 lemma1 {c10} env = impl⇒ ( λ cond → let open ≡-Reasoning in
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diff changeset
140 begin
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parents: 14
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141 (Equal (varn env) c10 ) ∧ true
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parents: 14
diff changeset
142 ≡⟨ ∧true ⟩
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parents: 14
diff changeset
143 Equal (varn env) c10
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parents: 14
diff changeset
144 ≡⟨ cond ⟩
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parents: 14
diff changeset
145 true
8d546766a9a8 Prim variable version done
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parents: 14
diff changeset
146 ∎ )
8d546766a9a8 Prim variable version done
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parents: 14
diff changeset
147
14
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148 proofs : (c10 : ℕ) → HTProof initCond (simple c10) (stmt2Cond {c10})
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parents: 10
diff changeset
149 proofs c10 =
15
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150 SeqRule {initCond} ( PrimRule (init-case {c10} ))
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parents: 14
diff changeset
151 $ PrimRule {stmt1Cond} {_} {stmt2Cond} (lemma1 {c10})
3
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152
8
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153 open import Data.Empty
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parents: 7
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154
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parents: 7
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155 open import Data.Nat.Properties
3
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156
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157 proof1 : (c10 : ℕ) → HTProof initCond (program c10 ) (termCond {c10})
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parents: 10
diff changeset
158 proof1 c10 =
15
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parents: 14
diff changeset
159 SeqRule {λ e → true} ( PrimRule (init-case {c10} ))
14
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160 $ SeqRule {λ e → Equal (varn e) c10} ( PrimRule lemma1 )
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parents: 10
diff changeset
161 $ WeakeningRule {λ e → (Equal (varn e) c10) ∧ (Equal (vari e) 0)} lemma2 (
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parents: 10
diff changeset
162 WhileRule {_} {λ e → Equal ((varn e) + (vari e)) c10}
7
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parents: 4
diff changeset
163 $ SeqRule (PrimRule {λ e → whileInv e ∧ lt zero (varn e) } lemma3 )
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parents: 4
diff changeset
164 $ PrimRule {whileInv'} {_} {whileInv} lemma4 ) lemma5
3
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165 where
14
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166 lemma21 : {env : Env } → {c10 : ℕ} → stmt2Cond env ≡ true → varn env ≡ c10
10
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parents: 8
diff changeset
167 lemma21 eq = Equal→≡ (∧-pi1 eq)
14
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parents: 10
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168 lemma22 : {env : Env } → {c10 : ℕ} → stmt2Cond {c10} env ≡ true → vari env ≡ 0
10
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parents: 8
diff changeset
169 lemma22 eq = Equal→≡ (∧-pi2 eq)
14
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parents: 10
diff changeset
170 lemma23 : {env : Env } → {c10 : ℕ} → stmt2Cond env ≡ true → varn env + vari env ≡ c10
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parents: 10
diff changeset
171 lemma23 {env} {c10} eq = let open ≡-Reasoning in
8
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diff changeset
172 begin
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173 varn env + vari env
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174 ≡⟨ cong ( \ x -> x + vari env ) (lemma21 eq ) ⟩
14
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diff changeset
175 c10 + vari env
15
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parents: 14
diff changeset
176 ≡⟨ cong ( \ x -> c10 + x) (lemma22 {env} {c10} eq ) ⟩
14
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diff changeset
177 c10 + 0
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178 ≡⟨ +-sym {c10} {0} ⟩
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diff changeset
179 0 + c10
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diff changeset
180 ≡⟨⟩
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diff changeset
181 c10
8
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diff changeset
182 ∎
14
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diff changeset
183 lemma2 : {c10 : ℕ} → Tautology stmt2Cond whileInv
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parents: 10
diff changeset
184 lemma2 {c10} env = bool-case (stmt2Cond env) (
8
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diff changeset
185 λ eq → let open ≡-Reasoning in
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diff changeset
186 begin
10
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187 (stmt2Cond env) ⇒ (whileInv env)
8
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diff changeset
188 ≡⟨⟩
14
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diff changeset
189 (stmt2Cond env) ⇒ ( Equal (varn env + vari env) c10 )
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parents: 10
diff changeset
190 ≡⟨ cong ( \ x -> (stmt2Cond {c10} env) ⇒ ( Equal x c10 ) ) ( lemma23 {env} eq ) ⟩
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parents: 10
diff changeset
191 (stmt2Cond env) ⇒ (Equal c10 c10)
15
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parents: 14
diff changeset
192 ≡⟨ cong ( \ x -> (stmt2Cond {c10} env) ⇒ x ) (≡→Equal refl ) ⟩
14
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diff changeset
193 (stmt2Cond {c10} env) ⇒ true
10
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parents: 8
diff changeset
194 ≡⟨ ⇒t ⟩
8
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diff changeset
195 true
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diff changeset
196 ∎
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parents: 7
diff changeset
197 ) (
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198 λ ne → let open ≡-Reasoning in
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diff changeset
199 begin
10
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diff changeset
200 (stmt2Cond env) ⇒ (whileInv env)
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diff changeset
201 ≡⟨ cong ( \ x -> x ⇒ (whileInv env) ) ne ⟩
14
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diff changeset
202 false ⇒ (whileInv {c10} env)
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diff changeset
203 ≡⟨ f⇒ {whileInv {c10} env} ⟩
8
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parents: 7
diff changeset
204 true
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parents: 7
diff changeset
205 ∎
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parents: 7
diff changeset
206 )
7
e7d6bdb6039d fix test1
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parents: 4
diff changeset
207 lemma3 : Axiom (λ e → whileInv e ∧ lt zero (varn e)) (λ env → record { varn = varn env ; vari = vari env + 1 }) whileInv'
10
bc819bdda374 proof completed
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parents: 8
diff changeset
208 lemma3 env = impl⇒ ( λ cond → let open ≡-Reasoning in
8
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parents: 7
diff changeset
209 begin
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parents: 7
diff changeset
210 whileInv' (record { varn = varn env ; vari = vari env + 1 })
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parents: 7
diff changeset
211 ≡⟨⟩
14
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parents: 10
diff changeset
212 Equal (varn env + (vari env + 1)) (suc c10) ∧ (lt 0 (varn env) )
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parents: 10
diff changeset
213 ≡⟨ cong ( λ z → Equal (varn env + (vari env + 1)) (suc c10) ∧ z ) (∧-pi2 cond ) ⟩
a622d1700a1b make 10 variable
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parents: 10
diff changeset
214 Equal (varn env + (vari env + 1)) (suc c10) ∧ true
10
bc819bdda374 proof completed
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parents: 8
diff changeset
215 ≡⟨ ∧true ⟩
14
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parents: 10
diff changeset
216 Equal (varn env + (vari env + 1)) (suc c10)
a622d1700a1b make 10 variable
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parents: 10
diff changeset
217 ≡⟨ cong ( \ x -> Equal x (suc c10) ) (sym (+-assoc (varn env) (vari env) 1)) ⟩
a622d1700a1b make 10 variable
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parents: 10
diff changeset
218 Equal ((varn env + vari env) + 1) (suc c10)
a622d1700a1b make 10 variable
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parents: 10
diff changeset
219 ≡⟨ cong ( \ x -> Equal x (suc c10) ) +1≡suc ⟩
a622d1700a1b make 10 variable
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parents: 10
diff changeset
220 Equal (suc (varn env + vari env)) (suc c10)
10
bc819bdda374 proof completed
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parents: 8
diff changeset
221 ≡⟨ sym Equal+1 ⟩
14
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parents: 10
diff changeset
222 Equal ((varn env + vari env) ) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
223 ≡⟨ ∧-pi1 cond ⟩
8
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parents: 7
diff changeset
224 true
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parents: 7
diff changeset
225 ∎ )
14
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parents: 10
diff changeset
226 lemma41 : (env : Env ) → {c10 : ℕ} → (varn env + vari env) ≡ (suc c10) → lt 0 (varn env) ≡ true → Equal ((varn env - 1) + vari env) c10 ≡ true
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parents: 10
diff changeset
227 lemma41 env {c10} c1 c2 = let open ≡-Reasoning in
10
bc819bdda374 proof completed
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parents: 8
diff changeset
228 begin
14
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parents: 10
diff changeset
229 Equal ((varn env - 1) + vari env) c10
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parents: 10
diff changeset
230 ≡⟨ cong ( λ z → Equal ((z - 1 ) + vari env ) c10 ) (sym (suc-predℕ=n c2) ) ⟩
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
231 Equal ((suc (predℕ {varn env} c2 ) - 1) + vari env) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
232 ≡⟨⟩
14
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
233 Equal ((predℕ {varn env} c2 ) + vari env) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
234 ≡⟨ Equal+1 ⟩
14
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
235 Equal ((suc (predℕ {varn env} c2 )) + vari env) (suc c10)
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
236 ≡⟨ cong ( λ z → Equal (z + vari env ) (suc c10) ) (suc-predℕ=n c2 ) ⟩
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
237 Equal (varn env + vari env) (suc c10)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
238 ≡⟨ cong ( λ z → (Equal z (suc c10) )) c1 ⟩
a622d1700a1b make 10 variable
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parents: 10
diff changeset
239 Equal (suc c10) (suc c10)
15
8d546766a9a8 Prim variable version done
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parents: 14
diff changeset
240 ≡⟨ ≡→Equal refl ⟩
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
241 true
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
242 ∎
14
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diff changeset
243 lemma4 : {c10 : ℕ} → Axiom whileInv' (λ env → record { varn = varn env - 1 ; vari = vari env }) whileInv
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parents: 10
diff changeset
244 lemma4 {c10} env = impl⇒ ( λ cond → let open ≡-Reasoning in
10
bc819bdda374 proof completed
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parents: 8
diff changeset
245 begin
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
246 whileInv (record { varn = varn env - 1 ; vari = vari env })
bc819bdda374 proof completed
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parents: 8
diff changeset
247 ≡⟨⟩
14
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parents: 10
diff changeset
248 Equal ((varn env - 1) + vari env) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
249 ≡⟨ lemma41 env (Equal→≡ (∧-pi1 cond)) (∧-pi2 cond) ⟩
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
250 true
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
251 ∎
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
252 )
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
253 lemma51 : (z : Env ) → neg (λ z → lt zero (varn z)) z ≡ true → varn z ≡ zero
17
b95a3cf9727c add Gears1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
254 lemma51 z cond with varn z
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
255 lemma51 z refl | zero = refl
b95a3cf9727c add Gears1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
256 lemma51 z () | suc x
14
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
257 lemma5 : {c10 : ℕ} → Tautology ((λ e → Equal (varn e + vari e) c10) and (neg (λ z → lt zero (varn z)))) termCond
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
258 lemma5 {c10} env = impl⇒ ( λ cond → let open ≡-Reasoning in
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
259 begin
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
260 termCond env
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
261 ≡⟨⟩
14
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
262 Equal (vari env) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
263 ≡⟨⟩
14
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
264 Equal (zero + vari env) c10
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
265 ≡⟨ cong ( λ z → Equal (z + vari env) c10 ) (sym ( lemma51 env ( ∧-pi2 cond ) )) ⟩
a622d1700a1b make 10 variable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
266 Equal (varn env + vari env) c10
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
267 ≡⟨ ∧-pi1 cond ⟩
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
268 true
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
269 ∎
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
270 )
3
6be8ee856666 add simple Hoare logic example
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
271
6be8ee856666 add simple Hoare logic example
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
272
10
bc819bdda374 proof completed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
273