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annotate whileTestGears.agda @ 63:222dd3869ab0
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 22 Dec 2019 09:15:44 +0900 |
parents | bfe7d83cf9ba |
children | 87e125b11999 |
rev | line source |
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4 | 1 module whileTestGears where |
2 | |
3 open import Function | |
4 open import Data.Nat | |
34 | 5 open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_) |
62 | 6 open import Data.Product |
4 | 7 open import Level renaming ( suc to succ ; zero to Zero ) |
8 open import Relation.Nullary using (¬_; Dec; yes; no) | |
9 open import Relation.Binary.PropositionalEquality | |
62 | 10 open import Agda.Builtin.Unit |
4 | 11 |
10 | 12 open import utilities |
13 open _/\_ | |
4 | 14 |
42 | 15 record Env : Set (succ Zero) where |
6 | 16 field |
17 varn : ℕ | |
18 vari : ℕ | |
42 | 19 open Env |
6 | 20 |
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21 whileTest : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Env → t) → t |
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22 whileTest c10 next = next (record {varn = c10 ; vari = 0 } ) |
4 | 23 |
24 {-# TERMINATING #-} | |
33 | 25 whileLoop : {l : Level} {t : Set l} → Env → (Code : Env → t) → t |
4 | 26 whileLoop env next with lt 0 (varn env) |
27 whileLoop env next | false = next env | |
28 whileLoop env next | true = | |
42 | 29 whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next |
4 | 30 |
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31 test1 : Env |
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32 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 )) |
4 | 33 |
34 | |
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35 proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 )) |
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36 proof1 = refl |
4 | 37 |
16 | 38 -- ↓PostCondition |
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39 whileTest' : {l : Level} {t : Set l} → {c10 : ℕ } → (Code : (env : Env ) → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t |
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40 whileTest' {_} {_} {c10} next = next env proof2 |
4 | 41 where |
42 | 42 env : Env |
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43 env = record {vari = 0 ; varn = c10 } |
16 | 44 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition |
4 | 45 proof2 = record {pi1 = refl ; pi2 = refl} |
11 | 46 |
47 open import Data.Empty | |
48 open import Data.Nat.Properties | |
49 | |
50 | |
16 | 51 {-# TERMINATING #-} -- ↓PreCondition(Invaliant) |
42 | 52 whileLoop' : {l : Level} {t : Set l} → (env : Env ) → {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) → (Code : Env → t) → t |
9 | 53 whileLoop' env proof next with ( suc zero ≤? (varn env) ) |
54 whileLoop' env proof next | no p = next env | |
14 | 55 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next |
4 | 56 where |
42 | 57 env1 = record env {varn = (varn env) - 1 ; vari = (vari env) + 1} |
11 | 58 1<0 : 1 ≤ zero → ⊥ |
59 1<0 () | |
14 | 60 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10 |
47 | 61 proof3 (s≤s lt) with varn env |
62 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p) | |
63 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in | |
64 begin | |
65 n' + (vari env + 1) | |
66 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ | |
67 n' + (1 + vari env ) | |
68 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ | |
69 (n' + 1) + vari env | |
70 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ | |
71 (suc n' ) + vari env | |
72 ≡⟨⟩ | |
73 varn env + vari env | |
74 ≡⟨ proof ⟩ | |
75 c10 | |
76 ∎ | |
6 | 77 |
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78 -- Condition to Invariant |
42 | 79 conversion1 : {l : Level} {t : Set l } → (env : Env ) → {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) |
80 → (Code : (env1 : Env ) → (varn env1 + vari env1 ≡ c10) → t) → t | |
14 | 81 conversion1 env {c10} p1 next = next env proof4 |
6 | 82 where |
14 | 83 proof4 : varn env + vari env ≡ c10 |
6 | 84 proof4 = let open ≡-Reasoning in |
85 begin | |
86 varn env + vari env | |
87 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩ | |
14 | 88 c10 + vari env |
89 ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩ | |
90 c10 + 0 | |
91 ≡⟨ +-sym {c10} {0} ⟩ | |
92 c10 | |
6 | 93 ∎ |
4 | 94 |
6 | 95 |
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96 proofGears : {c10 : ℕ } → Set |
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97 proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 )))) |
9 | 98 |
51 | 99 |
49 | 100 -- proofGearsMeta : {c10 : ℕ } → proofGears {c10} |
101 -- proofGearsMeta {c10} = {!!} -- net yet done | |
43 | 102 |
41 | 103 -- |
42 | 104 -- openended Env c <=> Context |
41 | 105 -- |
106 | |
107 open import Relation.Nullary | |
108 open import Relation.Binary | |
109 | |
53 | 110 record Envc : Set (succ Zero) where |
111 field | |
112 c10 : ℕ | |
113 varn : ℕ | |
114 vari : ℕ | |
115 open Envc | |
49 | 116 |
53 | 117 whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Envc → t) → t |
118 whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } ) | |
119 | |
120 whileLoopP : {l : Level} {t : Set l} → Envc → (next : Envc → t) → (exit : Envc → t) → t | |
49 | 121 whileLoopP env next exit with <-cmp 0 (varn env) |
122 whileLoopP env next exit | tri≈ ¬a b ¬c = exit env | |
123 whileLoopP env next exit | tri< a ¬b ¬c = | |
124 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) | |
125 | |
126 {-# TERMINATING #-} | |
53 | 127 loopP : {l : Level} {t : Set l} → Envc → (exit : Envc → t) → t |
49 | 128 loopP env exit = whileLoopP env (λ env → loopP env exit ) exit |
129 | |
53 | 130 whileTestPCall : (c10 : ℕ ) → Envc |
131 whileTestPCall c10 = whileTestP {_} {_} c10 (λ env → loopP env (λ env → env)) | |
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132 |
53 | 133 data whileTestState : Set where |
134 s1 : whileTestState | |
135 s2 : whileTestState | |
136 sf : whileTestState | |
49 | 137 |
53 | 138 whileTestStateP : whileTestState → Envc → Set |
139 whileTestStateP s1 env = (vari env ≡ 0) /\ (varn env ≡ c10 env) | |
140 whileTestStateP s2 env = (varn env + vari env ≡ c10 env) | |
141 whileTestStateP sf env = (vari env ≡ c10 env) | |
50 | 142 |
53 | 143 whileTestPwP : {l : Level} {t : Set l} → (c10 : ℕ) → ((env : Envc ) → whileTestStateP s1 env → t) → t |
144 whileTestPwP c10 next = next env record { pi1 = refl ; pi2 = refl } where | |
145 env : Envc | |
146 env = whileTestP c10 ( λ env → env ) | |
50 | 147 |
56 | 148 whileLoopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env |
53 | 149 → (next : (env : Envc ) → whileTestStateP s2 env → t) |
150 → (exit : (env : Envc ) → whileTestStateP sf env → t) → t | |
54 | 151 whileLoopPwP env s next exit with <-cmp 0 (varn env) |
55 | 152 whileLoopPwP env s next exit | tri≈ ¬a b ¬c = exit env (lem (sym b) s) |
153 where | |
154 lem : (varn env ≡ 0) → (varn env + vari env ≡ c10 env) → vari env ≡ c10 env | |
155 lem p1 p2 rewrite p1 = p2 | |
156 | |
56 | 157 whileLoopPwP env s next exit | tri< a ¬b ¬c = next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) (proof5 a) |
158 where | |
159 1<0 : 1 ≤ zero → ⊥ | |
160 1<0 () | |
161 proof5 : (suc zero ≤ (varn env)) → (varn env - 1) + (vari env + 1) ≡ c10 env | |
162 proof5 (s≤s lt) with varn env | |
163 proof5 (s≤s z≤n) | zero = ⊥-elim (1<0 a) | |
164 proof5 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in | |
165 begin | |
166 n' + (vari env + 1) | |
167 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ | |
168 n' + (1 + vari env ) | |
169 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ | |
170 (n' + 1) + vari env | |
171 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ | |
172 (suc n' ) + vari env | |
173 ≡⟨⟩ | |
174 varn env + vari env | |
175 ≡⟨ s ⟩ | |
176 c10 env | |
177 ∎ | |
51 | 178 |
62 | 179 |
180 -- induction にする | |
53 | 181 {-# TERMINATING #-} |
54 | 182 loopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t |
183 loopPwP env s exit = whileLoopPwP env s (λ env s → loopPwP env s exit ) exit | |
51 | 184 |
62 | 185 -- wP を Env のRel にする Env → Env → Set にしちゃう |
54 | 186 whileTestPCallwP : (c : ℕ ) → Set |
187 whileTestPCallwP c = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → vari env ≡ c ) ) where | |
188 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
55 | 189 conv e record { pi1 = refl ; pi2 = refl } = +zero |
190 | |
59 | 191 |
192 conv1 : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
193 conv1 e record { pi1 = refl ; pi2 = refl } = +zero | |
194 | |
195 -- = whileTestPwP (suc c) (λ env s → loopPwP env (conv1 env s) (λ env₁ s₁ → {!!})) | |
196 | |
61 | 197 |
62 | 198 data GComm : Set (succ Zero) where |
199 Skip : GComm | |
200 Abort : GComm | |
201 PComm : Set → GComm | |
202 -- Seq : GComm → GComm → GComm | |
203 -- If : whileTestState → GComm → GComm → GComm | |
204 while : whileTestState → GComm → GComm | |
61 | 205 |
62 | 206 gearsSem : {l : Level} {t : Set l} → {c10 : ℕ} → Envc → Envc → (Envc → (Envc → t) → t) → Set |
207 gearsSem pre post = {!!} | |
208 | |
209 unionInf : ∀ {l} -> (ℕ -> Rel Set l) -> Rel Set l | |
210 unionInf f a b = ∃ (λ (n : ℕ) → f n a b) | |
211 | |
212 comp : ∀ {l} → Rel Set l → Rel Set l → Rel Set (succ Zero Level.⊔ l) | |
213 comp r1 r2 a b = ∃ (λ (a' : Set) → r1 a a' × r2 a' b) | |
214 | |
215 -- repeat : ℕ -> rel set zero -> rel set zero | |
216 -- repeat ℕ.zero r = λ x x₁ → ⊤ | |
217 -- repeat (ℕ.suc m) r = comp (repeat m r) r | |
218 | |
219 GSemComm : {l : Level} {t : Set l} → GComm → Rel whileTestState (Zero) | |
220 GSemComm Skip = λ x x₁ → ⊤ | |
221 GSemComm Abort = λ x x₁ → ⊥ | |
222 GSemComm (PComm x) = λ x₁ x₂ → x | |
223 -- GSemComm (Seq con con₁ con₃) = λ x₁ x₂ → {!!} | |
224 -- GSemComm (If x con con₁) = {!!} | |
225 GSemComm (while x con) = λ x₁ x₂ → unionInf {Zero} (λ (n : ℕ) → {!!}) {!!} {!!} | |
226 | |
227 ProofConnect : {l : Level} {t : Set l} | |
228 → (pr1 : Envc → Set → Set) | |
229 → (Envc → Set → (Envc → Set → t)) | |
230 → (Envc → Set → Set) | |
231 ProofConnect prev f env post = {!!} -- with f env ({!!}) {!!} | |
60 | 232 |
233 Proof2 : (env : Envc) → (vari env ≡ c10 env) → vari env ≡ c10 env | |
234 Proof2 _ refl = refl | |
235 | |
236 | |
61 | 237 -- Proof1 : (env : Envc) → (s : varn env + vari env ≡ c10 env) → ((env : Envc) → (vari env ≡ c10 env) → vari env ≡ c10 env) → vari env ≡ c10 env |
60 | 238 Proof1 : (env : Envc) → (s : varn env + vari env ≡ c10 env) → loopPwP env s ( λ env s → vari env ≡ c10 env ) |
61 | 239 Proof1 env s = {!!} |
60 | 240 |
55 | 241 Proof : (c : ℕ ) → whileTestPCallwP c |
61 | 242 Proof c = {!!} |