Mercurial > hg > Members > ryokka > HoareLogic
annotate whileTestGears.agda @ 59:5c2cdcee9971
restore bad proof
author | ryokka |
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date | Sat, 21 Dec 2019 17:49:15 +0900 |
parents | 7523d5cd670b |
children | ad83c2d5e869 |
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 ( _≟_ ; _≤?_ ; _≤_ ; _<_) |
4 | 6 open import Level renaming ( suc to succ ; zero to Zero ) |
7 open import Relation.Nullary using (¬_; Dec; yes; no) | |
8 open import Relation.Binary.PropositionalEquality | |
9 | |
10 | 10 open import utilities |
11 open _/\_ | |
4 | 12 |
42 | 13 record Env : Set (succ Zero) where |
6 | 14 field |
15 varn : ℕ | |
16 vari : ℕ | |
42 | 17 open Env |
6 | 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 } ) |
4 | 21 |
22 {-# TERMINATING #-} | |
33 | 23 whileLoop : {l : Level} {t : Set l} → Env → (Code : Env → t) → t |
4 | 24 whileLoop env next with lt 0 (varn env) |
25 whileLoop env next | false = next env | |
26 whileLoop env next | true = | |
42 | 27 whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next |
4 | 28 |
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29 test1 : Env |
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30 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 )) |
4 | 31 |
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 |
4 | 35 |
16 | 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 |
4 | 39 where |
42 | 40 env : Env |
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41 env = record {vari = 0 ; varn = c10 } |
16 | 42 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition |
4 | 43 proof2 = record {pi1 = refl ; pi2 = refl} |
11 | 44 |
45 open import Data.Empty | |
46 open import Data.Nat.Properties | |
47 | |
48 | |
16 | 49 {-# TERMINATING #-} -- ↓PreCondition(Invaliant) |
42 | 50 whileLoop' : {l : Level} {t : Set l} → (env : Env ) → {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) → (Code : Env → t) → t |
9 | 51 whileLoop' env proof next with ( suc zero ≤? (varn env) ) |
52 whileLoop' env proof next | no p = next env | |
14 | 53 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next |
4 | 54 where |
42 | 55 env1 = record env {varn = (varn env) - 1 ; vari = (vari env) + 1} |
11 | 56 1<0 : 1 ≤ zero → ⊥ |
57 1<0 () | |
14 | 58 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10 |
47 | 59 proof3 (s≤s lt) with varn env |
60 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p) | |
61 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in | |
62 begin | |
63 n' + (vari env + 1) | |
64 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ | |
65 n' + (1 + vari env ) | |
66 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ | |
67 (n' + 1) + vari env | |
68 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ | |
69 (suc n' ) + vari env | |
70 ≡⟨⟩ | |
71 varn env + vari env | |
72 ≡⟨ proof ⟩ | |
73 c10 | |
74 ∎ | |
6 | 75 |
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parents:
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76 -- Condition to Invariant |
42 | 77 conversion1 : {l : Level} {t : Set l } → (env : Env ) → {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) |
78 → (Code : (env1 : Env ) → (varn env1 + vari env1 ≡ c10) → t) → t | |
14 | 79 conversion1 env {c10} p1 next = next env proof4 |
6 | 80 where |
14 | 81 proof4 : varn env + vari env ≡ c10 |
6 | 82 proof4 = let open ≡-Reasoning in |
83 begin | |
84 varn env + vari env | |
85 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩ | |
14 | 86 c10 + vari env |
87 ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩ | |
88 c10 + 0 | |
89 ≡⟨ +-sym {c10} {0} ⟩ | |
90 c10 | |
6 | 91 ∎ |
4 | 92 |
6 | 93 |
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parents:
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94 proofGears : {c10 : ℕ } → Set |
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parents:
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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 )))) |
9 | 96 |
51 | 97 |
49 | 98 -- proofGearsMeta : {c10 : ℕ } → proofGears {c10} |
99 -- proofGearsMeta {c10} = {!!} -- net yet done | |
43 | 100 |
41 | 101 -- |
42 | 102 -- openended Env c <=> Context |
41 | 103 -- |
104 | |
105 open import Relation.Nullary | |
106 open import Relation.Binary | |
107 | |
53 | 108 record Envc : Set (succ Zero) where |
109 field | |
110 c10 : ℕ | |
111 varn : ℕ | |
112 vari : ℕ | |
113 open Envc | |
49 | 114 |
53 | 115 whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Envc → t) → t |
116 whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } ) | |
117 | |
118 whileLoopP : {l : Level} {t : Set l} → Envc → (next : Envc → t) → (exit : Envc → t) → t | |
49 | 119 whileLoopP env next exit with <-cmp 0 (varn env) |
120 whileLoopP env next exit | tri≈ ¬a b ¬c = exit env | |
121 whileLoopP env next exit | tri< a ¬b ¬c = | |
122 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) | |
123 | |
124 {-# TERMINATING #-} | |
53 | 125 loopP : {l : Level} {t : Set l} → Envc → (exit : Envc → t) → t |
49 | 126 loopP env exit = whileLoopP env (λ env → loopP env exit ) exit |
127 | |
53 | 128 whileTestPCall : (c10 : ℕ ) → Envc |
129 whileTestPCall c10 = whileTestP {_} {_} c10 (λ env → loopP env (λ env → env)) | |
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130 |
53 | 131 data whileTestState : Set where |
132 s1 : whileTestState | |
133 s2 : whileTestState | |
134 sf : whileTestState | |
49 | 135 |
53 | 136 whileTestStateP : whileTestState → Envc → Set |
137 whileTestStateP s1 env = (vari env ≡ 0) /\ (varn env ≡ c10 env) | |
138 whileTestStateP s2 env = (varn env + vari env ≡ c10 env) | |
139 whileTestStateP sf env = (vari env ≡ c10 env) | |
50 | 140 |
53 | 141 whileTestPwP : {l : Level} {t : Set l} → (c10 : ℕ) → ((env : Envc ) → whileTestStateP s1 env → t) → t |
142 whileTestPwP c10 next = next env record { pi1 = refl ; pi2 = refl } where | |
143 env : Envc | |
144 env = whileTestP c10 ( λ env → env ) | |
50 | 145 |
56 | 146 whileLoopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env |
53 | 147 → (next : (env : Envc ) → whileTestStateP s2 env → t) |
148 → (exit : (env : Envc ) → whileTestStateP sf env → t) → t | |
54 | 149 whileLoopPwP env s next exit with <-cmp 0 (varn env) |
55 | 150 whileLoopPwP env s next exit | tri≈ ¬a b ¬c = exit env (lem (sym b) s) |
151 where | |
152 lem : (varn env ≡ 0) → (varn env + vari env ≡ c10 env) → vari env ≡ c10 env | |
153 lem p1 p2 rewrite p1 = p2 | |
154 | |
56 | 155 whileLoopPwP env s next exit | tri< a ¬b ¬c = next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) (proof5 a) |
156 where | |
157 1<0 : 1 ≤ zero → ⊥ | |
158 1<0 () | |
159 proof5 : (suc zero ≤ (varn env)) → (varn env - 1) + (vari env + 1) ≡ c10 env | |
160 proof5 (s≤s lt) with varn env | |
161 proof5 (s≤s z≤n) | zero = ⊥-elim (1<0 a) | |
162 proof5 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in | |
163 begin | |
164 n' + (vari env + 1) | |
165 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ | |
166 n' + (1 + vari env ) | |
167 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ | |
168 (n' + 1) + vari env | |
169 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ | |
170 (suc n' ) + vari env | |
171 ≡⟨⟩ | |
172 varn env + vari env | |
173 ≡⟨ s ⟩ | |
174 c10 env | |
175 ∎ | |
51 | 176 |
53 | 177 {-# TERMINATING #-} |
54 | 178 loopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t |
179 loopPwP env s exit = whileLoopPwP env s (λ env s → loopPwP env s exit ) exit | |
51 | 180 |
54 | 181 whileTestPCallwP : (c : ℕ ) → Set |
182 whileTestPCallwP c = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → vari env ≡ c ) ) where | |
183 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
55 | 184 conv e record { pi1 = refl ; pi2 = refl } = +zero |
185 | |
59 | 186 |
187 conv1 : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
188 conv1 e record { pi1 = refl ; pi2 = refl } = +zero | |
189 | |
190 -- = whileTestPwP (suc c) (λ env s → loopPwP env (conv1 env s) (λ env₁ s₁ → {!!})) | |
191 | |
57 | 192 {-# TERMINATING #-} |
55 | 193 Proof : (c : ℕ ) → whileTestPCallwP c |
59 | 194 Proof zero = whileTestPwP {_} {_} zero ( λ env s → loopPwP env (conv env s) ( λ env s → refl) ) |
195 where | |
196 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
197 conv e record { pi1 = refl ; pi2 = refl } = +zero | |
198 Proof (suc c) = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → lem )) | |
199 where | |
200 lem : whileLoopPwP (record { c10 = suc c ; varn = c ; vari = 0 + 1 }) ({!!}) | |
201 (λ env s → loopPwP env s (λ env₁ s₁ → vari env₁ ≡ suc c)) (λ env s3 → {!!}) | |
202 lem = {!!} | |
203 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
204 conv e record { pi1 = refl ; pi2 = refl } = +zero | |
205 | |
206 | |
207 {-- | |
208 -- whileLoopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env | |
209 -- → (next : (env : Envc ) → whileTestStateP s2 env → t) | |
210 -- → (exit : (env : Envc ) → whileTestStateP sf env → t) → t | |
211 | |
212 next : (whileTestGears.proof5 | |
213 (record { c10 = suc c ; varn = suc c ; vari = 0 }) | |
214 (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt) | |
215 (λ x → | |
216 Relation.Nullary.Reflects.invert (ofⁿ (λ ())) (≡⇒≡ᵇ 0 (suc c) x)) | |
217 (<⇒≯ (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt)) | |
218 (whileTestGears.conv (suc c) (whileTestP (suc c) (λ env₁ → env₁)) | |
219 (record { pi1 = refl ; pi2 = refl })) | |
220 (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c)) | |
221 (λ env₁ s₁ → vari env₁ ≡ suc c) | |
222 (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt)) | |
223 | |
224 exit : (whileTestGears.conv (suc c) (whileTestP (suc c) (λ env₁ → env₁)) | |
225 (record { pi1 = refl ; pi2 = refl })) | |
226 (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c)) | |
227 (λ env₁ s₁ → vari env₁ ≡ suc c) | |
228 (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt)) | |
229 (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c)) | |
230 (λ env₁ s₁ → vari env₁ ≡ suc c) | |
231 | |
232 | (<-cmp 0 c | |
233 | Relation.Nullary.Decidable.Core.map′ (≡ᵇ⇒≡ 0 c) (≡⇒≡ᵇ 0 c) | |
234 (Data.Bool.Properties.T? (0 ≡ᵇ c)) | |
235 | Data.Bool.Properties.T? (0 <ᵇ c)) | |
236 | |
237 --} |