Mercurial > hg > Members > ryokka > HoareLogic
annotate whileTestGears.agda @ 50:2edb44c5bf52
add s1~3, proofs
| author | ryokka |
|---|---|
| date | Wed, 18 Dec 2019 20:08:58 +0900 |
| parents | 91d6d8097a39 |
| children | 3f4f93ac841d |
| rev | line source |
|---|---|
| 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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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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94 proofGears : {c10 : ℕ } → Set |
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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 |
| 49 | 97 -- proofGearsMeta : {c10 : ℕ } → proofGears {c10} |
| 98 -- proofGearsMeta {c10} = {!!} -- net yet done | |
| 43 | 99 |
| 41 | 100 -- |
| 42 | 101 -- openended Env c <=> Context |
| 41 | 102 -- |
| 103 | |
| 104 open import Relation.Nullary | |
| 105 open import Relation.Binary | |
| 106 | |
| 49 | 107 whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Env → t) → t |
| 108 whileTestP c10 next = next (record {varn = c10 ; vari = 0 } ) | |
| 109 | |
| 110 whileLoopP : {l : Level} {t : Set l} → Env → (next : Env → t) → (exit : Env → t) → t | |
| 111 whileLoopP env next exit with <-cmp 0 (varn env) | |
| 112 whileLoopP env next exit | tri≈ ¬a b ¬c = exit env | |
| 113 whileLoopP env next exit | tri< a ¬b ¬c = | |
| 114 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) | |
| 115 | |
| 116 {-# TERMINATING #-} | |
| 117 loopP : {l : Level} {t : Set l} → Env → (exit : Env → t) → t | |
| 118 loopP env exit = whileLoopP env (λ env → loopP env exit ) exit | |
| 119 | |
| 120 whileTestPCall : {c10 : ℕ } → Set | |
| 121 whileTestPCall {c10} = whileTestP {_} {_} c10 (λ env → loopP env (λ env → ( vari env ≡ c10 ))) | |
| 122 | |
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8bf82026e4fe
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
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changeset
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123 data whileTestStateP (c10 i n : ℕ ) : Set where |
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parents:
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124 pstate1 : (i ≡ 0) /\ (n ≡ c10) → whileTestStateP c10 i n -- n ≡ c10 |
| 49 | 125 pstate2 : (0 < n) → (n < c10) → (n + i ≡ c10) → whileTestStateP c10 i n -- 0 < n < c10 |
| 126 pfinstate : (n ≡ 0 ) → (i ≡ c10 ) → whileTestStateP c10 i n -- n ≡ 0 | |
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127 |
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128 record EnvP : Set (succ Zero) where |
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129 field |
| 49 | 130 env : Env |
| 131 c10 : ℕ | |
| 132 cx : whileTestStateP c10 (vari env) (varn env) | |
| 133 | |
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parents:
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134 open EnvP |
| 33 | 135 |
| 50 | 136 s1 : (c10 : ℕ) → EnvP |
| 137 s1 c10 = record {env = record {vari = 0 ; varn = c10} ; c10 = c10 ; cx = pstate1 (record {pi1 = refl ; pi2 = refl})} | |
| 138 | |
| 139 s2 : (e : EnvP) → varn (env e) > 0 → varn (env e) < c10 e → EnvP | |
| 140 s2 e n>0 n<c10 with <-cmp 0 (varn (env e)) | |
| 141 s2 e n>0 n<c10 | tri< a ¬b ¬c = record { env = record { varn = varn (env e) - 1 ; vari = vari (env e) + 1 } ; c10 = c10 e ; cx = pstate2 {!!} {!!} {!!} } | |
| 142 s2 e n>0 n<c10 | tri≈ ¬a b ¬c = record { env = record { varn = varn (env e) - 1 ; vari = vari (env e) + 1 } ; c10 = c10 e ; cx = pfinstate {!!} {!!} } | |
| 143 | |
| 144 s3 : (e : EnvP) → varn (env e) ≡ 0 → vari (env e) ≡ c10 e | |
| 145 s3 record { env = record { varn = .0 ; vari = vari₁ } ; c10 = c11 ; cx = cx₁ } refl = {!!} | |
| 146 | |
| 147 proofs : (c : ℕ) (e : EnvP) → vari (env e) ≡ c10 e | |
| 148 proofs c e = s3 (s2 (s1 {!!}) {!!} {!!}) {!!} | |
| 149 | |
| 150 | |
| 49 | 151 whileTestPwithProof : {l : Level} {t : Set l} → (c10 : ℕ ) → (next : (e : EnvP ) → t) → t |
| 152 whileTestPwithProof {l} {t} c10 next = next record { env = env1 ; c10 = c10 ; cx = cx1 } where | |
| 153 env1 : Env | |
| 154 env1 = whileTestP c10 ( λ e → e ) | |
| 155 cx1 : whileTestStateP c10 (vari env1) (varn env1) | |
| 156 cx1 = pstate1 record { pi1 = refl ; pi2 = refl } | |
| 27 | 157 |
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parents:
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158 {-# TERMINATING #-} |
| 49 | 159 loopPwithProof : {l : Level} {t : Set l} → (e : EnvP ) → (exit : (e : EnvP ) → t ) → t |
| 160 loopPwithProof e exit = whileLoopP (env e) (λ e1 → loopPwithProof record e { env = e1 ; cx = {!!} } exit ) (λ env → exit {!!} ) where | |
| 161 lemma : {!!} | |
| 47 | 162 lemma = {!!} |
| 27 | 163 |
| 49 | 164 ConvP : (e : EnvP) → EnvP |
| 47 | 165 ConvP = {!!} |
| 166 | |
| 167 whileTestPProof : {c : ℕ } → Set | |
| 168 whileTestPProof {c} = whileTestPwithProof c | |
| 49 | 169 $ λ e → loopPwithProof e (λ e eq → vari (env e) ≡ c10 e ) (ConvP e ) |
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parents:
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170 |
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parents:
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171 whileTestPProofMeta : {c10 : ℕ } → whileTestPProof {c10} |
| 48 | 172 whileTestPProofMeta {c10} = {!!} |
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173 |
| 35 | 174 |