Mercurial > hg > Members > ryokka > HoareLogic
annotate whileTestGears.agda @ 54:3adf50622101
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 20 Dec 2019 16:13:52 +0900 |
| parents | 03235251b3a7 |
| children | 1be7bb658cf0 |
| 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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parents:
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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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parents:
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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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diff
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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:
43
diff
changeset
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94 proofGears : {c10 : ℕ } → Set |
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parents:
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diff
changeset
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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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parents:
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diff
changeset
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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 |
| 54 | 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) |
| 150 whileLoopPwP env s next exit | tri≈ ¬a b ¬c = exit env {!!} | |
| 151 whileLoopPwP env s next exit | tri< a ¬b ¬c = | |
| 53 | 152 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) {!!} |
| 51 | 153 |
| 53 | 154 {-# TERMINATING #-} |
| 54 | 155 loopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t |
| 156 loopPwP env s exit = whileLoopPwP env s (λ env s → loopPwP env s exit ) exit | |
| 51 | 157 |
| 54 | 158 whileTestPCallwP : (c : ℕ ) → Set |
| 159 whileTestPCallwP c = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → vari env ≡ c ) ) where | |
| 160 conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env | |
| 161 conv = {!!} | |
| 51 | 162 |
| 50 | 163 |
| 164 | |
| 27 | 165 |
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parents:
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166 |
| 35 | 167 |