Mercurial > hg > Members > ryokka > HoareLogic
annotate whileTestGears.agda @ 39:2eb30c8e5a20
whileLoopStep
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 14 Dec 2019 09:25:45 +0900 |
| parents | 7049fbaf5e18 |
| children | a9cac7960e81 |
| 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 |
| 6 | 13 record Env : Set where |
| 14 field | |
| 15 varn : ℕ | |
| 16 vari : ℕ | |
| 17 open Env | |
| 18 | |
| 33 | 19 whileTest : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Env → t) → t |
| 14 | 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 = | |
| 27 whileLoop (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next | |
| 28 | |
| 29 test1 : Env | |
| 14 | 30 test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 )) |
| 4 | 31 |
| 32 | |
| 14 | 33 proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 )) |
| 4 | 34 proof1 = refl |
| 35 | |
| 16 | 36 -- ↓PostCondition |
| 33 | 37 whileTest' : {l : Level} {t : Set l} → {c10 : ℕ } → (Code : (env : Env) → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t |
| 14 | 38 whileTest' {_} {_} {c10} next = next env proof2 |
| 4 | 39 where |
| 40 env : Env | |
| 14 | 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) |
| 33 | 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 |
| 55 env1 = record {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 |
| 9 | 59 proof3 (s≤s lt) with varn env |
| 11 | 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) ) ⟩ | |
| 13 | 67 (n' + 1) + vari env |
| 11 | 68 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ |
| 69 (suc n' ) + vari env | |
| 70 ≡⟨⟩ | |
| 71 varn env + vari env | |
| 72 ≡⟨ proof ⟩ | |
| 14 | 73 c10 |
| 11 | 74 ∎ |
| 6 | 75 |
| 16 | 76 -- Condition to Invaliant |
| 33 | 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 |
| 14 | 94 proofGears : {c10 : ℕ } → Set |
| 95 proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 )))) | |
| 9 | 96 |
| 33 | 97 data whileTestState (c10 : ℕ ) (env : Env ) : Set where |
| 98 error : whileTestState c10 env | |
| 99 state1 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → whileTestState c10 env | |
| 100 state2 : (varn env + vari env ≡ c10) → whileTestState c10 env | |
| 101 finstate : ((vari env) ≡ c10 ) → whileTestState c10 env | |
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102 |
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103 record Context : Set where |
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104 field |
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105 c10 : ℕ |
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106 whileDG : Env |
| 33 | 107 whileCond : whileTestState c10 whileDG |
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108 |
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109 open Context |
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110 |
| 33 | 111 whileTestContext : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
| 112 whileTestContext cxt next = next (record cxt { whileDG = record (whileDG cxt) {varn = c10 cxt ; vari = 0} ; whileCond = {!!} } ) | |
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113 |
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114 {-# TERMINATING #-} |
| 33 | 115 whileLoopContext : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
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116 whileLoopContext cxt next with lt 0 (varn (whileDG cxt) ) |
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117 whileLoopContext cxt next | false = next cxt |
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118 whileLoopContext cxt next | true = |
| 38 | 119 whileLoopContext (record cxt { whileDG = record {varn = (varn (whileDG cxt)) - 1 ; vari = (vari (whileDG cxt)) + 1} ; whileCond = state2 (proof cxt) } ) next |
| 120 where | |
| 121 proof : (cxt : Context) → (varn (whileDG cxt) - 1) + (vari (whileDG cxt) + 1) ≡ c10 cxt | |
| 122 proof cxt = {!!} | |
| 33 | 123 |
| 124 open import Relation.Nullary | |
| 125 open import Relation.Binary | |
| 126 | |
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127 |
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128 {-# TERMINATING #-} |
| 39 | 129 whileLoopStep : {l : Level} {t : Set l} → Env → (next : (e : Env ) → 1 ≤ varn e → t) (exit : (e : Env) → 0 ≡ varn e → t) → t |
| 33 | 130 whileLoopStep env next exit with <-cmp 0 (varn env) |
| 131 whileLoopStep env next exit | tri≈ _ eq _ = exit env eq | |
| 39 | 132 whileLoopStep env next exit | tri< gt ¬eq _ = next env gt |
| 133 whileLoopStep env next exit | tri> _ _ c = ⊥-elim (m<n⇒n≢0 {varn env} {0} c refl) | |
| 27 | 134 |
| 33 | 135 whileTestProof : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
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136 whileTestProof cxt next = next record cxt { whileDG = out ; whileCond = init } where |
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137 out : Env |
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138 out = whileTest (c10 cxt) ( λ e → e ) |
| 33 | 139 init : whileTestState (c10 cxt) out |
| 140 init = state1 record { pi1 = refl ; pi2 = refl } | |
| 27 | 141 |
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142 {-# TERMINATING #-} |
| 33 | 143 whileLoopProof : {l : Level} {t : Set l} → Context → (Code : Context → t) (Code : Context → t) → t |
| 31 | 144 whileLoopProof cxt next exit = whileLoopStep (whileDG cxt) |
| 33 | 145 ( λ env lt → next record cxt { whileDG = env ; whileCond = {!!} } ) |
| 146 ( λ env eq → exit record cxt { whileDG = env ; whileCond = exitCond env eq } ) where | |
| 147 proof5 : (e : Env ) → varn e + vari e ≡ c10 cxt → 0 ≡ varn e → vari e ≡ c10 cxt | |
| 34 | 148 proof5 record { varn = .0 ; vari = vari } refl refl = refl |
| 33 | 149 exitCond : (e : Env ) → 0 ≡ varn e → whileTestState (c10 cxt) e |
| 150 exitCond nenv eq1 with whileCond cxt | inspect whileDG cxt | |
| 151 ... | state2 cond | record { eq = eq2 } = finstate ( proof5 nenv {!!} eq1 ) | |
| 152 ... | _ | _ = error | |
| 27 | 153 |
| 33 | 154 whileConvProof : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
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155 whileConvProof cxt next = next record cxt { whileCond = postCond } where |
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156 proof4 : (e : Env ) → (vari e ≡ 0) /\ (varn e ≡ c10 cxt) → varn e + vari e ≡ c10 cxt |
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157 proof4 env p1 = let open ≡-Reasoning in |
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158 begin |
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159 varn env + vari env |
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160 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩ |
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161 c10 cxt + vari env |
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162 ≡⟨ cong ( λ n → c10 cxt + n ) (pi1 p1 ) ⟩ |
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163 c10 cxt + 0 |
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164 ≡⟨ +-sym {c10 cxt} {0} ⟩ |
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165 c10 cxt |
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166 ∎ |
| 33 | 167 postCond : whileTestState (c10 cxt) (whileDG cxt) |
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168 postCond with whileCond cxt |
| 33 | 169 ... | state1 cond = state2 (proof4 (whileDG cxt) cond ) |
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170 ... | _ = error |
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171 |
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172 {-# TERMINATING #-} |
| 33 | 173 loop : {l : Level} {t : Set l} → Context → (exit : Context → t) → t |
| 31 | 174 loop cxt exit = whileLoopProof cxt (λ cxt → loop cxt exit ) exit |
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175 |
| 35 | 176 |
| 36 | 177 {-# TERMINATING #-} |
| 37 | 178 loopProof : {l : Level} {t : Set l} {P Inv : Context → Set } → (c : Context) → (if : (c : Context) → Dec (P c)) |
| 179 → Inv c → (exit : (c2 : Context) → (P c2) → Inv c2 → t) | |
| 38 | 180 (f : Context → (exitn : (c2 : Context) → (P c2) → Inv c2 → t) → t) → t |
| 37 | 181 loopProof {l} {t} {P} {Inv} cxt if inv exit f = lem cxt inv |
| 36 | 182 where |
| 37 | 183 lem : (c : Context) → Inv c → t |
| 184 lem c inv with if c | |
| 38 | 185 lem c inv | no ¬p = f c (λ c1 inv1 exit1 → lem c1 exit1 ) |
| 186 lem c inv | yes p = exit c p inv | |
| 36 | 187 |
| 35 | 188 |
| 189 | |
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190 proofWhileGear : (c : ℕ) (cxt : Context) → whileTestProof (record cxt { c10 = c ; whileCond = error }) |
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191 $ λ cxt → whileConvProof cxt |
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192 $ λ cxt → loop cxt |
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193 $ λ cxt → vari (whileDG cxt) ≡ c10 cxt |
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194 proofWhileGear c cxt = {!!} |