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annotate whileTestGears.agda @ 44:5a3c9d087c7c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 15 Dec 2019 15:57:07 +0900 |
| parents | 8813f26da3b7 |
| children |
| 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 |
| 44 | 13 record Env (Cxt : Set) : Set (succ Zero) where |
| 6 | 14 field |
| 15 varn : ℕ | |
| 16 vari : ℕ | |
| 44 | 17 cx : Cxt |
| 18 open Env | |
| 6 | 19 |
| 44 | 20 whileTest : {l : Level} {t : Set l} {Cxt : Set} (c : Cxt ) → (c10 : ℕ) → (Code : Env Cxt → t) → t |
| 21 whileTest c c10 next = next (record {varn = c10 ; vari = 0 ; cx = c } ) | |
| 4 | 22 |
| 23 {-# TERMINATING #-} | |
| 44 | 24 whileLoop : {l : Level} {t : Set l} {Cxt : Set} {c : Cxt } → Env Cxt → (Code : Env Cxt → t) → t |
| 4 | 25 whileLoop env next with lt 0 (varn env) |
| 26 whileLoop env next | false = next env | |
| 27 whileLoop env next | true = | |
| 42 | 28 whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next |
| 4 | 29 |
| 44 | 30 test1 : {Cxt : Set } → (c : Cxt) → Env Cxt |
| 42 | 31 test1 c = whileTest c 10 (λ env → whileLoop env (λ env1 → env1 )) |
| 4 | 32 |
| 33 | |
| 44 | 34 proof1 : {Cxt : Set } (c : Cxt ) → whileTest c 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 )) |
| 42 | 35 proof1 c = refl |
| 4 | 36 |
| 16 | 37 -- ↓PostCondition |
| 44 | 38 whileTest' : {l : Level} {t : Set l} {Cxt : Set} → {c : Cxt} → {c10 : ℕ } → (Code : (env : Env Cxt ) → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t |
| 39 whileTest' {_} {_} {Cxt} {c} {c10} next = next env proof2 | |
| 4 | 40 where |
| 44 | 41 env : Env Cxt |
| 42 | 42 env = record {vari = 0 ; varn = c10 ; cx = c } |
| 16 | 43 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition |
| 4 | 44 proof2 = record {pi1 = refl ; pi2 = refl} |
| 11 | 45 |
| 46 open import Data.Empty | |
| 47 open import Data.Nat.Properties | |
| 48 | |
| 49 | |
| 16 | 50 {-# TERMINATING #-} -- ↓PreCondition(Invaliant) |
| 44 | 51 whileLoop' : {l : Level} {t : Set l} {Cxt : Set} {c : Cxt } → (env : Env Cxt ) → {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) → (Code : Env Cxt → t) → t |
| 9 | 52 whileLoop' env proof next with ( suc zero ≤? (varn env) ) |
| 53 whileLoop' env proof next | no p = next env | |
| 14 | 54 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next |
| 4 | 55 where |
| 42 | 56 env1 = record env {varn = (varn env) - 1 ; vari = (vari env) + 1} |
| 11 | 57 1<0 : 1 ≤ zero → ⊥ |
| 58 1<0 () | |
| 14 | 59 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10 |
| 42 | 60 proof3 = {!!} |
| 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 |
| 16 | 78 -- Condition to Invaliant |
| 44 | 79 conversion1 : {l : Level} {t : Set l } {Cxt : Set} {c : Cxt } → (env : Env Cxt ) → {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) |
| 80 → (Code : (env1 : Env Cxt ) → (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 |
| 44 | 96 proofGears : {c10 : ℕ } → { Cxt : Set } → (c : Cxt ) → Set |
| 97 proofGears {c10} c = whileTest' {_} {_} {_} {c} {c10} (λ n p1 → conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 → ( vari n2 ≡ c10 )))) | |
| 9 | 98 |
| 44 | 99 record Context (e : Set ) : Set (succ Zero) |
| 100 | |
| 101 data whileTestState {Cxt : Set } (c10 : ℕ ) (env : Env Cxt ) : Set where | |
| 33 | 102 error : whileTestState c10 env |
| 103 state1 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → whileTestState c10 env | |
| 104 state2 : (varn env + vari env ≡ c10) → whileTestState c10 env | |
| 105 finstate : ((vari env) ≡ c10 ) → whileTestState c10 env | |
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106 |
| 41 | 107 -- |
| 44 | 108 -- openended Env Cxt c <=> Context |
| 41 | 109 -- |
| 110 | |
| 44 | 111 record Context e where |
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112 field |
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113 c10 : ℕ |
| 44 | 114 whileDG : Env e |
| 33 | 115 whileCond : whileTestState c10 whileDG |
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116 |
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117 open Context |
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118 |
| 41 | 119 open import Relation.Nullary |
| 120 open import Relation.Binary | |
| 121 | |
| 122 -- | |
| 123 -- transparency of condition | |
| 124 -- | |
| 125 | |
| 44 | 126 whileCondP : Env {!!} → Set |
| 41 | 127 whileCondP env = varn env > 0 |
| 128 | |
| 129 whileDec : (cxt : Context) → Dec (whileCondP (whileDG cxt)) | |
| 130 whileDec cxt = {!!} | |
| 131 | |
| 33 | 132 whileTestContext : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
| 133 whileTestContext cxt next = next (record cxt { whileDG = record (whileDG cxt) {varn = c10 cxt ; vari = 0} ; whileCond = {!!} } ) | |
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134 |
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135 {-# TERMINATING #-} |
| 33 | 136 whileLoopContext : {l : Level} {t : Set l} → Context → (Code : Context → t) → t |
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137 whileLoopContext cxt next with lt 0 (varn (whileDG cxt) ) |
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138 whileLoopContext cxt next | false = next cxt |
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139 whileLoopContext cxt next | true = |
| 42 | 140 whileLoopContext (record cxt { whileDG = record (whileDG cxt) {varn = (varn (whileDG cxt)) - 1 ; vari = (vari (whileDG cxt)) + 1} ; whileCond = state2 (proof cxt) } ) next |
| 38 | 141 where |
| 142 proof : (cxt : Context) → (varn (whileDG cxt) - 1) + (vari (whileDG cxt) + 1) ≡ c10 cxt | |
| 143 proof cxt = {!!} | |
| 33 | 144 |
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145 {-# TERMINATING #-} |
| 44 | 146 whileLoopStep : {l : Level} {t : Set l} → Env {!!} → (next : (e : Env {!!} ) → t) (exit : (e : Env {!!} ) → t) → t |
| 33 | 147 whileLoopStep env next exit with <-cmp 0 (varn env) |
| 41 | 148 whileLoopStep env next exit | tri≈ _ eq _ = exit env |
| 42 | 149 whileLoopStep env next exit | tri< gt ¬eq _ = next record env {varn = (varn env) - 1 ; vari = (vari env) + 1} |
| 39 | 150 whileLoopStep env next exit | tri> _ _ c = ⊥-elim (m<n⇒n≢0 {varn env} {0} c refl) |
| 27 | 151 |
| 40 | 152 whileTestProof : {l : Level} {t : Set l} → Context → (Code : (cxt : Context ) → ¬ (vari (whileDG cxt) ≡ varn (whileDG cxt) ) → t) → t |
| 153 whileTestProof cxt next = next record cxt { whileDG = out ; whileCond = init } i!=n where | |
| 44 | 154 out : Env {!!} |
| 42 | 155 out = whileTest {!!} (c10 cxt) ( λ e → e ) |
| 33 | 156 init : whileTestState (c10 cxt) out |
| 157 init = state1 record { pi1 = refl ; pi2 = refl } | |
| 40 | 158 i!=n : ¬ vari out ≡ varn out |
| 159 i!=n eq = {!!} | |
| 27 | 160 |
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161 {-# TERMINATING #-} |
| 41 | 162 whileLoopProof : {l : Level} {t : Set l} → (cxt : Context ) → whileCondP (whileDG cxt) |
| 163 → (continue : (cxt : Context ) → whileCondP (whileDG cxt) → t) (exit : Context → ¬ whileCondP (whileDG cxt) → t) → t | |
| 40 | 164 whileLoopProof cxt i!=n next exit = whileLoopStep (whileDG cxt) |
| 41 | 165 ( λ env → next record cxt { whileDG = env ; whileCond = {!!} } {!!} ) |
| 166 ( λ env → exit record cxt { whileDG = env ; whileCond = exitCond env {!!} } {!!} ) where | |
| 44 | 167 proof5 : (e : Env {!!} ) → varn e + vari e ≡ c10 cxt → 0 ≡ varn e → vari e ≡ c10 cxt |
| 34 | 168 proof5 record { varn = .0 ; vari = vari } refl refl = refl |
| 44 | 169 exitCond : (e : Env {!!} ) → 0 ≡ varn e → whileTestState (c10 cxt) e |
| 33 | 170 exitCond nenv eq1 with whileCond cxt | inspect whileDG cxt |
| 171 ... | state2 cond | record { eq = eq2 } = finstate ( proof5 nenv {!!} eq1 ) | |
| 172 ... | _ | _ = error | |
| 27 | 173 |
| 40 | 174 whileConvProof : {l : Level} {t : Set l} → (cxt : Context ) → ¬ (vari (whileDG cxt) ≡ varn (whileDG cxt)) |
| 175 → ((cxt : Context ) → ¬ (vari (whileDG cxt) ≡ varn (whileDG cxt)) → t) → t | |
| 176 whileConvProof cxt i!=n next = next record cxt { whileCond = postCond } i!=n where | |
| 44 | 177 proof4 : (e : Env {!!} ) → (vari e ≡ 0) /\ (varn e ≡ c10 cxt) → varn e + vari e ≡ c10 cxt |
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178 proof4 env p1 = let open ≡-Reasoning in |
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179 begin |
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180 varn env + vari env |
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181 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩ |
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182 c10 cxt + vari env |
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183 ≡⟨ cong ( λ n → c10 cxt + n ) (pi1 p1 ) ⟩ |
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184 c10 cxt + 0 |
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185 ≡⟨ +-sym {c10 cxt} {0} ⟩ |
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186 c10 cxt |
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187 ∎ |
| 33 | 188 postCond : whileTestState (c10 cxt) (whileDG cxt) |
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189 postCond with whileCond cxt |
| 33 | 190 ... | state1 cond = state2 (proof4 (whileDG cxt) cond ) |
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191 ... | _ = error |
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192 |
| 41 | 193 {-# TERMINATING #-} |
| 194 loop : {l : Level} {t : Set l} → (cxt : Context ) → whileCondP (whileDG cxt) | |
| 195 → (exit : (cxt : Context ) → ¬ whileCondP (whileDG cxt) → t) → t | |
| 196 loop cxt i!=n exit = whileLoopProof cxt i!=n (λ cxt → loop cxt {!!} {!!} exit ) {!!} | |
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197 |
| 35 | 198 |
| 36 | 199 {-# TERMINATING #-} |
| 37 | 200 loopProof : {l : Level} {t : Set l} {P Inv : Context → Set } → (c : Context) → (if : (c : Context) → Dec (P c)) |
| 41 | 201 → ( (cont : (c2 : Context) → (P c2) → t) → (exit : (c2 : Context) → ¬ (P c2) → t) → t ) |
| 202 → t | |
| 203 loopProof {l} {t} {P} {Inv} cxt if f = {!!} | |
| 36 | 204 where |
| 41 | 205 lem : (c : Context) → t |
| 206 lem c with if c | |
| 207 lem c | no ¬p = {!!} -- f c (λ c1 _ → lem c1 ) | |
| 208 lem c | yes p = {!!} | |
| 36 | 209 |
| 42 | 210 whileLoopProof' : {l : Level} {t : Set l} |
| 211 → (cxt : Context ) → ( continue : (cxt : Context ) → whileCondP (whileDG cxt) → t) (exit : Context → ¬ whileCondP (whileDG cxt) → t) → t | |
| 212 whileLoopProof' = {!!} | |
| 35 | 213 |
| 41 | 214 proofWhileGear : (c : ℕ) (cxt : Context) → whileTestProof (record cxt { c10 = c ; whileCond = error }) |
| 215 $ λ cxt i!=n → whileConvProof cxt i!=n | |
| 42 | 216 $ λ cxt i+n=c10 → loopProof cxt whileDec {!!} -- whileLoopProof' |
| 217 $ λ cxt _ → {!!} | |
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218 proofWhileGear c cxt = {!!} |