Mercurial > hg > Members > soto > experimental
diff WhileTest.agda @ 13:724766af8b12
add impl
author | soto |
---|---|
date | Thu, 11 Feb 2021 19:33:35 +0900 |
parents | 77f0530f2eff |
children |
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--- a/WhileTest.agda Thu Feb 11 17:30:33 2021 +0900 +++ b/WhileTest.agda Thu Feb 11 19:33:35 2021 +0900 @@ -9,12 +9,28 @@ open import Function open import logic ---open import Data.Bool hiding ( _≟_ ) -- ; _≤?_ ; _≤_ ; _<_) ---open import Relation.Binary.PropositionalEquality +open import Data.Bool hiding ( _∧_ ; _≟_ ) -- ; _∧_ ; _≤_ ; _<_) +open import Data.Product open import Agda.Builtin.Unit +open import Relation.Nullary using (¬_; Dec; yes; no) + +open import Data.Empty +open import Data.Nat.Properties + +-- logicへ ++zero : { y : ℕ } → y + zero ≡ y ++zero {zero} = refl ++zero {suc y} = cong ( λ x → suc x ) ( +zero {y} ) + +-- utilへ +_-_ : ℕ → ℕ → ℕ +x - zero = x +zero - _ = zero +(suc x) - (suc y) = x - y record Env : Set (Suc Zero) where field + c10 : ℕ varn : ℕ vari : ℕ open Env @@ -22,23 +38,24 @@ data _implies_ (A B : Set ) : Set (Suc Zero) where proof : ( A → B ) → A implies B +implies2p : {A B : Set } → A implies B → A → B +implies2p (proof x) = x + data whileTestState : Set where s1 : whileTestState s2 : whileTestState sf : whileTestState -whileTestStateP : whileTestState → Env → ℕ → Set -whileTestStateP s1 env c10 = (vari env ≡ 0) ∧ (varn env ≡ c10) -whileTestStateP s2 env c10 = (varn env + vari env ≡ c10 ) -whileTestStateP sf env c10 = (vari env ≡ c10) +whileTestStateP : whileTestState → Env → Set +whileTestStateP s1 env = (vari env ≡ 0) ∧ (varn env ≡ c10 env) +whileTestStateP s2 env = (varn env + vari env ≡ c10 env) +whileTestStateP sf env = (vari env ≡ c10 env) record WhileTest {m : Level } {t : Set m } : Set (Suc m) where field env : Env whileInit : {m : Level } {t : Set m } → (c10 : ℕ) → (Env → t) → t - whileInit c10 next = next (record {varn = c10 ; vari = 0 } ) - whileInit-impl : (c10 : ℕ) → whileInit c10 (λ env → ⊤ implies (whileTestStateP s1 env c10) ) - whileInit-impl c = proof ( λ _ → record { proj1 = refl ; proj2 = refl } ) + whileInit c10 next = next (record {c10 = c10 ; varn = c10 ; vari = 0 } ) whileLoop : Env → (Code : Env → t) → t whileLoop env next = whileLoop1 (varn env) env where whileLoop1 : ℕ → Env → t @@ -47,10 +64,58 @@ whileTest : (c10 : ℕ) → (Env → t) → t whileTest c10 next = whileInit c10 $ λ env → whileLoop env next + loopPP : (n : ℕ) → (input : Env ) → (n ≡ varn input) → Env + loopPP zero input refl = input + loopPP (suc n) input refl = loopPP n (record input { varn = pred (varn input) ; vari = suc (vari input)}) refl + + -- init + whileInit-impl : (c10 : ℕ) → whileInit c10 (λ env → ⊤ implies (whileTestStateP s1 env) ) + whileInit-impl c = proof ( λ _ → record { proj1 = refl ; proj2 = refl } ) + whileTestPSemSound : (c : ℕ ) (output : Env ) → output ≡ whileInit c (λ e → e) → ⊤ implies ((vari output ≡ 0) ∧ (varn output ≡ c)) + whileTestPSemSound c output refl = whileInit-impl c + -- init → loop + whileConvPSemSound : {l : Level} → (input : Env) → (whileTestStateP s1 input ) implies (whileTestStateP s2 input) + whileConvPSemSound input = proof λ x → (conv input x) where + conv : (env : Env ) → (vari env ≡ 0) ∧ (varn env ≡ c10 env ) → varn env + vari env ≡ c10 env + conv e record { proj1 = refl ; proj2 = refl } = +zero + -- loop → loop + whileLoopPSem : {l : Level} {t : Set l} → (input : Env ) → whileTestStateP s2 input + → (next : (output : Env ) → (whileTestStateP s2 input ) implies (whileTestStateP s2 output) → t) + → (exit : (output : Env ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output) → t) → t + whileLoopPSem env s next exit with varn env | s + ... | zero | _ = exit env (proof (λ z → z)) + ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof λ x → +-suc varn (vari env) ) + -- loop → fin + + loopPPSem : (input output : Env ) → output ≡ loopPP (varn input) input refl + → (whileTestStateP s2 input ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output) + loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p + where + lem : (n : ℕ) → (env : Env) → n + suc (vari env) ≡ suc (n + vari env) + lem n env = +-suc (n) (vari env) + loopPPSemInduct : (n : ℕ) → (current : Env) → (eq : n ≡ varn current) → (loopeq : output ≡ loopPP n current eq) + → (whileTestStateP s2 current ) → (whileTestStateP s2 current ) implies (whileTestStateP sf output) + loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (λ x → refl) + loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) = + whileLoopPSem current refl + (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl) + (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl) + + + whileLoopPSemSound : {l : Level} → (input output : Env ) + → whileTestStateP s2 input + → output ≡ loopPP (varn input) input refl + → (whileTestStateP s2 input ) implies ( whileTestStateP sf output ) + whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre + + whileTestSound : {l : Level} (c : ℕ) → (output : Env) → ⊤ → whileTestStateP sf output + whileTestSound {l} c record { c10 = c10 ; varn = varn ; vari = vari } top = + implies2p (whileLoopPSemSound {l} (record { c10 = c ; varn = c ; vari = zero }) (record { c10 = c10 ; varn = c ; vari = vari}) (+zero) {!!}) (implies2p (whileConvPSemSound {l} (record { c10 = c ; varn = c ; vari = zero })) (implies2p (whileTestPSemSound c (whileInit c (λ e → e)) refl) top)) + open WhileTest createWhileTest : {m : Level} {t : Set m } → WhileTest {m} {t} -createWhileTest = record { env = record { varn = 0; vari = 0 } } +createWhileTest = record { env = record { c10 = 0; varn = 0; vari = 0 } } test2 : ℕ test2 = whileTest createWhileTest 10 $ λ e → vari e