Mercurial > hg > Members > ryokka > HoareLogic
diff HoareSoundness.agda @ 24:e668962ac31a
rename modules
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 25 Dec 2018 08:45:06 +0900 |
parents | Hoare.agda@e88ad1d70faf |
children | 222dd3869ab0 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/HoareSoundness.agda Tue Dec 25 08:45:06 2018 +0900 @@ -0,0 +1,200 @@ +{-# OPTIONS --universe-polymorphism #-} + +open import Level +open import Data.Nat +open import Data.Product +open import Data.Bool +open import Data.Empty +open import Data.Sum +open import Relation.Binary +open import Relation.Nullary +open import Relation.Binary.Core +open import Relation.Binary.PropositionalEquality +open import RelOp +open import utilities + +module HoareSoundness + (Cond : Set) + (PrimComm : Set) + (neg : Cond -> Cond) + (_/\_ : Cond -> Cond -> Cond) + (Tautology : Cond -> Cond -> Set) + (State : Set) + (SemCond : Cond -> State -> Set) + (tautValid : (b1 b2 : Cond) -> Tautology b1 b2 -> + (s : State) -> SemCond b1 s -> SemCond b2 s) + (respNeg : (b : Cond) -> (s : State) -> + Iff (SemCond (neg b) s) (¬ SemCond b s)) + (respAnd : (b1 b2 : Cond) -> (s : State) -> + Iff (SemCond (b1 /\ b2) s) + ((SemCond b1 s) × (SemCond b2 s))) + (PrimSemComm : ∀ {l} -> PrimComm -> Rel State l) + (Axiom : Cond -> PrimComm -> Cond -> Set) + (axiomValid : ∀ {l} -> (bPre : Cond) -> (pcm : PrimComm) -> (bPost : Cond) -> + (ax : Axiom bPre pcm bPost) -> (s1 s2 : State) -> + SemCond bPre s1 -> PrimSemComm {l} pcm s1 s2 -> SemCond bPost s2) where + +open import Hoare PrimComm Cond Axiom Tautology _/\_ neg + +open import RelOp +module RelOpState = RelOp State + +NotP : {S : Set} -> Pred S -> Pred S +NotP X s = ¬ X s + +_\/_ : Cond -> Cond -> Cond +b1 \/ b2 = neg (neg b1 /\ neg b2) + +_==>_ : Cond -> Cond -> Cond +b1 ==> b2 = neg (b1 \/ b2) + +when : {X Y Z : Set} -> (X -> Z) -> (Y -> Z) -> + X ⊎ Y -> Z +when f g (inj₁ x) = f x +when f g (inj₂ y) = g y + +-- semantics of commands +SemComm : Comm -> Rel State (Level.zero) +SemComm Skip = RelOpState.deltaGlob +SemComm Abort = RelOpState.emptyRel +SemComm (PComm pc) = PrimSemComm pc +SemComm (Seq c1 c2) = RelOpState.comp (SemComm c1) (SemComm c2) +SemComm (If b c1 c2) + = RelOpState.union + (RelOpState.comp (RelOpState.delta (SemCond b)) + (SemComm c1)) + (RelOpState.comp (RelOpState.delta (NotP (SemCond b))) + (SemComm c2)) +SemComm (While b c) + = RelOpState.unionInf + (λ (n : ℕ) -> + RelOpState.comp (RelOpState.repeat + n + (RelOpState.comp + (RelOpState.delta (SemCond b)) + (SemComm c))) + (RelOpState.delta (NotP (SemCond b)))) + +Satisfies : Cond -> Comm -> Cond -> Set +Satisfies bPre cm bPost + = (s1 : State) -> (s2 : State) -> + SemCond bPre s1 -> SemComm cm s1 s2 -> SemCond bPost s2 + +Soundness : {bPre : Cond} -> {cm : Comm} -> {bPost : Cond} -> + HTProof bPre cm bPost -> Satisfies bPre cm bPost +Soundness (PrimRule {bPre} {cm} {bPost} pr) s1 s2 q1 q2 + = axiomValid bPre cm bPost pr s1 s2 q1 q2 +Soundness {.bPost} {.Skip} {bPost} (SkipRule .bPost) s1 s2 q1 q2 + = substId1 State {Level.zero} {State} {s1} {s2} (proj₂ q2) (SemCond bPost) q1 +Soundness {bPre} {.Abort} {bPost} (AbortRule .bPre .bPost) s1 s2 q1 () +Soundness (WeakeningRule {bPre} {bPre'} {cm} {bPost'} {bPost} tautPre pr tautPost) + s1 s2 q1 q2 + = let hyp : Satisfies bPre' cm bPost' + hyp = Soundness pr + r1 : SemCond bPre' s1 + r1 = tautValid bPre bPre' tautPre s1 q1 + r2 : SemCond bPost' s2 + r2 = hyp s1 s2 r1 q2 + in tautValid bPost' bPost tautPost s2 r2 +Soundness (SeqRule {bPre} {cm1} {bMid} {cm2} {bPost} pr1 pr2) + s1 s2 q1 q2 + = let hyp1 : Satisfies bPre cm1 bMid + hyp1 = Soundness pr1 + hyp2 : Satisfies bMid cm2 bPost + hyp2 = Soundness pr2 + sMid : State + sMid = proj₁ q2 + r1 : SemComm cm1 s1 sMid × SemComm cm2 sMid s2 + r1 = proj₂ q2 + r2 : SemComm cm1 s1 sMid + r2 = proj₁ r1 + r3 : SemComm cm2 sMid s2 + r3 = proj₂ r1 + r4 : SemCond bMid sMid + r4 = hyp1 s1 sMid q1 r2 + in hyp2 sMid s2 r4 r3 +Soundness (IfRule {cmThen} {cmElse} {bPre} {bPost} {b} pThen pElse) + s1 s2 q1 q2 + = let hypThen : Satisfies (bPre /\ b) cmThen bPost + hypThen = Soundness pThen + hypElse : Satisfies (bPre /\ neg b) cmElse bPost + hypElse = Soundness pElse + rThen : RelOpState.comp + (RelOpState.delta (SemCond b)) + (SemComm cmThen) s1 s2 -> + SemCond bPost s2 + rThen = λ h -> + let t1 : SemCond b s1 × SemComm cmThen s1 s2 + t1 = (proj₂ (RelOpState.deltaRestPre + (SemCond b) + (SemComm cmThen) s1 s2)) h + t2 : SemCond (bPre /\ b) s1 + t2 = (proj₂ (respAnd bPre b s1)) + (q1 , proj₁ t1) + in hypThen s1 s2 t2 (proj₂ t1) + rElse : RelOpState.comp + (RelOpState.delta (NotP (SemCond b))) + (SemComm cmElse) s1 s2 -> + SemCond bPost s2 + rElse = λ h -> + let t10 : (NotP (SemCond b) s1) × + (SemComm cmElse s1 s2) + t10 = proj₂ (RelOpState.deltaRestPre + (NotP (SemCond b)) (SemComm cmElse) s1 s2) + h + t6 : SemCond (neg b) s1 + t6 = proj₂ (respNeg b s1) (proj₁ t10) + t7 : SemComm cmElse s1 s2 + t7 = proj₂ t10 + t8 : SemCond (bPre /\ neg b) s1 + t8 = proj₂ (respAnd bPre (neg b) s1) + (q1 , t6) + in hypElse s1 s2 t8 t7 + in when rThen rElse q2 +Soundness (WhileRule {cm'} {bInv} {b} pr) s1 s2 q1 q2 + = proj₂ (respAnd bInv (neg b) s2) t20 + where + hyp : Satisfies (bInv /\ b) cm' bInv + hyp = Soundness pr + n : ℕ + n = proj₁ q2 + Rel1 : ℕ -> Rel State (Level.zero) + Rel1 = λ m -> + RelOpState.repeat + m + (RelOpState.comp (RelOpState.delta (SemCond b)) + (SemComm cm')) + t1 : RelOpState.comp + (Rel1 n) + (RelOpState.delta (NotP (SemCond b))) s1 s2 + t1 = proj₂ q2 + t15 : (Rel1 n s1 s2) × (NotP (SemCond b) s2) + t15 = proj₂ (RelOpState.deltaRestPost + (NotP (SemCond b)) (Rel1 n) s1 s2) + t1 + t16 : Rel1 n s1 s2 + t16 = proj₁ t15 + t17 : NotP (SemCond b) s2 + t17 = proj₂ t15 + lem1 : (m : ℕ) -> (ss2 : State) -> Rel1 m s1 ss2 -> + SemCond bInv ss2 + lem1 ℕ.zero ss2 h + = substId1 State (proj₂ h) (SemCond bInv) q1 + lem1 (ℕ.suc n) ss2 h + = let hyp2 : (z : State) -> Rel1 n s1 z -> + SemCond bInv z + hyp2 = lem1 n + s20 : State + s20 = proj₁ h + t21 : Rel1 n s1 s20 + t21 = proj₁ (proj₂ h) + t22 : (SemCond b s20) × (SemComm cm' s20 ss2) + t22 = proj₂ (RelOpState.deltaRestPre + (SemCond b) (SemComm cm') s20 ss2) + (proj₂ (proj₂ h)) + t23 : SemCond (bInv /\ b) s20 + t23 = proj₂ (respAnd bInv b s20) + (hyp2 s20 t21 , proj₁ t22) + in hyp s20 ss2 t23 (proj₂ t22) + t20 : SemCond bInv s2 × SemCond (neg b) s2 + t20 = lem1 n s2 t16 , proj₂ (respNeg b s2) t17