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1 Soundness : {bPre : Cond} -> {cm : Comm} -> {bPost : Cond} ->
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2 HTProof bPre cm bPost -> Satisfies bPre cm bPost
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3 Soundness (PrimRule {bPre} {cm} {bPost} pr) s1 s2 q1 q2
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4 = axiomValid bPre cm bPost pr s1 s2 q1 q2
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5 Soundness {.bPost} {.Skip} {bPost} (SkipRule .bPost) s1 s2 q1 q2
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6 = substId1 State {Level.zero} {State} {s1} {s2} (proj₂ q2) (SemCond bPost) q1
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7 Soundness {bPre} {.Abort} {bPost} (AbortRule .bPre .bPost) s1 s2 q1 ()
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8 Soundness (WeakeningRule {bPre} {bPre'} {cm} {bPost'} {bPost} tautPre pr tautPost)
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9 s1 s2 q1 q2
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10 = let hyp : Satisfies bPre' cm bPost'
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11 hyp = Soundness pr
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12 in tautValid bPost' bPost tautPost s2 (hyp s1 s2 (tautValid bPre bPre' tautPre s1 q1) q2)
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13 Soundness (SeqRule {bPre} {cm1} {bMid} {cm2} {bPost} pr1 pr2)
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14 s1 s2 q1 q2
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15 = let hyp1 : Satisfies bPre cm1 bMid
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16 hyp1 = Soundness pr1
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17 hyp2 : Satisfies bMid cm2 bPost
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18 hyp2 = Soundness pr2
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19 in hyp2 (proj₁ q2) s2 (hyp1 s1 (proj₁ q2) q1 (proj₁ (proj₂ q2))) (proj₂ (proj₂ q2))
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20 Soundness (IfRule {cmThen} {cmElse} {bPre} {bPost} {b} pThen pElse)
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21 s1 s2 q1 q2
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22 = let hypThen : Satisfies (bPre /\ b) cmThen bPost
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23 hypThen = Soundness pThen
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24 hypElse : Satisfies (bPre /\ neg b) cmElse bPost
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25 hypElse = Soundness pElse
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26 rThen : RelOpState.comp (RelOpState.delta (SemCond b))
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27 (SemComm cmThen) s1 s2 -> SemCond bPost s2
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28 rThen = λ h -> hypThen s1 s2 ((proj₂ (respAnd bPre b s1)) (q1 , proj₁ t1))
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29 (proj₂ ((proj₂ (RelOpState.deltaRestPre (SemCond b) (SemComm cmThen) s1 s2)) h))
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30 rElse : RelOpState.comp (RelOpState.delta (NotP (SemCond b)))
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31 (SemComm cmElse) s1 s2 -> SemCond bPost s2
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32 rElse = λ h ->
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33 let t10 : (NotP (SemCond b) s1) × (SemComm cmElse s1 s2)
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34 t10 = proj₂ (RelOpState.deltaRestPre
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35 (NotP (SemCond b)) (SemComm cmElse) s1 s2) h
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36 in hypElse s1 s2 (proj₂ (respAnd bPre (neg b) s1)
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37 (q1 , (proj₂ (respNeg b s1) (proj₁ t10)))) (proj₂ t10)
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38 in when rThen rElse q2
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39 Soundness (WhileRule {cm'} {bInv} {b} pr) s1 s2 q1 q2
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40 = proj₂ (respAnd bInv (neg b) s2)
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41 (lem1 (proj₁ q2) s2 (proj₁ t15) , proj₂ (respNeg b s2) (proj₂ t15))
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42 where
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43 hyp : Satisfies (bInv /\ b) cm' bInv
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44 hyp = Soundness pr
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45 Rel1 : ℕ -> Rel State (Level.zero)
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46 Rel1 = λ m ->
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47 RelOpState.repeat
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48 m
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49 (RelOpState.comp (RelOpState.delta (SemCond b))
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50 (SemComm cm'))
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51 t15 : (Rel1 (proj₁ q2) s1 s2) × (NotP (SemCond b) s2)
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52 t15 = proj₂ (RelOpState.deltaRestPost
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53 (NotP (SemCond b)) (Rel1 (proj₁ q2)) s1 s2) (proj₂ q2)
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54 lem1 : (m : ℕ) -> (ss2 : State) -> Rel1 m s1 ss2 -> SemCond bInv ss2
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55 lem1 zero ss2 h = substId1 State (proj₂ h) (SemCond bInv) q1
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56 lem1 (suc n) ss2 h
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57 = let hyp2 : (z : State) -> Rel1 (proj₁ q2) s1 z ->
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58 SemCond bInv z
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59 hyp2 = lem1 n
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60 t22 : (SemCond b (proj₁ h)) × (SemComm cm' (proj₁ h) ss2)
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61 t22 = proj₂ (RelOpState.deltaRestPre (SemCond b) (SemComm cm') (proj₁ h) ss2)
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62 (proj₂ (proj₂ h))
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63 t23 : SemCond (bInv /\ b) (proj₁ h)
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64 t23 = proj₂ (respAnd bInv b (proj₁ h))
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65 (hyp2 (proj₁ h) (proj₁ (proj₂ h)) , proj₁ t22)
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66 in hyp (proj₁ h) ss2 t23 (proj₂ t22)
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