Section soundness.
Existing Instance unit_kMD.
Context {P: ProgrammingLanguage}
{MD: Model}
{J: Join model}
{SA: SeparationAlgebra model}
{R: Relation model}
{po_R: PreOrder Krelation}
{DCSA: DownwardsClosedSeparationAlgebra model}
{UCSA: UpwardsClosedSeparationAlgebra model}
{Res: Resource}
{Ac: Action}
{Acr: Action_resource Ac Res}
{nAcr: NormalAction_resource Ac Res}
{TS: TraceSemantics P (resources * model) Ac}
{AIr: ActionInterpret_resource model Ac Res ac_sem}
{SAAIr: @SAActionInterpret_resource (resources * model) Ac ac_sem (@prod_Join _ _ (Pred_Join resource) J) (fun a => ~ is_resources_action a)}
{KAIr: @KActionInterpret_resource (resources * model) Ac ac_sem (RelProd (discPred_R resource) R)}.
Instance KSAAIr: @KSAActionInterpret_resource (resources * model) Ac ac_sem (@prod_Join _ _ (Pred_Join resource) J) (RelProd (discPred_R resource) R) (fun a => ~ is_resources_action a) :=
ordered_and_frame_AIr _ _ _.
Context {L: Language} {minL: MinimunLanguage L} {pL: PropositionalLanguage L} {SL: SeparationLanguage L} {SM: Semantics L MD} {kiSM: KripkeIntuitionisticSemantics L MD tt SM} {fsSM: FlatSemantics.SeparatingSemantics L MD tt SM}.
Class LegalInvariants (Inv: resource * (model -> Prop) -> Prop): Prop := {
at_most_one_invariant: forall r I1 I2, Inv (r, I1) -> Inv (r, I2) -> I1 = I2;
invariant_mono: forall r I,  Inv (r, I) -> forall s1 s2, I s1 -> s1 <= s2 -> I s2;
invariant_precise: forall r I, Inv (r, I) -> forall s,
(exists s', (fun s' => exists f, I f /\ join s' f s) s') ->
(exists s', greatest (fun s' => exists f, I f /\ join s' f s) s')
}.
(* Definition ThreadLocal_KSA_AIr: forall (Inv: resource * (model -> Prop) -> Prop) {INV: LegalInvariants Inv}, @KSAActionInterpret_resource (resources * model) Ac (ThreadLocal_ActionInterpret_resource ac_sem Inv) (@prod_Join _ _ (Pred_Join resource) J) (RelProd (discPred_R resource) R). Check @lift_join. *)

Lemma hoare_parallel_partial_sound
{CPP: ConcurrentProgrammingLanguage_Sparallel P}
{AcP: Action_Parallel Ac}
{nAcPr: NormalAction_Parallel_resource Ac Res}
{c2tP: Command2Traces_Sparallel_resource P model Ac Res c2t}
{AIPr: ActionInterpret_Parallel_resource model Ac Res ac_sem}:
forall Inv c1 P1 Q1 c2 P2 Q2 (INV: LegalInvariants Inv),
guarded_triple_partial_valid Inv P1 c1 Q1 ->
guarded_triple_partial_valid Inv P2 c2 Q2 ->
guarded_triple_partial_valid Inv (P1 * P2) (Sparallel c1 c2) (Q1 * Q2).