Program Definition new_MachineSemantics (op_m:option Mem.mem):
        @ConcurSemantics G nat schedule event_trace machine_state mem res (*@semC Sem*).(** The class of Hybrid Machines parameterized by:
        - Threadwise semantics
        - Scheduler granularity *)
    Class HybridMachine:=
      {
        MachineSemantics: schedule -> option res ->
                          CoreSemantics MachState mem
        ; ConcurMachineSemantics: option mem ->
                                  @ConcurSemantics G nat (seq.seq nat) event_trace t mem res (*@semC Sem*)
        ; initial_schedule: forall m m' main vals U p st n,
            initial_core (MachineCoreSemantics U p) n m st m' main vals ->
            exists c, st = (U, nil, c) (*XXX:this seems wrong *)
      }.(** The class of Hybrid Machines parameterized by:
        - Threadwise semantics
        - Scheduler granularity *)
    Class HybridMachine:=
      {
        MachineSemantics: schedule -> option res ->
                          CoreSemantics MachState mem
        ; ConcurMachineSemantics: option mem ->
                                  @ConcurSemantics G nat (seq.seq nat) event_trace t mem res (*@semC Sem*)
        ; initial_schedule: forall m m' main vals U p st n,
            initial_core (MachineCoreSemantics U p) n m st m' main vals ->
            exists c, st = (U, nil, c) (*XXX:this seems wrong *)
      }.(** The class of Hybrid Machines parameterized by:
        - Threadwise semantics
        - Scheduler granularity *)
    Class HybridMachine:=
      {
        MachineSemantics: schedule -> option res ->
                          CoreSemantics MachState mem
        ; ConcurMachineSemantics: option mem ->
                                  @ConcurSemantics G nat (seq.seq nat) event_trace t mem res (*@semC Sem*)
        ; initial_schedule: forall m m' main vals U p st n,
            initial_core (MachineCoreSemantics U p) n m st m' main vals ->
            exists c, st = (U, nil, c) (*XXX:this seems wrong *)
      }.(** The class of Hybrid Machines parameterized by:
        - Threadwise semantics
        - Scheduler granularity *)
    Class HybridMachine:=
      {
        MachineSemantics: schedule -> option res ->
                          CoreSemantics MachState mem
        ; ConcurMachineSemantics: option mem ->
                                  @ConcurSemantics G nat (seq.seq nat) event_trace t mem res (*@semC Sem*)
        ; initial_schedule: forall m m' main vals U p st n,
            initial_core (MachineCoreSemantics U p) n m st m' main vals ->
            exists c, st = (U, nil, c) (*XXX:this seems wrong *)
      }.Instance DilMem : DiluteMem :=
        {| diluteMem := fun x => x |}.Instance scheduler : Scheduler :=
        {| isCoarse := true;
           yield := fun x => x |}.Notation thread_pool := t.Notation C:= (semC).Notation G:= (semG).Notation machine_state := thread_pool.Notation schedule := (seq nat).Notation event_trace := (seq machine_event).Definition HybridCoarseMachine : HybridMachine :=
        @Build_HybridMachine resources Sem ThreadPool _ _ _
                             (MachineCoreSemantics)
                             (new_MachineSemantics)
                             (hybrid_initial_schedule).(** Schedule safety of the coarse-grained machine*)
      Inductive csafe (st : MachState) (m : mem) : nat -> Prop :=
      | Safe_0: csafe st m 0
      | HaltedSafe: forall n, halted_machine st -> csafe st m n
      | CoreSafe : forall tp' m' n tr
                     (Hstep: MachStep st m (fst (fst st),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: csafe (fst (fst st),(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n)
      | AngelSafe: forall tp' m' n (tr: event_trace)
                     (Hstep: MachStep st m (schedSkip (fst (fst st)),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: forall U'', csafe (U'',(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive csafe (st : MachState) (m : mem) : nat -> Prop :=
      | Safe_0: csafe st m 0
      | HaltedSafe: forall n, halted_machine st -> csafe st m n
      | CoreSafe : forall tp' m' n tr
                     (Hstep: MachStep st m (fst (fst st),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: csafe (fst (fst st),(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n)
      | AngelSafe: forall tp' m' n (tr: event_trace)
                     (Hstep: MachStep st m (schedSkip (fst (fst st)),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: forall U'', csafe (U'',(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive csafe (st : MachState) (m : mem) : nat -> Prop :=
      | Safe_0: csafe st m 0
      | HaltedSafe: forall n, halted_machine st -> csafe st m n
      | CoreSafe : forall tp' m' n tr
                     (Hstep: MachStep st m (fst (fst st),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: csafe (fst (fst st),(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n)
      | AngelSafe: forall tp' m' n (tr: event_trace)
                     (Hstep: MachStep st m (schedSkip (fst (fst st)),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: forall U'', csafe (U'',(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive csafe (st : MachState) (m : mem) : nat -> Prop :=
      | Safe_0: csafe st m 0
      | HaltedSafe: forall n, halted_machine st -> csafe st m n
      | CoreSafe : forall tp' m' n tr
                     (Hstep: MachStep st m (fst (fst st),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: csafe (fst (fst st),(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n)
      | AngelSafe: forall tp' m' n (tr: event_trace)
                     (Hstep: MachStep st m (schedSkip (fst (fst st)),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: forall U'', csafe (U'',(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive csafe (st : MachState) (m : mem) : nat -> Prop :=
      | Safe_0: csafe st m 0
      | HaltedSafe: forall n, halted_machine st -> csafe st m n
      | CoreSafe : forall tp' m' n tr
                     (Hstep: MachStep st m (fst (fst st),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: csafe (fst (fst st),(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n)
      | AngelSafe: forall tp' m' n (tr: event_trace)
                     (Hstep: MachStep st m (schedSkip (fst (fst st)),(snd (fst st)) ++ tr,tp') m')
                     (Hsafe: forall U'', csafe (U'',(snd (fst st)) ++ tr,tp') m' n),
          csafe st m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).(** Schedule safety of the coarse-grained machine*)
      Inductive concur_safe U tp (m : mem) : nat -> Prop :=
      | concur_Safe_0: concur_safe U tp m 0
      | concur_HaltedSafe: forall n, halted_machine (U, nil, tp) -> concur_safe U tp m n
      | concur_Internal : forall tp' m' n
                     (Hstep: internal_step U tp m tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m U tr' tp' m')
                     (Hsafe: concur_safe U tp' m' n),
          concur_safe U tp m (S n)
      | concur_External_Angel: forall tp' m' n (tr tr': event_trace)
                     (Hstep: external_step U tr tp m (schedSkip U) tr' tp' m')
                     (Hsafe: forall U'', concur_safe U'' tp' m' n),
          concur_safe U tp m (S n).

Lemma csafe_reduce: forall sched tp tr mem n m, csafe (sched, tr, tp) mem n -> m <= n -> csafe (sched, tr, tp) mem m.