Inductive mem_event_erasure : mem_event -> mem_event -> Prop :=
  | WriteErasure: forall b ofs mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Write b ofs mvals) (Write b ofs mvals')
  | ReadErasure: forall b ofs sz mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Read b ofs sz mvals) (Read b ofs sz mvals')
  | AllocErasure: forall b ofs sz,
      mem_event_erasure (Alloc b ofs sz) (Alloc b ofs sz)
  | FreeErasure: forall ls,
      mem_event_erasure (Free ls) (Free ls).Inductive mem_event_erasure : mem_event -> mem_event -> Prop :=
  | WriteErasure: forall b ofs mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Write b ofs mvals) (Write b ofs mvals')
  | ReadErasure: forall b ofs sz mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Read b ofs sz mvals) (Read b ofs sz mvals')
  | AllocErasure: forall b ofs sz,
      mem_event_erasure (Alloc b ofs sz) (Alloc b ofs sz)
  | FreeErasure: forall ls,
      mem_event_erasure (Free ls) (Free ls).Inductive mem_event_erasure : mem_event -> mem_event -> Prop :=
  | WriteErasure: forall b ofs mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Write b ofs mvals) (Write b ofs mvals')
  | ReadErasure: forall b ofs sz mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Read b ofs sz mvals) (Read b ofs sz mvals')
  | AllocErasure: forall b ofs sz,
      mem_event_erasure (Alloc b ofs sz) (Alloc b ofs sz)
  | FreeErasure: forall ls,
      mem_event_erasure (Free ls) (Free ls).Inductive mem_event_erasure : mem_event -> mem_event -> Prop :=
  | WriteErasure: forall b ofs mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Write b ofs mvals) (Write b ofs mvals')
  | ReadErasure: forall b ofs sz mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Read b ofs sz mvals) (Read b ofs sz mvals')
  | AllocErasure: forall b ofs sz,
      mem_event_erasure (Alloc b ofs sz) (Alloc b ofs sz)
  | FreeErasure: forall ls,
      mem_event_erasure (Free ls) (Free ls).Inductive mem_event_erasure : mem_event -> mem_event -> Prop :=
  | WriteErasure: forall b ofs mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Write b ofs mvals) (Write b ofs mvals')
  | ReadErasure: forall b ofs sz mvals mvals',
      memval_erasure_list mvals mvals' ->
      mem_event_erasure (Read b ofs sz mvals) (Read b ofs sz mvals')
  | AllocErasure: forall b ofs sz,
      mem_event_erasure (Alloc b ofs sz) (Alloc b ofs sz)
  | FreeErasure: forall ls,
      mem_event_erasure (Free ls) (Free ls).Inductive mem_event_list_erasure: list mem_event -> list mem_event -> Prop :=
  | nilMemEvent: mem_event_list_erasure nil nil
  | consMemEvent: forall mev mev' ls ls'
                   (Hev_erasure: mem_event_erasure mev mev')
                   (Hls_erasure: mem_event_list_erasure ls ls'),
      mem_event_list_erasure (mev :: ls) (mev' :: ls').Inductive mem_event_list_erasure: list mem_event -> list mem_event -> Prop :=
  | nilMemEvent: mem_event_list_erasure nil nil
  | consMemEvent: forall mev mev' ls ls'
                   (Hev_erasure: mem_event_erasure mev mev')
                   (Hls_erasure: mem_event_list_erasure ls ls'),
      mem_event_list_erasure (mev :: ls) (mev' :: ls').Inductive mem_event_list_erasure: list mem_event -> list mem_event -> Prop :=
  | nilMemEvent: mem_event_list_erasure nil nil
  | consMemEvent: forall mev mev' ls ls'
                   (Hev_erasure: mem_event_erasure mev mev')
                   (Hls_erasure: mem_event_list_erasure ls ls'),
      mem_event_list_erasure (mev :: ls) (mev' :: ls').(** Removing the footprints from a [sync_event] *)
  Definition eraseSyncEvent ev :=
    match ev with
    | Events.release addr _ => Events.release addr None
    | Events.acquire addr _ => Events.acquire addr None
    | Events.spawn addr _ _ => Events.spawn addr None None
    | _ => ev
    end.Inductive event_erasure : Events.machine_event -> Events.machine_event -> Prop :=
  | InternalErasure: forall tid mev mev',
      mem_event_erasure mev mev' ->
      event_erasure (Events.internal tid mev) (Events.internal tid mev')
  | ExternalErasure: forall tid ev,
      event_erasure (Events.external tid ev)
                    (Events.external tid (eraseSyncEvent ev)).Inductive event_erasure : Events.machine_event -> Events.machine_event -> Prop :=
  | InternalErasure: forall tid mev mev',
      mem_event_erasure mev mev' ->
      event_erasure (Events.internal tid mev) (Events.internal tid mev')
  | ExternalErasure: forall tid ev,
      event_erasure (Events.external tid ev)
                    (Events.external tid (eraseSyncEvent ev)).Inductive event_erasure : Events.machine_event -> Events.machine_event -> Prop :=
  | InternalErasure: forall tid mev mev',
      mem_event_erasure mev mev' ->
      event_erasure (Events.internal tid mev) (Events.internal tid mev')
  | ExternalErasure: forall tid ev,
      event_erasure (Events.external tid ev)
                    (Events.external tid (eraseSyncEvent ev)).Inductive trace_erasure : list Events.machine_event ->
                            list Events.machine_event -> Prop :=
  | NilErasure: trace_erasure nil nil
  | ConsErasure: forall ev ev' tr tr'
                   (Hev_erasure: event_erasure ev ev')
                   (Htr_erasure: trace_erasure tr tr'),
      trace_erasure (ev :: tr) (ev' :: tr').Inductive trace_erasure : list Events.machine_event ->
                            list Events.machine_event -> Prop :=
  | NilErasure: trace_erasure nil nil
  | ConsErasure: forall ev ev' tr tr'
                   (Hev_erasure: event_erasure ev ev')
                   (Htr_erasure: trace_erasure tr tr'),
      trace_erasure (ev :: tr) (ev' :: tr').Inductive trace_erasure : list Events.machine_event ->
                            list Events.machine_event -> Prop :=
  | NilErasure: trace_erasure nil nil
  | ConsErasure: forall ev ev' tr tr'
                   (Hev_erasure: event_erasure ev ev')
                   (Htr_erasure: trace_erasure tr tr'),
      trace_erasure (ev :: tr) (ev' :: tr').

(* Specifications: *)
Lemma mem_event_list_erasure_cat: forall tr1 tr1' tr2 tr2', mem_event_list_erasure tr1 tr1' -> mem_event_list_erasure tr2 tr2' -> mem_event_list_erasure (tr1 ++ tr2) (tr1' ++ tr2').
Lemma trace_erasure_cat: forall tr1 tr1' tr2 tr2', trace_erasure tr1 tr1' -> trace_erasure tr2 tr2' -> trace_erasure (tr1 ++ tr2) (tr1' ++ tr2').