Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.Record SM_wd (mu:SM_Injection):Prop := {
  disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
  disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;

  local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
               (locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
  extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
               (extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);

  pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
              exists b2 z, local_of mu b1 = Some(b2,z) /\
                           pubBlocksTgt mu b2 = true;
  frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
              exists b2 z, extern_of mu b1 = Some(b2,z) /\
                           frgnBlocksTgt mu b2 = true;

  pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
                               locBlocksTgt mu b = true;
  frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
                              extBlocksTgt mu b = true
}.

Lemma pubSrc: forall mu (WD: SM_wd mu) b1, pubBlocksSrc mu b1 = true -> exists b2 z, pub_of mu b1 = Some(b2,z) /\ pubBlocksTgt mu b2 = true.