Inductive alloc_variablesT (g: genv): PTree.t (block * type) -> mem -> list (ident * type) ->
                                      PTree.t (block * type) -> mem -> (list mem_event) -> Prop :=
    alloc_variablesT_nil : forall e m, alloc_variablesT g e m nil e m nil
  | alloc_variablesT_cons :
      forall e m id ty vars m1 b1 m2 e2 T,
        Mem.alloc m 0 (@sizeof g ty) = (m1, b1) ->
        alloc_variablesT g (PTree.set id (b1, ty) e) m1 vars e2 m2 T ->
        alloc_variablesT g e m ((id, ty) :: vars) e2 m2 (Alloc b1 0 (@sizeof g ty) :: T).Inductive alloc_variablesT (g: genv): PTree.t (block * type) -> mem -> list (ident * type) ->
                                      PTree.t (block * type) -> mem -> (list mem_event) -> Prop :=
    alloc_variablesT_nil : forall e m, alloc_variablesT g e m nil e m nil
  | alloc_variablesT_cons :
      forall e m id ty vars m1 b1 m2 e2 T,
        Mem.alloc m 0 (@sizeof g ty) = (m1, b1) ->
        alloc_variablesT g (PTree.set id (b1, ty) e) m1 vars e2 m2 T ->
        alloc_variablesT g e m ((id, ty) :: vars) e2 m2 (Alloc b1 0 (@sizeof g ty) :: T).Inductive alloc_variablesT (g: genv): PTree.t (block * type) -> mem -> list (ident * type) ->
                                      PTree.t (block * type) -> mem -> (list mem_event) -> Prop :=
    alloc_variablesT_nil : forall e m, alloc_variablesT g e m nil e m nil
  | alloc_variablesT_cons :
      forall e m id ty vars m1 b1 m2 e2 T,
        Mem.alloc m 0 (@sizeof g ty) = (m1, b1) ->
        alloc_variablesT g (PTree.set id (b1, ty) e) m1 vars e2 m2 T ->
        alloc_variablesT g e m ((id, ty) :: vars) e2 m2 (Alloc b1 0 (@sizeof g ty) :: T).

Lemma alloc_variablesT_ax2 g: forall e m l e' m' (A:alloc_variables g e m l e' m'), exists T, alloc_variablesT g e m l e' m' T.