Definition alloc_left_sm (mu: SM_Injection) b1 b2 delta: SM_Injection :=
  Build_SM_Injection (fun b => eq_block b b1 || locBlocksSrc mu b)
                     (locBlocksTgt mu) (*b2 is already in locBlocksTgt!*)
                     (pubBlocksSrc mu) (pubBlocksTgt mu)
                     (fun b => if eq_block b b1 then Some(b2, delta)
                               else local_of mu b)
                     (extBlocksSrc mu) (extBlocksTgt mu)
                     (frgnBlocksSrc mu) (frgnBlocksTgt mu) (extern_of mu).

Lemma alloc_left_sm_as_inj_same: forall mu b1 b2 delta (WD: SM_wd mu) (NEW1: DomSrc mu b1 = false), as_inj (alloc_left_sm mu b1 b2 delta) b1 = Some(b2,delta).