Remark symbol_address_inject:
  forall id ofs, Val.inject f (Genv.symbol_address genv id ofs) (Genv.symbol_address genv id ofs).

(* Specifications: *)
Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b.
Lemma eval_addressing_inject: forall addr vl1 vl2 v1, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 -> exists v2, eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_operation_inject: forall op vl1 vl2 v1 m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 -> exists v2, eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b.
Lemma eval_addressing_inject: forall addr vl1 vl2 v1, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 -> exists v2, eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_operation_inject: forall op vl1 vl2 v1 m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 -> exists v2, eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b.
Lemma eval_addressing_inject: forall addr vl1 vl2 v1, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 -> exists v2, eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_operation_inject: forall op vl1 vl2 v1 m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 -> exists v2, eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b.
Lemma eval_addressing_inject: forall addr vl1 vl2 v1, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 -> exists v2, eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2 /\ Val.inject f v1 v2.
Lemma eval_operation_inject: forall op vl1 vl2 v1 m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 -> exists v2, eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2 /\ Val.inject f v1 v2.