Ltac InvApproxRegs :=
match goal with
| [ H: _ :: _ = _ :: _ |- _ ] =>
injection H; clear H; intros; InvApproxRegs
| [ H: ?v = AE.get ?r ae |- _ ] =>
generalize (MATCH r); rewrite <- H; clear H; intro; InvApproxRegs
| _ => idtac
end.
Ltac SimplVM :=
match goal with
| [ H: vmatch _ ?v (I ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vint n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (F ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vfloat n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (FS ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vsingle n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (Ptr(Gl ?id ?ofs)) |- _ ] =>
let E := fresh in
assert (E: Val.lessdef v (Genv.symbol_address ge id ofs)) by (eapply vmatch_ptr_gl; eauto);
clear H; SimplVM
| [ H: vmatch _ ?v (Ptr(Stk ?ofs)) |- _ ] =>
let E := fresh in
assert (E: Val.lessdef v (Vptr sp ofs)) by (eapply vmatch_ptr_stk; eauto);
clear H; SimplVM
| _ => idtac
end.

(* Specifications: *)
Lemma const_for_result_correct:
forall a op v,
const_for_result a = Some op ->
vmatch bc v a ->
exists v', eval_operation ge (Vptr sp Ptrofs.zero) op nil m = Some v' /\ Val.lessdef v v'.
Lemma eval_static_shift_correct:
forall s n, eval_shift s (Vint n) = Vint (eval_static_shift s n).
Lemma cond_strength_reduction_correct:
forall cond args vl,
vl = map (fun r => AE.get r ae) args ->
let (cond', args') := cond_strength_reduction cond args vl in
eval_condition cond' rs##args' m = eval_condition cond rs##args m.
Lemma make_cmp_base_correct:
forall c args vl,
vl = map (fun r => AE.get r ae) args ->
let (op', args') := make_cmp_base c args vl in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some v
/\ Val.lessdef (Val.of_optbool (eval_condition c rs##args m)) v.
Lemma make_cmp_correct:
forall c args vl,
vl = map (fun r => AE.get r ae) args ->
let (op', args') := make_cmp c args vl in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some v
/\ Val.lessdef (Val.of_optbool (eval_condition c rs##args m)) v.
Lemma make_addimm_correct:
forall n r,
let (op, args) := make_addimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.add rs#r (Vint n)) v.
Lemma make_shlimm_correct:
forall n r1 r2,
rs#r2 = Vint n ->
let (op, args) := make_shlimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shl rs#r1 (Vint n)) v.
Lemma make_shrimm_correct:
forall n r1 r2,
rs#r2 = Vint n ->
let (op, args) := make_shrimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shr rs#r1 (Vint n)) v.
Lemma make_shruimm_correct:
forall n r1 r2,
rs#r2 = Vint n ->
let (op, args) := make_shruimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shru rs#r1 (Vint n)) v.
Lemma make_mulimm_correct:
forall n r1 r2,
rs#r2 = Vint n ->
let (op, args) := make_mulimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mul rs#r1 (Vint n)) v.
Lemma make_mla_mulimm_correct:
forall n1 r1 r2 r3,
rs#r1 = Vint n1 ->
let (op, args) := make_mla_mulimm n1 r1 r2 r3 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.add (Val.mul (Vint n1) rs#r2) rs#r3) v.
Lemma make_mla_addimm_correct:
forall n3 r1 r2 r3,
rs#r3 = Vint n3 ->
let (op, args) := make_mla_addimm n3 r1 r2 r3 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.add (Val.mul rs#r1 rs#r2) (Vint n3)) v.
Lemma make_mla_bothimm_correct:
forall n1 n3 r1 r2 r3,
rs#r1 = Vint n1 ->
rs#r3 = Vint n3 ->
let (op, args) := make_mla_bothimm n1 n3 r1 r2 r3 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.add (Val.mul (Vint n1) rs#r2) (Vint n3)) v.
Lemma make_divimm_correct:
forall n r1 r2 v,
Val.divs rs#r1 rs#r2 = Some v ->
rs#r2 = Vint n ->
let (op, args) := make_divimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some w /\ Val.lessdef v w.
Lemma make_divuimm_correct:
forall n r1 r2 v,
Val.divu rs#r1 rs#r2 = Some v ->
rs#r2 = Vint n ->
let (op, args) := make_divuimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some w /\ Val.lessdef v w.
Lemma make_andimm_correct:
forall n r x,
vmatch bc rs#r x ->
let (op, args) := make_andimm n r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.and rs#r (Vint n)) v.
Lemma make_orimm_correct:
forall n r,
let (op, args) := make_orimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.or rs#r (Vint n)) v.
Lemma make_xorimm_correct:
forall n r,
let (op, args) := make_xorimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.xor rs#r (Vint n)) v.
Lemma make_mulfimm_correct:
forall n r1 r2,
rs#r2 = Vfloat n ->
let (op, args) := make_mulfimm n r1 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulf rs#r1 rs#r2) v.
Lemma make_mulfimm_correct_2:
forall n r1 r2,
rs#r1 = Vfloat n ->
let (op, args) := make_mulfimm n r2 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulf rs#r1 rs#r2) v.
Lemma make_mulfsimm_correct:
forall n r1 r2,
rs#r2 = Vsingle n ->
let (op, args) := make_mulfsimm n r1 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulfs rs#r1 rs#r2) v.
Lemma make_mulfsimm_correct_2:
forall n r1 r2,
rs#r1 = Vsingle n ->
let (op, args) := make_mulfsimm n r2 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulfs rs#r1 rs#r2) v.
Lemma make_cast8signed_correct:
forall r x,
vmatch bc rs#r x ->
let (op, args) := make_cast8signed r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.sign_ext 8 rs#r) v.
Lemma make_cast16signed_correct:
forall r x,
vmatch bc rs#r x ->
let (op, args) := make_cast16signed r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.sign_ext 16 rs#r) v.
Lemma op_strength_reduction_correct:
forall op args vl v,
vl = map (fun r => AE.get r ae) args ->
eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v ->
let (op', args') := op_strength_reduction op args vl in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some w /\ Val.lessdef v w.
Lemma addr_strength_reduction_correct:
forall addr args vl res,
vl = map (fun r => AE.get r ae) args ->
eval_addressing ge (Vptr sp Ptrofs.zero) addr rs##args = Some res ->
let (addr', args') := addr_strength_reduction addr args vl in
exists res', eval_addressing ge (Vptr sp Ptrofs.zero) addr' rs##args' = Some res' /\ Val.lessdef res res'.