# Module CSEdomain

The abstract domain for value numbering, used in common subexpression elimination.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Values.
Require Import Memory.
Require Import Op.
Require Import Registers.
Require Import RTL.

Value numbers are represented by positive integers. Equations are of the form valnum = rhs or valnum >= rhs, where the right-hand sides rhs are either arithmetic operations or memory loads, = is strict equality of values, and >= is the "more defined than" relation over values.

Definition valnum := positive.

Inductive rhs : Type :=
| Op: operation -> list valnum -> rhs

Inductive equation : Type :=
| Eq (v: valnum) (strict: bool) (r: rhs).

Definition eq_valnum: forall (x y: valnum), {x=y}+{x<>y} := peq.

Definition eq_list_valnum: forall (x y: list valnum), {x=y}+{x<>y} := list_eq_dec peq.

Definition eq_rhs (x y: rhs) : {x=y}+{x<>y}.
Proof.
generalize chunk_eq eq_operation eq_addressing eq_valnum eq_list_valnum.
decide equality.
Defined.

A value numbering is a collection of equations between value numbers plus a partial map from registers to value numbers. Additionally, we maintain the next unused value number, so as to easily generate fresh value numbers. We also maintain a reverse mapping from value numbers to registers, redundant with the mapping from registers to value numbers, in order to speed up some operations.

Record numbering : Type := mknumbering {
num_next: valnum; (* first unused value number *)
num_eqs: list equation; (* valid equations *)
num_reg: PTree.t valnum; (* mapping register to valnum *)
num_val: PMap.t (list reg) (* reverse mapping valnum to regs containing it *)
}.

Definition empty_numbering :=
{| num_next := 1%positive;
num_eqs := nil;
num_reg := PTree.empty _;
num_val := PMap.init nil |}.

A numbering is well formed if all value numbers mentioned are below num_next. Moreover, the num_val reverse mapping must be consistent with the num_reg direct mapping.

Definition valnums_rhs (r: rhs): list valnum :=
match r with
| Op op vl => vl
end.

Definition wf_rhs (next: valnum) (r: rhs) : Prop :=
forall v, In v (valnums_rhs r) -> Plt v next.

Definition wf_equation (next: valnum) (e: equation) : Prop :=
match e with Eq l str r => Plt l next /\ wf_rhs next r end.

Record wf_numbering (n: numbering) : Prop := {
wf_num_eqs: forall e,
In e n.(num_eqs) -> wf_equation n.(num_next) e;
wf_num_reg: forall r v,
PTree.get r n.(num_reg) = Some v -> Plt v n.(num_next);
wf_num_val: forall r v,
In r (PMap.get v n.(num_val)) -> PTree.get r n.(num_reg) = Some v
}.

Hint Resolve wf_num_eqs wf_num_reg wf_num_val: cse.

Satisfiability of numberings. A numbering holds in a concrete execution state if there exists a valuation assigning values to value numbers that satisfies the equations and register mapping of the numbering.

Definition valuation := valnum -> val.

Inductive rhs_eval_to (valu: valuation) (ge: genv) (sp: val) (m: mem):
rhs -> val -> Prop :=
| op_eval_to: forall op vl v,
eval_operation ge sp op (map valu vl) m = Some v ->
rhs_eval_to valu ge sp m (Op op vl) v
Mem.loadv chunk m a = Some v ->

Inductive equation_holds (valu: valuation) (ge: genv) (sp: val) (m: mem):
equation -> Prop :=
| eq_holds_strict: forall l r,
rhs_eval_to valu ge sp m r (valu l) ->
equation_holds valu ge sp m (Eq l true r)
| eq_holds_lessdef: forall l r v,
rhs_eval_to valu ge sp m r v -> Val.lessdef v (valu l) ->
equation_holds valu ge sp m (Eq l false r).

Record numbering_holds (valu: valuation) (ge: genv) (sp: val)
(rs: regset) (m: mem) (n: numbering) : Prop := {
num_holds_wf:
wf_numbering n;
num_holds_eq: forall eq,
In eq n.(num_eqs) -> equation_holds valu ge sp m eq;
num_holds_reg: forall r v,
n.(num_reg)!r = Some v -> rs#r = valu v
}.

Hint Resolve num_holds_wf num_holds_eq num_holds_reg: cse.

Lemma empty_numbering_holds:
forall valu ge sp rs m,
numbering_holds valu ge sp rs m empty_numbering.
Proof.
intros; split; simpl; intros.
- split; simpl; intros.