Module Registers

Pseudo-registers and register states. This library defines the type of pseudo-registers (also known as "temporaries" in compiler literature) used in the RTL intermediate language. We also define finite sets and finite maps over pseudo-registers.

Require Import Coqlib.
Require Import AST.
Require Import Maps.
Require Import Ordered.
Require FSetAVL.
Require Import Values.

Definition reg: Type := positive.

Module Reg.

Definition eq := peq.

Definition typenv := PMap.t typ.

End Reg.

Mappings from registers to some type.

Module Regmap := PMap.

Set Implicit Arguments.

Definition regmap_optget
    (A: Type) (or: option reg) (dfl: A) (rs: Regmap.t A) : A :=
  match or with
  | None => dfl
  | Some r => Regmap.get r rs
  end.

Definition regmap_optset
    (A: Type) (or: option reg) (v: A) (rs: Regmap.t A) : Regmap.t A :=
  match or with
  | None => rs
  | Some r => Regmap.set r v rs
  end.

Definition regmap_setres
    (A: Type) (res: builtin_res reg) (v: A) (rs: Regmap.t A) : Regmap.t A :=
  match res with
  | BR r => Regmap.set r v rs
  | _ => rs
  end.

Notation "a # b" := (Regmap.get b a) (at level 1).
Notation "a ## b" := (List.map (fun r => Regmap.get r a) b) (at level 1).
Notation "a # b <- c" := (Regmap.set b c a) (at level 1, b at next level).

Pointwise "less defined than" relation between register maps.

Definition regs_lessdef (rs1 rs2: Regmap.t val) : Prop :=
  forall r, Val.lessdef (rs1#r) (rs2#r).

Lemma regs_lessdef_regs:
  forall rs1 rs2, regs_lessdef rs1 rs2 ->
  forall rl, Val.lessdef_list rs1##rl rs2##rl.

Proof.

induction rl; constructor; auto.
Qed.

Lemma set_reg_lessdef:
forall r v1 v2 rs1 rs2,
Val.lessdef v1 v2 -> regs_lessdef rs1 rs2 -> regs_lessdef (rs1#r <- v1) (rs2#r <- v2).

Proof.

intros; red; intros. repeat rewrite Regmap.gsspec.
destruct (peq r0 r); auto.
Qed.

Lemma set_res_lessdef:
  forall res v1 v2 rs1 rs2,
  Val.lessdef v1 v2 -> regs_lessdef rs1 rs2 ->
  regs_lessdef (regmap_setres res v1 rs1) (regmap_setres res v2 rs2).

Proof.

intros. destruct res; simpl; auto. apply set_reg_lessdef; auto.
Qed.

Sets of registers

Module Regset := FSetAVL.Make(OrderedPositive).