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Require Export Coq.Strings.String Coq.Lists.List Coq.Init.Logic.
Require Export Bedrock.Word.
Require Import Coq.ZArith.ZArith Coq.NArith.NArith Coq.Numbers.Natural.Peano.NPeano Coq.NArith.Ndec.
Require Import Coq.Arith.Compare_dec Coq.omega.Omega.
Require Import Coq.Structures.OrderedType Coq.Structures.OrderedTypeEx.
Require Import Coq.FSets.FMapPositive Coq.FSets.FMapFullAVL Coq.Logic.JMeq.
Require Import Crypto.Assembly.QhasmUtil Crypto.Assembly.QhasmCommon.
Require Export Crypto.Util.FixCoqMistakes.
(* We want to use pairs and triples as map keys: *)
Module Pair_as_OT <: UsualOrderedType.
Definition t := (nat * nat)%type.
Definition eq := @eq t.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt (a b: t) :=
if (Nat.eq_dec (fst a) (fst b))
then lt (snd a) (snd b)
else lt (fst a) (fst b).
Lemma conv: forall {x0 x1 y0 y1: nat},
(x0 = y0 /\ x1 = y1) <-> (x0, x1) = (y0, y1).
Proof.
intros; split; intros.
- destruct H; destruct H0; subst; intuition.
- inversion_clear H; intuition.
Qed.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
intros; destruct x as [x0 x1], y as [y0 y1], z as [z0 z1];
unfold lt in *; simpl in *;
destruct (Nat.eq_dec x0 y0), (Nat.eq_dec y0 z0), (Nat.eq_dec x0 z0);
omega.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
intros; destruct x as [x0 x1], y as [y0 y1];
unfold lt, eq in *; simpl in *;
destruct (Nat.eq_dec x0 y0); subst; intuition;
inversion H0; subst; omega.
Qed.
Definition compare x y : Compare lt eq x y.
destruct x as [x0 x1], y as [y0 y1];
destruct (Nat_as_OT.compare x0 y0);
unfold Nat_as_OT.lt, Nat_as_OT.eq in *.
- apply LT; abstract (unfold lt; simpl; destruct (Nat.eq_dec x0 y0); intuition auto with zarith).
- destruct (Nat_as_OT.compare x1 y1);
unfold Nat_as_OT.lt, Nat_as_OT.eq in *.
+ apply LT; abstract (unfold lt; simpl; destruct (Nat.eq_dec x0 y0); intuition).
+ apply EQ; abstract (unfold lt; simpl; subst; intuition auto with relations).
+ apply GT; abstract (unfold lt; simpl; destruct (Nat.eq_dec y0 x0); intuition auto with zarith).
- apply GT; abstract (unfold lt; simpl; destruct (Nat.eq_dec y0 x0); intuition auto with zarith).
Defined.
Definition eq_dec (a b: t): {a = b} + {a <> b}.
destruct (compare a b);
destruct a as [a0 a1], b as [b0 b1].
- right; abstract (
unfold lt in *; simpl in *;
destruct (Nat.eq_dec a0 b0); intuition;
inversion H; intuition auto with zarith).
- left; abstract (inversion e; intuition).
- right; abstract (
unfold lt in *; simpl in *;
destruct (Nat.eq_dec b0 a0); intuition;
inversion H; intuition auto with zarith).
Defined.
End Pair_as_OT.
Module Triple_as_OT <: UsualOrderedType.
Definition t := (nat * nat * nat)%type.
Definition get0 (x: t) := fst (fst x).
Definition get1 (x: t) := snd (fst x).
Definition get2 (x: t) := snd x.
Definition eq := @eq t.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Definition lt (a b: t) :=
if (Nat.eq_dec (get0 a) (get0 b))
then
if (Nat.eq_dec (get1 a) (get1 b))
then lt (get2 a) (get2 b)
else lt (get1 a) (get1 b)
else lt (get0 a) (get0 b).
Lemma conv: forall {x0 x1 x2 y0 y1 y2: nat},
(x0 = y0 /\ x1 = y1 /\ x2 = y2) <-> (x0, x1, x2) = (y0, y1, y2).
Proof.
intros; split; intros.
- destruct H; destruct H0; subst; intuition.
- inversion_clear H; intuition.
Qed.
Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.
intros; unfold lt in *;
destruct (Nat.eq_dec (get0 x) (get0 y)),
(Nat.eq_dec (get1 x) (get1 y)),
(Nat.eq_dec (get0 y) (get0 z)),
(Nat.eq_dec (get1 y) (get1 z)),
(Nat.eq_dec (get0 x) (get0 z)),
(Nat.eq_dec (get1 x) (get1 z));
omega.
Qed.
Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.
intros; unfold lt, eq in *;
destruct (Nat.eq_dec (get0 x) (get0 y)),
(Nat.eq_dec (get1 x) (get1 y));
subst; intuition;
inversion H0; subst; omega.
Qed.
Ltac compare_tmp x y :=
abstract (
unfold Nat_as_OT.lt, Nat_as_OT.eq, lt, eq in *;
destruct (Nat.eq_dec (get0 x) (get0 y));
destruct (Nat.eq_dec (get1 x) (get1 y));
simpl; intuition auto with zarith).
Ltac compare_eq x y :=
abstract (
unfold Nat_as_OT.lt, Nat_as_OT.eq, lt, eq, get0, get1 in *;
destruct x as [x x2], y as [y y2];
destruct x as [x0 x1], y as [y0 y1];
simpl in *; subst; intuition).
Definition compare x y : Compare lt eq x y.
destruct (Nat_as_OT.compare (get0 x) (get0 y)).
- apply LT; compare_tmp x y.
- destruct (Nat_as_OT.compare (get1 x) (get1 y)).
+ apply LT; compare_tmp x y.
+ destruct (Nat_as_OT.compare (get2 x) (get2 y)).
* apply LT; compare_tmp x y.
* apply EQ; compare_eq x y.
* apply GT; compare_tmp y x.
+ apply GT; compare_tmp y x.
- apply GT; compare_tmp y x.
Defined.
Definition eq_dec (a b: t): {a = b} + {a <> b}.
destruct (compare a b);
destruct a as [a a2], b as [b b2];
destruct a as [a0 a1], b as [b0 b1].
- right; abstract (
unfold lt, get0, get1, get2 in *; simpl in *;
destruct (Nat.eq_dec a0 b0), (Nat.eq_dec a1 b1);
intuition; inversion H; intuition auto with zarith).
- left; abstract (inversion e; intuition).
- right; abstract (
unfold lt, get0, get1, get2 in *; simpl in *;
destruct (Nat.eq_dec b0 a0), (Nat.eq_dec b1 a1);
intuition; inversion H; intuition auto with zarith).
Defined.
End Triple_as_OT.
Module StateCommon.
Export ListNotations.
Module NatM := FMapFullAVL.Make(Nat_as_OT).
Module PairM := FMapFullAVL.Make(Pair_as_OT).
Module TripleM := FMapFullAVL.Make(Triple_as_OT).
Definition NatNMap: Type := NatM.t N.
Definition PairNMap: Type := PairM.t N.
Definition TripleNMap: Type := TripleM.t N.
Definition LabelMap: Type := NatM.t nat.
End StateCommon.
Module ListState.
Export StateCommon.
Definition ListState (n: nat) := ((list (word n)) * TripleNMap * (option bool))%type.
Definition emptyState {n}: ListState n := ([], TripleM.empty N, None).
Definition getVar {n: nat} (name: nat) (st: ListState n): option (word n) :=
nth_error (fst (fst st)) name.
Definition getList {n: nat} (st: ListState n): list (word n) :=
fst (fst st).
Definition setList {n: nat} (lst: list (word n)) (st: ListState n): ListState n :=
(lst, snd (fst st), snd st).
Definition getMem {n: nat} (name index: nat) (st: ListState n): option (word n) :=
omap (TripleM.find (n, name, index) (snd (fst st))) (fun v => Some (NToWord n v)).
Definition setMem {n: nat} (name index: nat) (v: word n) (st: ListState n): ListState n :=
(fst (fst st), TripleM.add (n, name, index) (wordToN v) (snd (fst st)), snd st).
Definition getCarry {n: nat} (st: ListState n): option bool := (snd st).
Definition setCarry {n: nat} (v: bool) (st: ListState n): ListState n :=
(fst st, Some v).
Definition setCarryOpt {n: nat} (v: option bool) (st: ListState n): ListState n :=
match v with
| Some v' => (fst st, v)
| None => st
end.
End ListState.
Module FullState.
Export StateCommon.
(* The Big Definition *)
Inductive State :=
| fullState (regState: PairNMap)
(stackState: PairNMap)
(memState: TripleNMap)
(retState: list nat)
(carry: Carry): State.
Definition emptyState: State :=
fullState (PairM.empty N) (PairM.empty N) (TripleM.empty N) [] None.
(* Register *)
Definition getReg {n} (r: Reg n) (state: State): option (word n) :=
match state with
| fullState regS _ _ _ _ =>
match (PairM.find (n, regName r) regS) with
| Some v => Some (NToWord n v)
| None => None
end
end.
Definition setReg {n} (r: Reg n) (value: word n) (state: State): State :=
match state with
| fullState regS stackS memS retS carry =>
fullState (PairM.add (n, regName r) (wordToN value) regS)
stackS memS retS carry
end.
(* Stack *)
Definition getStack {n} (s: Stack n) (state: State): option (word n) :=
match state with
| fullState _ stackS _ _ _ =>
match (PairM.find (n, stackName s) stackS) with
| Some v => Some (NToWord n v)
| None => None
end
end.
Definition setStack {n} (s: Stack n) (value: word n) (state: State): State :=
match state with
| fullState regS stackS memS retS carry =>
fullState regS
(PairM.add (n, stackName s) (wordToN value) stackS)
memS retS carry
end.
(* Memory *)
Definition getMem {n m} (x: Mem n m) (i: Index m) (state: State): option (word n) :=
match state with
| fullState _ _ memS _ _ =>
match (TripleM.find (n, memName x, proj1_sig i) memS) with
| Some v => Some (NToWord n v)
| None => None
end
end.
Definition setMem {n m} (x: Mem n m) (i: Index m) (value: word n) (state: State): State :=
match state with
| fullState regS stackS memS retS carry =>
fullState regS stackS
(TripleM.add (n, memName x, proj1_sig i) (wordToN value) memS)
retS carry
end.
(* Return Pointers *)
Definition pushRet (x: nat) (state: State): State :=
match state with
| fullState regS stackS memS retS carry =>
fullState regS stackS memS (cons x retS) carry
end.
Definition popRet (state: State): option (State * nat) :=
match state with
| fullState regS stackS memS [] carry => None
| fullState regS stackS memS (r :: rs) carry =>
Some (fullState regS stackS memS rs carry, r)
end.
(* Carry State Manipulations *)
Definition getCarry (state: State): Carry :=
match state with
| fullState _ _ _ _ b => b
end.
Definition setCarry (value: bool) (state: State): State :=
match state with
| fullState regS stackS memS retS carry =>
fullState regS stackS memS retS (Some value)
end.
Definition setCarryOpt (value: option bool) (state: State): State :=
match value with
| Some c' => setCarry c' state
| _ => state
end.
End FullState.
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