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Require Import QhasmCommon QhasmUtil State.
Require Import ZArith Sumbool.
Require Import Bedrock.Word.
Require Import Logic.Eqdep_dec ProofIrrelevance.
Module EvalUtil.
Definition evalTest {n} (o: TestOp) (a b: word n): bool :=
let c := (N.compare (wordToN a) (wordToN b)) in
let eqBit := match c with | Eq => true | _ => false end in
let ltBit := match c with | Lt => true | _ => false end in
let gtBit := match c with | Gt => true | _ => false end in
match o with
| TEq => eqBit
| TLt => ltBit
| TLe => orb (eqBit) (ltBit)
| TGt => gtBit
| TGe => orb (eqBit) (gtBit)
end.
Definition evalIntOp {b} (io: IntOp) (x y: word b): (word b) * option bool :=
match io with
| Add =>
let v := (wordToN x + wordToN y)%N in
let c := (overflows b (&x + &y)%N)%w in
match c as c' return c' = c -> _ with
| right _ => fun _ => (NToWord b v, Some false)
| left _ => fun _ => (NToWord b v, Some true)
end eq_refl
| Sub => (wminus x y, None)
| Xor => (wxor x y, None)
| And => (wand x y, None)
| Or => (wor x y, None)
end.
Definition evalCarryOp {b} (io: CarryOp) (x y: word b) (c: bool): (word b) * bool :=
match io with
| AddWidthCarry =>
let c' := (overflows b (&x + &y + (if c then 1 else 0))%N)%w in
let v := addWithCarry x y c in
match c' as c'' return c' = c'' -> _ with
| right _ => fun _ => (v, false)
| left _ => fun _ => (v, true)
end eq_refl
end.
Definition highBits {n} (m: nat) (x: word n) := snd (break m x).
Definition evalDualOp {n} (duo: DualOp) (x y: word n): word n * word n.
refine match duo with
| Mult => (x ^* y,
@extend _ n _ ((highBits (n/2) x) ^* (highBits (n/2) y)))
end; abstract omega.
Defined.
Definition evalRotOp {b} (ro: RotOp) (x: word b) (n: nat) :=
match ro with
| Shl => NToWord b (N.shiftl_nat (wordToN x) n)
| Shr => NToWord b (N.shiftr_nat (wordToN x) n)
end.
(* Width decideability *)
Definition getWidth (n: nat): option (Width n) :=
match n with
| 32 => Some W32
| 64 => Some W64
| _ => None
end.
Lemma getWidth_eq {n} (a: Width n): Some a = getWidth n.
Proof. induction a; unfold getWidth; simpl; intuition. Qed.
Lemma width_eq {n} (a b: Width n): a = b.
Proof.
assert (Some a = Some b) as H by (
replace (Some a) with (getWidth n) by (rewrite getWidth_eq; intuition);
replace (Some b) with (getWidth n) by (rewrite getWidth_eq; intuition);
intuition).
inversion H; intuition.
Qed.
(* Mapping Conversions *)
Definition wordToM {n: nat} {spec: Width n} (w: word n): Mapping n :=
constM _ (constant spec w).
Definition regToM {n: nat} {spec: Width n} (r: Reg n): Mapping n :=
regM _ r.
Definition stackToM {n: nat} {spec: Width n} (s: Stack n): Mapping n :=
stackM _ s.
Definition constToM {n: nat} {spec: Width n} (c: Const n): Mapping n :=
constM _ c.
Definition mapping_dec {n} (a b: Mapping n): {a = b} + {a <> b}.
refine (match (a, b) as p' return (a, b) = p' -> _ with
| (regM v, regM v') => fun _ =>
if (Nat.eq_dec (regName v) (regName v'))
then left _
else right _
| (stackM v, stackM v') => fun _ =>
if (Nat.eq_dec (stackName v) (stackName v'))
then left _
else right _
| (constM v, constM v') => fun _ =>
if (Nat.eq_dec (constValueN v) (constValueN v'))
then left _
else right _
| (memM _ v i, memM _ v' i') => fun _ =>
if (Nat.eq_dec (memName v) (memName v'))
then if (Nat.eq_dec (memLength v) (memLength v'))
then if (Nat.eq_dec (proj1_sig i) (proj1_sig i'))
then left _ else right _ else right _ else right _
| _ => fun _ => right _
end (eq_refl (a, b))); abstract (
inversion_clear _H;
unfold regName, stackName, constValueN, memName, memLength in *;
intuition; try inversion H;
destruct v, v'; subst;
try assert (w = w0) by (apply width_eq); do 3 first [
contradict _H0; inversion H1
| rewrite <- (natToWord_wordToNat w0);
rewrite <- (natToWord_wordToNat w2);
assert (w = w1) by (apply width_eq); subst;
rewrite _H0
| apply (inj_pair2_eq_dec _ Nat.eq_dec) in H3
| inversion H; subst; intuition
| destruct i, i'; simpl in _H2; subst;
replace l with l0 by apply proof_irrelevance
| intuition ]).
Defined.
Definition dec_lt (a b: nat): {(a < b)%nat} + {(a >= b)%nat}.
assert ({(a <? b)%nat = true} + {(a <? b)%nat <> true})
by abstract (destruct (a <? b)%nat; intuition);
destruct H.
- left; abstract (apply Nat.ltb_lt in e; intuition).
- right; abstract (rewrite Nat.ltb_lt in n; intuition).
Defined.
Fixpoint stackNames {n} (lst: list (Mapping n)): list nat :=
match lst with
| nil => nil
| cons c cs =>
match c with
| stackM v => cons (stackName v) (stackNames cs)
| _ => stackNames cs
end
end.
Fixpoint regNames {n} (lst: list (Mapping n)): list nat :=
match lst with
| nil => nil
| cons c cs =>
match c with
| regM v => cons (regName v) (regNames cs)
| _ => regNames cs
end
end.
End EvalUtil.
Module QhasmEval.
Export EvalUtil FullState.
Definition evalCond (c: Conditional) (state: State): option bool :=
match c with
| CTrue => Some true
| CZero n r =>
omap (getReg r state) (fun v =>
if (Nat.eq_dec O (wordToNat v))
then Some true
else Some false)
| CReg n o a b =>
omap (getReg a state) (fun va =>
omap (getReg b state) (fun vb =>
Some (evalTest o va vb)))
| CConst n o a c =>
omap (getReg a state) (fun va =>
Some (evalTest o va (constValueW c)))
end.
Definition evalOperation (o: Operation) (state: State): option State :=
match o with
| IOpConst _ o r c =>
omap (getReg r state) (fun v =>
let (v', co) := (evalIntOp o v (constValueW c)) in
Some (setCarryOpt co (setReg r v' state)))
| IOpReg _ o a b =>
omap (getReg a state) (fun va =>
omap (getReg b state) (fun vb =>
let (v', co) := (evalIntOp o va vb) in
Some (setCarryOpt co (setReg a v' state))))
| IOpStack _ o a b =>
omap (getReg a state) (fun va =>
omap (getStack b state) (fun vb =>
let (v', co) := (evalIntOp o va vb) in
Some (setCarryOpt co (setReg a v' state))))
| IOpMem _ _ o r m i =>
omap (getReg r state) (fun va =>
omap (getMem m i state) (fun vb =>
let (v', co) := (evalIntOp o va vb) in
Some (setCarryOpt co (setReg r v' state))))
| DOp _ o a b (Some x) =>
omap (getReg a state) (fun va =>
omap (getReg b state) (fun vb =>
let (low, high) := (evalDualOp o va vb) in
Some (setReg x high (setReg a low state))))
| DOp _ o a b None =>
omap (getReg a state) (fun va =>
omap (getReg b state) (fun vb =>
let (low, high) := (evalDualOp o va vb) in
Some (setReg a low state)))
| ROp _ o r i =>
omap (getReg r state) (fun v =>
let v' := (evalRotOp o v i) in
Some (setReg r v' state))
| COp _ o a b =>
omap (getReg a state) (fun va =>
omap (getReg b state) (fun vb =>
match (getCarry state) with
| None => None
| Some c0 =>
let (v', c') := (evalCarryOp o va vb c0) in
Some (setCarry c' (setReg a v' state))
end))
end.
Definition evalAssignment (a: Assignment) (state: State): option State :=
match a with
| ARegMem _ _ r m i =>
omap (getMem m i state) (fun v => Some (setReg r v state))
| AMemReg _ _ m i r =>
omap (getReg r state) (fun v => Some (setMem m i v state))
| AStackReg _ a b =>
omap (getReg b state) (fun v => Some (setStack a v state))
| ARegStack _ a b =>
omap (getStack b state) (fun v => Some (setReg a v state))
| ARegReg _ a b =>
omap (getReg b state) (fun v => Some (setReg a v state))
| AConstInt _ r c =>
Some (setReg r (constValueW c) state)
end.
End QhasmEval.
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