frap/CompilerCorrectness.v

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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 9: Compiler Correctness
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap.
Set Implicit Arguments.
(* In this chapter, we'll work with a small variation on the imperative language
* from the previous chapter. *)
Inductive arith : Set :=
| Const (n : nat)
| Var (x : var)
| Plus (e1 e2 : arith)
| Minus (e1 e2 : arith)
| Times (e1 e2 : arith).
Inductive cmd :=
| Skip
| Assign (x : var) (e : arith)
| Sequence (c1 c2 : cmd)
| If (e : arith) (then_ else_ : cmd)
| While (e : arith) (body : cmd)
| Output (e : arith).
(* The last constructor above is the new one, for generating an _output_ value,
* say to display in a terminal. By including this operation, we create
* interesting differences between the behaviors of different nonterminating
* programs. A correct compiler should preserve these differences. *)
(* The next span of notations and definitions is the same as last chapter. *)
Coercion Const : nat >-> arith.
Coercion Var : var >-> arith.
Infix "+" := Plus : arith_scope.
Infix "-" := Minus : arith_scope.
Infix "*" := Times : arith_scope.
Delimit Scope arith_scope with arith.
Notation "x <- e" := (Assign x e%arith) (at level 75).
Infix ";;" := Sequence (at level 76). (* This one changed slightly, to avoid parsing clashes. *)
Notation "'when' e 'then' then_ 'else' else_ 'done'" := (If e%arith then_ else_) (at level 75, e at level 0).
Notation "'while' e 'loop' body 'done'" := (While e%arith body) (at level 75).
Definition valuation := fmap var nat.
Fixpoint interp (e : arith) (v : valuation) : nat :=
match e with
| Const n => n
| Var x =>
match v $? x with
| None => 0
| Some n => n
end
| Plus e1 e2 => interp e1 v + interp e2 v
| Minus e1 e2 => interp e1 v - interp e2 v
| Times e1 e2 => interp e1 v * interp e2 v
end.
Inductive context :=
| Hole
| CSeq (C : context) (c : cmd).
Inductive plug : context -> cmd -> cmd -> Prop :=
| PlugHole : forall c, plug Hole c c
| PlugSeq : forall c C c' c2,
plug C c c'
-> plug (CSeq C c2) c (Sequence c' c2).
(* Here's our first difference. We add a new parameter to [step0], giving a
* _label_ that records which _externally visible effect_ the step has. For
* this language, output is the only externally visible effect, so a label
* records an optional output value. *)
Inductive step0 : valuation * cmd -> option nat -> valuation * cmd -> Prop :=
| Step0Assign : forall v x e,
step0 (v, Assign x e) None (v $+ (x, interp e v), Skip)
| Step0Seq : forall v c2,
step0 (v, Sequence Skip c2) None (v, c2)
| Step0IfTrue : forall v e then_ else_,
interp e v <> 0
-> step0 (v, If e then_ else_) None (v, then_)
| Step0IfFalse : forall v e then_ else_,
interp e v = 0
-> step0 (v, If e then_ else_) None (v, else_)
| Step0WhileTrue : forall v e body,
interp e v <> 0
-> step0 (v, While e body) None (v, Sequence body (While e body))
| Step0WhileFalse : forall v e body,
interp e v = 0
-> step0 (v, While e body) None (v, Skip)
| Step0Output : forall v e,
step0 (v, Output e) (Some (interp e v)) (v, Skip).
Inductive cstep : valuation * cmd -> option nat -> valuation * cmd -> Prop :=
| CStep : forall C v c l v' c' c1 c2,
plug C c c1
-> step0 (v, c) l (v', c')
-> plug C c' c2
-> cstep (v, c1) l (v', c2).
(* To characterize correct compilation, it is helpful to define a relation to
* capture which output _traces_ a command might generate. Note that, for us, a
* trace is a list of output values, where [None] labels are simply dropped. *)
Inductive generate : valuation * cmd -> list nat -> Prop :=
| GenDone : forall vc,
generate vc []
| GenSilent : forall vc vc' ns,
cstep vc None vc'
-> generate vc' ns
-> generate vc ns
| GenOutput : forall vc n vc' ns,
cstep vc (Some n) vc'
-> generate vc' ns
-> generate vc (n :: ns).
Hint Constructors plug step0 cstep generate.
Definition traceInclusion (vc1 vc2 : valuation * cmd) :=
forall ns, generate vc1 ns -> generate vc2 ns.
Infix "<|" := traceInclusion (at level 70).
Definition traceEquivalence (vc1 vc2 : valuation * cmd) :=
vc1 <| vc2 /\ vc2 <| vc1.
Infix "=|" := traceEquivalence (at level 70).
(** * Basic Simulation Arguments and Optimizing Expressions *)
Fixpoint cfoldArith (e : arith) : arith :=
match e with
| Const _ => e
| Var _ => e
| Plus e1 e2 =>
let e1' := cfoldArith e1 in
let e2' := cfoldArith e2 in
match e1', e2' with
| Const n1, Const n2 => Const (n1 + n2)
| _, _ => Plus e1' e2'
end
| Minus e1 e2 =>
let e1' := cfoldArith e1 in
let e2' := cfoldArith e2 in
match e1', e2' with
| Const n1, Const n2 => Const (n1 - n2)
| _, _ => Minus e1' e2'
end
| Times e1 e2 =>
let e1' := cfoldArith e1 in
let e2' := cfoldArith e2 in
match e1', e2' with
| Const n1, Const n2 => Const (n1 * n2)
| _, _ => Times e1' e2'
end
end.
Theorem cfoldArith_ok : forall v e,
interp (cfoldArith e) v = interp e v.
Proof.
induct e; simplify; try equality;
repeat (match goal with
| [ |- context[match ?E with _ => _ end] ] => cases E
| [ H : _ = interp _ _ |- _ ] => rewrite <- H
end; simplify); subst; ring.
Qed.
Fixpoint cfoldExprs (c : cmd) : cmd :=
match c with
| Skip => c
| Assign x e => Assign x (cfoldArith e)
| Sequence c1 c2 => Sequence (cfoldExprs c1) (cfoldExprs c2)
| If e then_ else_ => If (cfoldArith e) (cfoldExprs then_) (cfoldExprs else_)
| While e body => While (cfoldArith e) (cfoldExprs body)
| Output e => Output (cfoldArith e)
end.
Theorem skip_or_step : forall v c,
c = Skip
\/ exists v' l c', cstep (v, c) l (v', c').
Proof.
induct c; simplify; first_order; subst;
try match goal with
| [ H : cstep _ _ _ |- _ ] => invert H
end;
try match goal with
| [ |- context[cstep (?v, If ?e _ _)] ] => cases (interp e v ==n 0)
| [ |- context[cstep (?v, While ?e _)] ] => cases (interp e v ==n 0)
end; eauto 10.
Qed.
Lemma deterministic0 : forall vc l vc',
step0 vc l vc'
-> forall l' vc'', step0 vc l' vc''
-> l = l' /\ vc'' = vc'.
Proof.
invert 1; invert 1; simplify; propositional.
Qed.
Theorem plug_function : forall C c1 c2, plug C c1 c2
-> forall c2', plug C c1 c2'
-> c2 = c2'.
Proof.
induct 1; invert 1; eauto.
apply IHplug in H5.
equality.
Qed.
Lemma peel_cseq : forall C1 C2 c (c1 c2 : cmd),
C1 = C2 /\ c1 = c2
-> CSeq C1 c = CSeq C2 c /\ c1 = c2.
Proof.
equality.
Qed.
Hint Resolve peel_cseq.
Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2
-> forall l vc1, step0 (v, c1) l vc1
-> forall C' c1', plug C' c1' c2
-> forall l' vc1', step0 (v, c1') l' vc1'
-> C' = C /\ c1' = c1.
Proof.
induct 1; invert 1; invert 1; invert 1; auto;
try match goal with
| [ H : plug _ _ _ |- _ ] => invert1 H
end; eauto.
Qed.
Theorem deterministic : forall vc l vc',
cstep vc l vc'
-> forall l' vc'', cstep vc l' vc''
-> l = l' /\ vc' = vc''.
Proof.
invert 1; invert 1; simplify.
eapply plug_deterministic in H0; eauto.
invert H0.
eapply deterministic0 in H1; eauto.
propositional; subst; auto.
invert H0.
auto.
eapply plug_function in H2; eauto.
equality.
Qed.
Section simulation.
Variable R : valuation * cmd -> valuation * cmd -> Prop.
Hypothesis one_step : forall vc1 vc2, R vc1 vc2
-> forall vc1' l, cstep vc1 l vc1'
-> exists vc2', cstep vc2 l vc2' /\ R vc1' vc2'.
Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
-> c2 = Skip.
Lemma simulation_fwd' : forall vc1 ns, generate vc1 ns
-> forall vc2, R vc1 vc2
-> generate vc2 ns.
Proof.
induct 1; simplify; eauto.
eapply one_step in H; eauto.
first_order.
eauto.
eapply one_step in H1; eauto.
first_order.
eauto.
Qed.
Theorem simulation_fwd : forall vc1 vc2, R vc1 vc2
-> vc1 <| vc2.
Proof.
unfold traceInclusion; eauto using simulation_fwd'.
Qed.
Lemma simulation_bwd' : forall vc2 ns, generate vc2 ns
-> forall vc1, R vc1 vc2
-> generate vc1 ns.
Proof.
induct 1; simplify; eauto.
cases vc1; cases vc.
assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
first_order; subst.
apply agree_on_termination in H1; subst.
invert H.
invert H3.
invert H4.
specialize (one_step H1 H2).
first_order.
eapply deterministic in H; eauto.
propositional; subst.
eauto.
cases vc1; cases vc.
assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
first_order; subst.
apply agree_on_termination in H1; subst.
invert H.
invert H3.
invert H4.
specialize (one_step H1 H2).
first_order.
eapply deterministic in H; eauto.
propositional; subst.
eauto.
Qed.
Theorem simulation_bwd : forall vc1 vc2, R vc1 vc2
-> vc2 <| vc1.
Proof.
unfold traceInclusion; eauto using simulation_bwd'.
Qed.
Theorem simulation : forall vc1 vc2, R vc1 vc2
-> vc1 =| vc2.
Proof.
simplify; split; auto using simulation_fwd, simulation_bwd.
Qed.
End simulation.
Lemma cfoldExprs_ok' : forall v1 c1 l v2 c2,
step0 (v1, c1) l (v2, c2)
-> step0 (v1, cfoldExprs c1) l (v2, cfoldExprs c2).
Proof.
invert 1; simplify;
try match goal with
| [ _ : context[interp ?e ?v] |- _ ] => rewrite <- (cfoldArith_ok v e) in *
| [ |- context[interp ?e ?v] ] => rewrite <- (cfoldArith_ok v e)
end; eauto.
Qed.
Fixpoint cfoldExprsContext (C : context) : context :=
match C with
| Hole => Hole
| CSeq C c => CSeq (cfoldExprsContext C) (cfoldExprs c)
end.
Lemma plug_cfoldExprs1 : forall C c1 c2, plug C c1 c2
-> plug (cfoldExprsContext C) (cfoldExprs c1) (cfoldExprs c2).
Proof.
induct 1; simplify; eauto.
Qed.
Hint Resolve plug_cfoldExprs1.
Lemma cfoldExprs_ok : forall v c,
(v, c) =| (v, cfoldExprs c).
Proof.
simplify.
apply simulation with (R := fun vc1 vc2 => fst vc1 = fst vc2
/\ snd vc2 = cfoldExprs (snd vc1));
simplify; propositional.
invert H0; simplify; subst.
apply cfoldExprs_ok' in H3.
cases vc2; simplify; subst.
eauto 7.
Qed.
(** * Simulations That Allow Skipping Steps *)
Fixpoint cfold (c : cmd) : cmd :=
match c with
| Skip => c
| Assign x e => Assign x (cfoldArith e)
| Sequence c1 c2 => Sequence (cfold c1) (cfold c2)
| If e then_ else_ =>
let e' := cfoldArith e in
match e' with
| Const n => if n ==n 0 then cfold else_ else cfold then_
| _ => If e' (cfold then_) (cfold else_)
end
| While e body => While (cfoldArith e) (cfold body)
| Output e => Output (cfoldArith e)
end.
Notation silent_cstep := (fun a b => cstep a None b).
Lemma silent_generate_fwd : forall ns vc vc',
silent_cstep^* vc vc'
-> generate vc ns
-> generate vc' ns.
Proof.
induct 1; simplify; eauto.
invert H1; auto.
eapply deterministic in H; eauto.
propositional; subst.
auto.
eapply deterministic in H; eauto.
equality.
Qed.
Lemma silent_generate_bwd : forall ns vc vc',
silent_cstep^* vc vc'
-> generate vc' ns
-> generate vc ns.
Proof.
induct 1; eauto.
Qed.
Lemma generate_Skip : forall v a ns,
generate (v, Skip) (a :: ns)
-> False.
Proof.
induct 1; simplify.
invert H.
invert H3.
invert H4.
invert H.
invert H3.
invert H4.
Qed.
Hint Resolve silent_generate_fwd silent_generate_bwd generate_Skip.
Section simulation_skipping.
Variable R : nat -> valuation * cmd -> valuation * cmd -> Prop.
Hypothesis one_step : forall n vc1 vc2, R n vc1 vc2
-> forall vc1' l, cstep vc1 l vc1'
-> (exists n', n = S n' /\ l = None /\ R n' vc1' vc2)
\/ exists n' vc2', cstep vc2 l vc2' /\ R n' vc1' vc2'.
Hypothesis agree_on_termination : forall n v1 v2 c2, R n (v1, Skip) (v2, c2)
-> c2 = Skip.
Lemma simulation_skipping_fwd' : forall vc1 ns, generate vc1 ns
-> forall n vc2, R n vc1 vc2
-> generate vc2 ns.
Proof.
induct 1; simplify; eauto.
eapply one_step in H; eauto.
first_order.
eauto.
eapply one_step in H1; eauto.
first_order.
equality.
eauto.
Qed.
Theorem simulation_skipping_fwd : forall n vc1 vc2, R n vc1 vc2
-> vc1 <| vc2.
Proof.
unfold traceInclusion; eauto using simulation_skipping_fwd'.
Qed.
Lemma match_step : forall n vc2 l vc2' vc1,
cstep vc2 l vc2'
-> R n vc1 vc2
-> exists vc1' vc1'' n', silent_cstep^* vc1 vc1'
/\ cstep vc1' l vc1''
/\ R n' vc1'' vc2'.
Proof.
induct n; simplify.
cases vc1; cases vc2.
assert (c = Skip \/ exists v' l' c', cstep (v, c) l' (v', c')) by apply skip_or_step.
first_order; subst.
apply agree_on_termination in H0; subst.
invert H.
invert H2.
invert H3.
eapply one_step in H0; eauto.
first_order; subst.
equality.
eapply deterministic in H; eauto.
first_order; subst.
eauto 6.
cases vc1; cases vc2.
assert (c = Skip \/ exists v' l' c', cstep (v, c) l' (v', c')) by apply skip_or_step.
first_order; subst.
apply agree_on_termination in H0; subst.
invert H.
invert H2.
invert H3.
eapply one_step in H0; eauto.
first_order; subst.
invert H0.
eapply IHn in H3; eauto.
first_order.
eauto 8.
eapply deterministic in H; eauto.
first_order; subst.
eauto 6.
Qed.
Lemma simulation_skipping_bwd' : forall ns vc2, generate vc2 ns
-> forall n vc1, R n vc1 vc2
-> generate vc1 ns.
Proof.
induct 1; simplify; eauto.
eapply match_step in H1; eauto.
first_order.
eauto.
eapply match_step in H1; eauto.
first_order.
eauto.
Qed.
Theorem simulation_skipping_bwd : forall n vc1 vc2, R n vc1 vc2
-> vc2 <| vc1.
Proof.
unfold traceInclusion; eauto using simulation_skipping_bwd'.
Qed.
Theorem simulation_skipping : forall n vc1 vc2, R n vc1 vc2
-> vc1 =| vc2.
Proof.
simplify; split; eauto using simulation_skipping_fwd, simulation_skipping_bwd.
Qed.
End simulation_skipping.
Fixpoint countIfs (c : cmd) : nat :=
match c with
| Skip => 0
| Assign _ _ => 0
| Sequence c1 c2 => countIfs c1 + countIfs c2
| If _ c1 c2 => 1 + countIfs c1 + countIfs c2
| While _ c1 => countIfs c1
| Output _ => 0
end.
Hint Extern 1 (_ < _) => linear_arithmetic.
Lemma cfold_ok' : forall v1 c1 l v2 c2,
step0 (v1, c1) l (v2, c2)
-> step0 (v1, cfold c1) l (v2, cfold c2)
\/ (l = None /\ v1 = v2 /\ cfold c1 = cfold c2 /\ countIfs c2 < countIfs c1).
Proof.
invert 1; simplify;
try match goal with
| [ _ : context[interp ?e ?v] |- _ ] => rewrite <- (cfoldArith_ok v e) in *
| [ |- context[interp ?e ?v] ] => rewrite <- (cfoldArith_ok v e)
end;
repeat match goal with
| [ |- context[match ?E with _ => _ end] ] => cases E; subst; simplify
end; propositional; eauto.
Qed.
Fixpoint cfoldContext (C : context) : context :=
match C with
| Hole => Hole
| CSeq C c => CSeq (cfoldContext C) (cfold c)
end.
Lemma plug_cfold1 : forall C c1 c2, plug C c1 c2
-> plug (cfoldContext C) (cfold c1) (cfold c2).
Proof.
induct 1; simplify; eauto.
Qed.
Hint Resolve plug_cfold1.
Lemma plug_samefold : forall C c1 c1',
plug C c1 c1'
-> forall c2 c2', plug C c2 c2'
-> cfold c1 = cfold c2
-> cfold c1' = cfold c2'.
Proof.
induct 1; invert 1; simplify; propositional.
f_equal; eauto.
Qed.
Hint Resolve plug_samefold.
Lemma plug_countIfs : forall C c1 c1',
plug C c1 c1'
-> forall c2 c2', plug C c2 c2'
-> countIfs c1 < countIfs c2
-> countIfs c1' < countIfs c2'.
Proof.
induct 1; invert 1; simplify; propositional.
apply IHplug in H5; linear_arithmetic.
Qed.
Hint Resolve plug_countIfs.
Lemma cfold_ok : forall v c,
(v, c) =| (v, cfold c).
Proof.
simplify.
apply simulation_skipping with (R := fun n vc1 vc2 => fst vc1 = fst vc2
/\ snd vc2 = cfold (snd vc1)
/\ countIfs (snd vc1) < n)
(n := S (countIfs c));
simplify; propositional; auto.
invert H0; simplify; subst.
apply cfold_ok' in H4.
propositional; subst.
cases vc2; simplify; subst.
eauto 11.
cases vc2; simplify; subst.
cases n; try linear_arithmetic.
assert (countIfs c2 < n).
eapply plug_countIfs in H2; eauto.
eauto.
eauto 10.
Qed.
(** * Simulations That Allow Taking Multiple Matching Steps *)
Fixpoint tempVar (n : nat) : string :=
match n with
| O => "_tmp"
| S n' => tempVar n' ++ "'"
end%string.
Fixpoint noUnderscoreVar (x : var) : bool :=
match x with
| String "_" _ => false
| _ => true
end.
Lemma append_assoc : forall a b c,
(a ++ (b ++ c) = (a ++ b) ++ c)%string.
Proof.
induct a; simplify; equality.
Qed.
Lemma append_assoc_String : forall a b,
(String a b = String a "" ++ b)%string.
Proof.
induct b; simplify; equality.
Qed.
Lemma noUnderscoreVar_tempVar' : forall n,
exists s, tempVar n = ("_tmp" ++ s)%string.
Proof.
induct n; simplify; first_order.
exists ""; auto.
rewrite H.
exists (x ++ "'")%string.
repeat match goal with
| [ |- context[String ?c ?x] ] =>
match x with
| "" => fail 1
| _ => rewrite (append_assoc_String c x)
end
end.
repeat rewrite append_assoc.
reflexivity.
Qed.
Theorem noUnderscoreVar_tempVar : forall x,
noUnderscoreVar x = true
-> forall n, x <> tempVar n.
Proof.
unfold not; simplify.
subst.
pose proof (noUnderscoreVar_tempVar' n).
first_order.
rewrite H0 in H.
simplify.
equality.
Qed.
Lemma tempVar_inj' : forall s1 s2,
(s1 ++ "'" = s2 ++ "'")%string
-> s1 = s2.
Proof.
induct s1; simplify.
cases s2; simplify; try equality.
invert H.
cases s2; simplify; equality.
cases s2; simplify.
invert H.
cases s1; simplify; equality.
invert H.
f_equal; auto.
Qed.
Theorem tempVar_inj : forall n1 n2,
tempVar n1 = tempVar n2
-> n1 = n2.
Proof.
induct n1; simplify; cases n2; simplify; try equality.
repeat match goal with
| [ _ : context[(?s ++ "'")%string] |- _ ] => cases s; simplify; try equality
end.
repeat match goal with
| [ _ : context[(?s ++ "'")%string] |- _ ] => cases s; simplify; try equality
end.
auto using tempVar_inj'.
Qed.
Fixpoint noUnderscoreArith (e : arith) : bool :=
match e with
| Const _ => true
| Var x => noUnderscoreVar x
| Plus e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
| Minus e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
| Times e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
end.
Fixpoint noUnderscore (c : cmd) : bool :=
match c with
| Skip => true
| Assign x e => noUnderscoreVar x && noUnderscoreArith e
| Sequence c1 c2 => noUnderscore c1 && noUnderscore c2
| If e then_ else_ => noUnderscoreArith e && noUnderscore then_ && noUnderscore else_
| While e body => noUnderscoreArith e && noUnderscore body
| Output e => noUnderscoreArith e
end.
Fixpoint flattenArith (tempCount : nat) (dst : var) (e : arith) : nat * cmd :=
match e with
| Const _
| Var _ => (tempCount, Assign dst e)
| Plus e1 e2 =>
let x1 := tempVar tempCount in
let (tempCount, c1) := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar tempCount in
let (tempCount, c2) := flattenArith (S tempCount) x2 e2 in
(tempCount, Sequence c1 (Sequence c2 (Assign dst (Plus x1 x2))))
| Minus e1 e2 =>
let x1 := tempVar tempCount in
let (tempCount, c1) := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar tempCount in
let (tempCount, c2) := flattenArith (S tempCount) x2 e2 in
(tempCount, Sequence c1 (Sequence c2 (Assign dst (Minus x1 x2))))
| Times e1 e2 =>
let x1 := tempVar tempCount in
let (tempCount, c1) := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar tempCount in
let (tempCount, c2) := flattenArith (S tempCount) x2 e2 in
(tempCount, Sequence c1 (Sequence c2 (Assign dst (Times x1 x2))))
end.
Fixpoint flatten (c : cmd) : cmd :=
match c with
| Skip => c
| Assign x e => snd (flattenArith 0 x e)
| Sequence c1 c2 => Sequence (flatten c1) (flatten c2)
| If e then_ else_ => If e (flatten then_) (flatten else_)
| While e body => While e (flatten body)
| Output _ => c
end.
Section simulation_multiple.
Variable R : valuation * cmd -> valuation * cmd -> Prop.
Hypothesis one_step : forall vc1 vc2, R vc1 vc2
-> forall vc1' l, cstep vc1 l vc1'
-> exists vc2' vc2'',
silent_cstep^* vc2 vc2'
/\ cstep vc2' l vc2''
/\ R vc1' vc2''.
Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
-> c2 = Skip.
Lemma simulation_multiple_fwd' : forall vc1 ns, generate vc1 ns
-> forall vc2, R vc1 vc2
-> generate vc2 ns.
Proof.
induct 1; simplify; eauto.
eapply one_step in H; eauto.
first_order.
eauto.
eapply one_step in H1; eauto.
first_order.
eauto.
Qed.
Theorem simulation_multiple_fwd : forall vc1 vc2, R vc1 vc2
-> vc1 <| vc2.
Proof.
unfold traceInclusion; eauto using simulation_multiple_fwd'.
Qed.
(* A version of [generate] that counts how many steps run *)
Inductive generateN : nat -> valuation * cmd -> list nat -> Prop :=
| GenDoneN : forall vc,
generateN 0 vc []
| GenSilentN : forall sc vc vc' ns,
cstep vc None vc'
-> generateN sc vc' ns
-> generateN (S sc) vc ns
| GenOutputN : forall sc vc n vc' ns,
cstep vc (Some n) vc'
-> generateN sc vc' ns
-> generateN (S sc) vc (n :: ns).
Hint Constructors generateN.
Lemma generateN_fwd : forall sc vc ns,
generateN sc vc ns
-> generate vc ns.
Proof.
induct 1; eauto.
Qed.
Hint Resolve generateN_fwd.
Lemma generateN_bwd : forall vc ns,
generate vc ns
-> exists sc, generateN sc vc ns.
Proof.
induct 1; first_order; eauto.
Qed.
Lemma generateN_silent_cstep : forall sc vc ns,
generateN sc vc ns
-> forall vc', silent_cstep^* vc vc'
-> exists sc', sc' <= sc /\ generateN sc' vc' ns.
Proof.
clear; induct 1; simplify; eauto.
invert H1; eauto.
eapply deterministic in H; eauto.
propositional; subst.
apply IHgenerateN in H3.
first_order.
eauto.
invert H1; eauto.
eapply deterministic in H; eauto.
equality.
Qed.
Lemma simulation_multiple_bwd' : forall sc sc', sc' < sc
-> forall vc2 ns, generateN sc' vc2 ns
-> forall vc1, R vc1 vc2
-> generate vc1 ns.
Proof.
induct sc; simplify.
linear_arithmetic.
cases sc'.
invert H0.
auto.
cases vc1; cases vc2.
assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
first_order; subst.
apply agree_on_termination in H1; subst.
cases ns; auto.
exfalso; eauto.
eapply one_step in H1; eauto.
first_order.
eapply generateN_silent_cstep in H0; eauto.
first_order.
invert H5; auto.
eapply deterministic in H3; eauto.
propositional; subst.
econstructor.
eauto.
eapply IHsc; try eassumption.
linear_arithmetic.
eapply deterministic in H3; eauto.
propositional; subst.
eapply GenOutput.
eauto.
eapply IHsc; try eassumption.
linear_arithmetic.
Qed.
Theorem simulation_multiple_bwd : forall vc1 vc2, R vc1 vc2
-> vc2 <| vc1.
Proof.
unfold traceInclusion; simplify.
apply generateN_bwd in H0.
first_order.
eauto using simulation_multiple_bwd'.
Qed.
Theorem simulation_multiple : forall vc1 vc2, R vc1 vc2
-> vc1 =| vc2.
Proof.
simplify; split; auto using simulation_multiple_fwd, simulation_multiple_bwd.
Qed.
End simulation_multiple.
2017-03-19 18:04:51 +00:00
Definition agree (v v' : valuation) :=
forall x,
noUnderscoreVar x = true
-> v $? x = v' $? x.
Ltac bool :=
simplify;
repeat match goal with
| [ H : _ && _ = true |- _ ] => apply andb_true_iff in H; propositional
end.
Lemma interp_agree : forall v v', agree v v'
-> forall e, noUnderscoreArith e = true
-> interp e v = interp e v'.
Proof.
induct e; bool; try equality.
unfold agree in H.
specialize (H _ H0).
rewrite H.
equality.
Qed.
Lemma agree_add : forall v v' x n,
agree v v'
-> agree (v $+ (x, n)) (v' $+ (x, n)).
Proof.
unfold agree; simplify.
apply H in H0.
cases (x ==v x0); simplify; auto.
Qed.
Lemma agree_add_tempVar_fwd : forall v v' n nv,
agree v v'
-> agree (v $+ (tempVar n, nv)) v'.
Proof.
unfold agree; simplify.
cases (x ==v tempVar n); simplify; subst; auto.
eapply noUnderscoreVar_tempVar in H0.
propositional.
Qed.
Lemma agree_add_tempVar_bwd : forall v v' n nv,
agree (v $+ (tempVar n, nv)) v'
-> agree v v'.
Proof.
unfold agree; simplify.
specialize (H _ H0).
cases (x ==v tempVar n); simplify; subst; auto.
eapply noUnderscoreVar_tempVar in H0.
propositional.
Qed.
Lemma agree_refl : forall v,
agree v v.
Proof.
first_order.
Qed.
Hint Resolve agree_add agree_add_tempVar_fwd agree_add_tempVar_bwd agree_refl.
Hint Extern 1 (_ >= _) => linear_arithmetic.
Lemma silent_csteps_front : forall c v1 v2 c1 c2,
silent_cstep^* (v1, c1) (v2, c2)
-> silent_cstep^* (v1, c1;; c) (v2, c2;; c).
Proof.
induct 1; eauto.
invert H.
eauto 6.
Qed.
Hint Resolve silent_csteps_front.
Lemma tempVar_contra : forall n1 n2,
tempVar n1 = tempVar n2
-> n1 <> n2
-> False.
Proof.
pose proof tempVar_inj.
first_order.
Qed.
Hint Resolve tempVar_contra.
Hint Extern 1 (_ <> _) => linear_arithmetic.
Lemma flatten_Assign : forall e dst tempCount,
noUnderscoreArith e = true
-> (forall n, n >= tempCount -> dst <> tempVar n)
-> forall v1 v2, agree v1 v2
-> exists v c v2',
fst (flattenArith tempCount dst e) >= tempCount
/\ silent_cstep^* (v2, snd (flattenArith tempCount dst e)) (v, c)
/\ cstep (v, c) None (v2', Skip)
/\ agree (v1 $+ (dst, interp e v1)) v2'
/\ v2' $? dst = Some (interp e v1)
/\ (forall n, n < tempCount -> dst <> tempVar n -> v2' $? tempVar n = v2 $? tempVar n).
Proof.
induct e; bool.
do 3 eexists.
split.
auto.
split.
eauto.
split.
eauto.
propositional; auto.
simplify; auto.
simplify.
cases (dst ==v tempVar n0); simplify; subst; auto.
do 3 eexists.
split.
auto.
split.
eauto.
split.
eauto.
propositional; auto.
simplify.
unfold agree in H1.
apply H1 in H.
rewrite H.
eauto.
simplify.
unfold agree in H1.
apply H1 in H.
rewrite H.
equality.
cases (dst ==v tempVar n); simplify; subst; auto.
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
cases (flattenArith (S tempCount) (tempVar tempCount) e1); simplify.
first_order.
eapply IHe2 with (dst := tempVar n) (tempCount := S n) in H5; eauto; clear IHe2.
cases (flattenArith (S n) (tempVar n) e2); simplify.
first_order.
eexists; exists (dst <- tempVar tempCount + tempVar n); eexists.
split.
auto.
split.
apply trc_trans with (y := (x1, c0;; dst <- tempVar tempCount + tempVar n)).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H12 by eauto.
eauto.
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
cases (flattenArith (S tempCount) (tempVar tempCount) e1); simplify.
first_order.
eapply IHe2 with (dst := tempVar n) (tempCount := S n) in H5; eauto; clear IHe2.
cases (flattenArith (S n) (tempVar n) e2); simplify.
first_order.
eexists; exists (dst <- tempVar tempCount - tempVar n); eexists.
split.
auto.
split.
apply trc_trans with (y := (x1, c0;; dst <- tempVar tempCount - tempVar n)).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H12 by eauto.
eauto.
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
cases (flattenArith (S tempCount) (tempVar tempCount) e1); simplify.
first_order.
eapply IHe2 with (dst := tempVar n) (tempCount := S n) in H5; eauto; clear IHe2.
cases (flattenArith (S n) (tempVar n) e2); simplify.
first_order.
eexists; exists (dst <- tempVar tempCount * tempVar n); eexists.
split.
auto.
split.
apply trc_trans with (y := (x1, c0;; dst <- tempVar tempCount * tempVar n)).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H11.
rewrite H12 by eauto.
rewrite H6.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H12 by eauto.
eauto.
Qed.
Lemma flatten_ok' : forall v1 c1 l v2 c2,
step0 (v1, c1) l (v2, c2)
-> noUnderscore c1 = true
-> forall v1', agree v1 v1'
-> exists v c v2', silent_cstep^* (v1', flatten c1) (v, c)
/\ cstep (v, c) l (v2', flatten c2)
/\ agree v2 v2'.
Proof.
invert 1; simplify; bool;
repeat erewrite interp_agree in * by eassumption; eauto 10.
assert (Hnu : noUnderscoreArith e = true) by assumption.
eapply flatten_Assign with (tempCount := 0) (dst := x) in Hnu; eauto.
first_order.
do 3 eexists.
split.
eassumption.
split.
eassumption.
erewrite <- interp_agree; eauto.
simplify.
eauto using noUnderscoreVar_tempVar.
Qed.
Lemma noUnderscore_plug : forall C c0 c1,
plug C c0 c1
-> noUnderscore c1 = true
-> noUnderscore c0 = true.
Proof.
induct 1; bool; auto.
Qed.
Hint Immediate noUnderscore_plug.
Lemma silent_csteps_plug : forall C c1 c1',
plug C c1 c1'
-> forall v1 v2 c2 c2', plug C c2 c2'
-> silent_cstep^* (v1, c1) (v2, c2)
-> silent_cstep^* (v1, c1') (v2, c2').
Proof.
induct 1; invert 1; eauto.
Qed.
Hint Resolve silent_csteps_plug.
Fixpoint flattenContext (C : context) : context :=
match C with
| Hole => Hole
| CSeq C c => CSeq (flattenContext C) (flatten c)
end.
Lemma plug_flatten : forall C c1 c2, plug C c1 c2
-> plug (flattenContext C) (flatten c1) (flatten c2).
Proof.
induct 1; simplify; eauto.
Qed.
Hint Resolve plug_flatten.
Lemma plug_total : forall c C, exists c', plug C c c'.
Proof.
induct C; first_order; eauto.
Qed.
Lemma plug_cstep : forall C c1 c1', plug C c1 c1'
-> forall c2 c2', plug C c2 c2'
-> forall v l v', cstep (v, c1) l (v', c2)
-> cstep (v, c1') l (v', c2').
Proof.
induct 1; invert 1; first_order; eauto.
eapply IHplug in H0; eauto.
first_order.
invert H0.
eauto.
Qed.
Hint Resolve plug_cstep.
Lemma step0_noUnderscore : forall v c l v' c',
step0 (v, c) l (v', c')
-> noUnderscore c = true
-> noUnderscore c' = true.
Proof.
invert 1; bool; auto.
rewrite H0, H1.
reflexivity.
Qed.
Hint Resolve step0_noUnderscore.
Fixpoint noUnderscoreContext (C : context) : bool :=
match C with
| Hole => true
| CSeq C' c => noUnderscoreContext C' && noUnderscore c
end.
Lemma noUnderscore_plug_context : forall C c0 c1,
plug C c0 c1
-> noUnderscore c1 = true
-> noUnderscoreContext C = true.
Proof.
induct 1; bool; auto.
rewrite H0, H2; reflexivity.
Qed.
Lemma noUnderscore_plug_fwd : forall C c0 c1,
plug C c0 c1
-> noUnderscoreContext C = true
-> noUnderscore c0 = true
-> noUnderscore c1 = true.
Proof.
induct 1; bool; auto.
rewrite H4, H3; reflexivity.
Qed.
Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd.
Lemma flatten_ok : forall v c,
noUnderscore c = true
-> (v, c) =| (v, flatten c).
Proof.
simplify.
apply simulation_multiple with (R := fun vc1 vc2 => noUnderscore (snd vc1) = true
/\ agree (fst vc1) (fst vc2)
/\ snd vc2 = flatten (snd vc1));
simplify; propositional; eauto.
invert H1; simplify; subst.
assert (noUnderscore c2 = true) by eauto.
eapply flatten_ok' in H5; eauto.
first_order.
cases vc2; simplify; subst.
pose proof (plug_total x0 (flattenContext C)).
first_order.
do 2 eexists.
split.
eapply silent_csteps_plug; try apply H4; eauto.
eauto 6.
Qed.