(** Formal Reasoning About Programs
* Chapter 9: Compiler Correctness
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap.
Set Implicit Arguments.
(* Let's look at another example of what we can model with operational
* semantics: correctness of compiler transformations. Our inspiration here is
* the seminal project CompCert, which uses Coq to verify a realistic C
* compiler. We will adopt the same *simulation*-based techniques as CompCert,
* albeit on a simpler language and with simpler compiler phases. We'll stick
* to transformations from the source language to itself, since that's enough to
* illustrate the big ideas. Here's the object language that we'll use, which
* is _almost_ the same as from Chapter 7. *)
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 from Chapter 7. *)
Coercion Const : nat >-> arith.
Coercion Var : var >-> arith.
(*Declare Scope arith_scope.*)
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. Including this element makes our semantics
* a _labelled transition system_. *)
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).
(* It's easy to push labels through steps with context. *)
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 and/or terminating markers. We drop silent
* labels as we run, and we use [Some n] for outputting of [n] and [None] for
* termination. *)
Inductive generate : valuation * cmd -> list (option nat) -> Prop :=
| GenDone : forall vc,
generate vc []
| GenSkip : forall v,
generate (v, Skip) [None]
| 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 (Some n :: ns).
Hint Constructors plug step0 cstep generate : core.
(* Notice that [generate] is defined so that, for any two of a starting state's
* traces, one is a prefix of the other. The same wouldn't necessarily hold if
* our language were nondeterministic. *)
(* We define trace inclusion to capture the notion of
* _preserving all behaviors_. *)
Definition traceInclusion (vc1 vc2 : valuation * cmd) :=
forall ns, generate vc1 ns -> generate vc2 ns.
Infix "<|" := traceInclusion (at level 70).
(* And trace equivalence captures _having the same behaviors_. *)
Definition traceEquivalence (vc1 vc2 : valuation * cmd) :=
vc1 <| vc2 /\ vc2 <| vc1.
Infix "=|" := traceEquivalence (at level 70).
(* Trace equivalence is an appropriate notion of correctness, to relate the
* "before" and "after" programs of a compiler transformation. Note that here
* we break from our usual modus operandi, as we will not be using invariants to
* characterize correctness! This kind of trace equivalence is one of the other
* big concepts that competes with invariants to unify different correctness
* notions. *)
(* Here's a simple example program, which outputs how many days have elapsed at
* the end of each one-month period (with a simplified notion of "month"!). *)
Definition daysPerWeek := 7.
Definition weeksPerMonth := 4.
Definition daysPerMonth := (daysPerWeek * weeksPerMonth)%arith.
(* We are purposely building an expression with arithmetic that can be resolved
* at compile time, to give our optimizations something to do. *)
Example month_boundaries_in_days :=
"acc" <- 0;;
while 1 loop
when daysPerMonth then
"acc" <- "acc" + daysPerMonth;;
Output "acc"
else
(* Oh no! We must have calculated it wrong, since we got zero! *)
(* And, yes, we know this spot can never be reached. Some of our
* optimizations will prove it for us! *)
Skip
done
done.
(* Moderately laboriously, we can prove a particular example trace for this
* program, including its first two outputs. Traces of all lengths exist,
* because the program does not terminate, generating new output infinitely
* often. *)
Hint Extern 1 (interp _ _ = _) => simplify; equality : core.
Hint Extern 1 (interp _ _ <> _) => simplify; equality : core.
Theorem first_few_values :
generate ($0, month_boundaries_in_days) [Some 28; Some 56].
Proof.
unfold month_boundaries_in_days.
eapply GenSilent.
eapply CStep with (C := CSeq Hole _); eauto.
eapply GenSilent.
eapply CStep with (C := Hole); eauto.
eapply GenSilent.
eapply CStep with (C := Hole); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq Hole _); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq (CSeq Hole _) _); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq Hole _); eauto.
eapply GenOutput.
eapply CStep with (C := CSeq Hole _); eauto.
replace 28 with (interp "acc"
($0 $+ ("acc", interp 0 $0)
$+ ("acc", interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))))); eauto.
eapply GenSilent.
eapply CStep with (C := Hole); eauto.
eapply GenSilent.
eapply CStep with (C := Hole); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq Hole _); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq (CSeq Hole _) _); eauto.
eapply GenSilent.
eapply CStep with (C := CSeq Hole _); eauto.
eapply GenOutput.
eapply CStep with (C := CSeq Hole _); eauto.
replace 56 with (interp "acc"
($0 $+ ("acc", interp 0 $0) $+ ("acc",
interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))) $+ ("acc",
interp ("acc" + daysPerMonth)%arith
($0 $+ ("acc", interp 0 $0) $+ ("acc",
interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))))))); eauto.
constructor.
Qed.
(** * Basic Simulation Arguments and Optimizing Expressions *)
(* Let's define an optimization that just simplifies 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.
(* Here's what our optimization does to the example program. *)
Compute cfoldExprs month_boundaries_in_days.
(* It's actually not obvious how to prove trace equivalence for this kind of
* optimization, and we should be on the lookout for general principles that
* help us avoid rehashing the same argument structure for each optimization.
* To let us prove such principles, we first establish a few key properties of
* the object language. *)
(* First, any program that isn't a [Skip] can make progress. *)
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.
(* Now, a set of (boring) lemmas relevant to contexts: *)
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 : core.
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.
(* Finally, the big theorem we are after: the [cstep] relation is
* deterministic. *)
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 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.
match goal with
| [ H : step0 _ _ _, H' : step0 _ _ _ |- _ ] => eapply deterministic0 in H; [ | apply H' ]
end.
propositional; subst; auto.
invert H0.
auto.
eapply plug_function in H2; eauto.
equality.
Qed.
(* OK, we are ready for the first variant of today's big proof technique,
* _simulation_. The method is much like with invariants. Recall that, in our
* old workhorse technique, we pick a predicate over states, and we show that
* all steps preserve it. Simulation is very similar, but now we have a
* two-argument predicate or _relation_ between states of two systems. The
* relation is a simulation when it is able to track execution in one system by
* playing appropriate steps in the other. For deterministic systems like ours,
* the existence of a simulation implies trace equivalence. *)
Section simulation.
(* Here's the kind of relation we assume. *)
Variable R : valuation * cmd -> valuation * cmd -> Prop.
(* Starting from two related states, when the lefthand one makes a step, the
* righthand one can make a matching step, such that the new states are also
* related. *)
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'.
(* When a righthand command is related to [Skip], it must be [Skip], too. *)
Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
-> c2 = Skip.
(* First (easy) step: [R] implies left-to-right trace inclusion. *)
Lemma simulation_fwd' : forall vc1 ns, generate vc1 ns
-> forall vc2, R vc1 vc2
-> generate vc2 ns.
Proof.
induct 1; simplify; eauto.
cases vc2.
apply agree_on_termination in H; subst.
auto.
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.
(* Second (slightly harder) step: [R] implies right-to-left trace
* inclusion. *)
Lemma simulation_bwd' : forall vc2 ns, generate vc2 ns
-> forall vc1, R vc1 vc2
-> generate vc1 ns.
Proof.
induct 1; simplify; eauto.
cases vc1.
assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
first_order; subst.
auto.
eapply one_step in H; eauto.
first_order.
invert H.
invert H4.
invert H5.
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.
(* Put them together and we have trace equivalence. *)
Theorem simulation : forall vc1 vc2, R vc1 vc2
-> vc1 =| vc2.
Proof.
simplify; split; auto using simulation_fwd, simulation_bwd.
Qed.
End simulation.
(* Now to prove our particular optimization. First, original steps can be
* lifted into optimized steps. *)
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.
(* It helps to add optimization on contexts, as a proof device. *)
Fixpoint cfoldExprsContext (C : context) : context :=
match C with
| Hole => Hole
| CSeq C c => CSeq (cfoldExprsContext C) (cfoldExprs c)
end.
(* The optimization can be applied over [plug]. *)
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 : core.
(* The main correctness property! *)
Theorem cfoldExprs_ok : forall v c,
(v, c) =| (v, cfoldExprs c).
Proof.
simplify.
(* Notice our clever choice of a simulation relation here, much as we often
* choose strengthened invariants. We basically just recast the theorem
* statement as a two-state predicate using equality. *)
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 *)
(* Here's a reasonable variant of the last optimization: when an [If] test
* expression reduces to a constant, replace the [If] with whichever branch is
* guaranteed to run. *)
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.
(* Here's how our running example optimizes further. *)
Compute cfold month_boundaries_in_days.
(* It will be helpful to have a shorthand for steps that don't generate output.
* [Notation] is a useful way to introduce a shorthand so that it looks exactly
* the same as its expansion, to all Coq tactics. *)
Notation silent_cstep := (fun a b => cstep a None b).
(* Silent steps have a few interesting properties, proved here. *)
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.
invert H.
invert H3.
invert H4.
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) (Some 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 : core.
(* You might have noticed that our old notion of simulation doesn't apply to the
* new optimization. The reason is that, because the optimized program skips
* some steps, some steps in the source program may not have matching steps in
* the optimized program. Let's extend simulation to allow skipped steps. *)
Section simulation_skipping.
(* Now the relation takes a number as an argument. The idea is that, for
* [R n vc1 vc2], at most [n] steps of [vc1] may go unmatched by [vc2], before
* we finally find one that matches. It is an interesting exercise to work
* out why, without tracking such quantities, this notion of simulation
* wouldn't imply trace equivalence! *)
Variable R : nat -> valuation * cmd -> valuation * cmd -> Prop.
(* Now this key hypothesis has two cases. *)
Hypothesis one_step : forall n vc1 vc2, R n vc1 vc2
-> forall vc1' l, cstep vc1 l vc1'
(* Case 1: Skipping a (silent!) step, decreasing [n] *)
-> (exists n', n = S n' /\ l = None /\ R n' vc1' vc2)
(* Case 2: Matching the step like before; note how [n]
* resets to an arbitrary new limit! *)
\/ 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.
(* The forward direction is just as easy to prove. *)
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.
cases vc2.
apply agree_on_termination in H.
subst.
auto.
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.
(* This one isn't so obvious: a step on the right can now be matched by
* _one or more_ steps on the left, preserving [R]. *)
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 step_to_termination : forall vc v,
silent_cstep^* vc (v, Skip)
-> generate vc [None].
Proof.
clear; induct 1; eauto.
Qed.
Hint Resolve step_to_termination : core.
Lemma R_Skip : forall n vc1 v,
R n vc1 (v, Skip)
-> exists v', silent_cstep^* vc1 (v', Skip).
Proof.
induct n; simplify.
cases vc1.
assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
first_order; subst.
eauto.
eapply one_step in H; eauto.
first_order.
equality.
invert H.
invert H4.
invert H5.
cases vc1.
assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
first_order; subst.
eauto.
eapply one_step in H; eauto.
first_order; subst.
invert H.
apply IHn in H2.
first_order.
eauto.
invert H.
invert H4.
invert H5.
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.
cases vc1.
apply R_Skip in H; first_order.
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.
Hint Extern 1 (_ < _) => linear_arithmetic : core.
Hint Extern 1 (_ >= _) => linear_arithmetic : core.
Hint Extern 1 (_ <> _) => linear_arithmetic : core.
(* We will need to do some bookkeeping of [n] values. This function is the
* trick, as we only need to skip steps based on removing [If]s from the code.
* That means the number of [If]s in a program is an upper bound on skipped
* steps. (It's not a tight bound, because some [If]s may be in branches that
* are themselves removed by the optimization!) *)
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.
(* Our notion of [step0] porting must now allow some skipped steps, also showing
* that they decrease [If] count. *)
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.
(* Now some fiddling with contexts: *)
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 : core.
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 : core.
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 : core.
Hint Extern 1 (interp ?e _ = _) =>
match goal with
| [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H
end : core.
Hint Extern 1 (interp ?e _ <> _) =>
match goal with
| [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H
end : core.
(* The final proof is fairly straightforward now. *)
Lemma cfold_ok : forall v c,
(v, c) =| (v, cfold c).
Proof.
simplify.
(* Note the use of [countIfs] in the simulation relation. *)
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 *)
(* Some optimizations actually transform code toward lower-level languages.
* Let's take the example of breaking compound expressions into step-by-step
* computations using new temporary variables. *)
(* We'll use this function to give us an infinite supply of disjoint
* temporaries. *)
Fixpoint tempVar (n : nat) : string :=
match n with
| O => "_tmp"
| S n' => tempVar n' ++ "'"
end%string.
Compute tempVar 0.
Compute tempVar 1.
Compute tempVar 2.
(* With that kind of temporary, we need to watch our for name clashes with
* variables that already exist in a program. These Boolean functions check for
* lack of clashes. We also prove some properties that will come in handy
* later. *)
Definition 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
| Minus e1 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.
(* It's good to verify that our example program makes the grade. *)
Compute noUnderscore month_boundaries_in_days.
(* Now here's the optimization. First, we flatten expressions. The idea is
* that argument [tempCount] gives us the index of the next temporary we should
* use, also guaranteeing that earlier code only uses lower-numbered
* temporaries. Argument [dst] is a variable where we should write the result
* of the expression. A return value is a command that has the
* effect of writing the value of [e] into [dst]. *)
Fixpoint flattenArith (tempCount : nat) (dst : var) (e : arith) : cmd :=
match e with
| Const _
| Var _ => Assign dst e
| Plus e1 e2 =>
let x1 := tempVar tempCount in
let c1 := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar (S tempCount) in
let c2 := flattenArith (S (S tempCount)) x2 e2 in
Sequence c1 (Sequence c2 (Assign dst (Plus x1 x2)))
| Minus e1 e2 =>
let x1 := tempVar tempCount in
let c1 := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar (S tempCount) in
let c2 := flattenArith (S (S tempCount)) x2 e2 in
Sequence c1 (Sequence c2 (Assign dst (Minus x1 x2)))
| Times e1 e2 =>
let x1 := tempVar tempCount in
let c1 := flattenArith (S tempCount) x1 e1 in
let x2 := tempVar (S tempCount) in
let c2 := flattenArith (S (S tempCount)) x2 e2 in
Sequence c1 (Sequence c2 (Assign dst (Times x1 x2)))
end.
(* For simplicity, the main optimization only flattens variables in
* assignments. *)
Fixpoint flatten (c : cmd) : cmd :=
match c with
| Skip => c
| Assign x e => 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.
(* Here's what it does on our example. *)
Compute flatten month_boundaries_in_days.
(* The alert reader may noticed that, yet again, we picked a transformation that
* our existing simulation relations can't handle directly, at least if we put
* the original system on the left and the compiled version on the right. Now
* we need a single step on the left to be matched by _one or more_ steps on the
* right. *)
Section simulation_multiple.
(* At least we can remove that pesky numeric parameter of [R]. *)
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.
(* The forward direction is easy, as usual. *)
Lemma simulation_multiple_fwd' : forall vc1 ns, generate vc1 ns
-> forall vc2, R vc1 vc2
-> generate vc2 ns.
Proof.
induct 1; simplify; eauto.
cases vc2.
apply agree_on_termination in H; subst.
auto.
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.
(* The backward proof essentially proceeds by strong induction on
* _how many steps it took to generate a trace_, which we facilitate by
* defining a [generate] variant parameterized by a step count. *)
Inductive generateN : nat -> valuation * cmd -> list (option nat) -> Prop :=
| GenDoneN : forall vc,
generateN 0 vc []
| GenSkupN : forall v,
generateN 0 (v, Skip) [None]
| 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 (Some n :: ns).
(* We won't comment on the other proof details, though they could be
* interesting reading. *)
Hint Constructors generateN : core.
Lemma generateN_fwd : forall sc vc ns,
generateN sc vc ns
-> generate vc ns.
Proof.
induct 1; eauto.
Qed.
Hint Resolve generateN_fwd : core.
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 H; eauto.
invert H0.
invert H3.
invert H4.
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 termination_is_last : forall sc vc ns,
generateN sc vc (None :: ns)
-> ns = [].
Proof.
induct 1; eauto.
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.
assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
first_order; subst.
auto.
eapply one_step in H1; eauto.
first_order.
invert H1.
invert H2.
invert H5.
invert H6.
invert H4.
invert H7.
invert H8.
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.
cases o.
exfalso; eauto.
eapply termination_is_last in H0; subst.
auto.
eapply one_step in H1; eauto.
first_order.
eapply generateN_silent_cstep in H0; eauto.
first_order.
invert H5; auto.
invert H3.
invert H7.
invert H8.
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.
(* Now, to verify our particular flattening translation. First, one wrinkle is
* that, by writing to new temporary variables, valuations will _not_ be exactly
* the same across the sides of our relation. Here is the sense in which we
* need the sides to agree: *)
Definition agree (v v' : valuation) :=
forall x,
noUnderscoreVar x = true
-> v $? x = v' $? x.
(* That is, they only need to agree on variables that aren't legal
* temporaries. *)
(* There now follow a whole bunch of thrilling lemmas about agreement. *)
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_add_tempVar_bwd_prime : forall v v' n nv,
agree (v $+ (tempVar n ++ "'", nv)%string) v'
-> agree v v'.
Proof.
simplify.
change (tempVar n ++ "'")%string with (tempVar (S n)) in *.
eauto using agree_add_tempVar_bwd.
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_add_tempVar_bwd_prime agree_refl : core.
(* And here are two more unremarkable lemmas. *)
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 : core.
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 : core.
Lemma self_prime_contra : forall s,
(s ++ "'")%string = s -> False.
Proof.
induct s; simplify; equality.
Qed.
Hint Resolve self_prime_contra : core.
(* We've now proved all properties of [tempVar] that we need, so let's ask Coq
* not to reduce applications of it anymore, to keep goals simpler. *)
Opaque tempVar.
(* This is our workhorse lemma, establishing correct compilation of assignments
* with [flattenArith]. *)
Lemma flatten_Assign : forall e dst tempCount,
noUnderscoreArith e = true
(* We compile an expression [e] with no variable clashes. *)
-> (forall n, n >= tempCount -> dst <> tempVar n)
(* Our destination variable [dst] is distinct from any temporary that we give
* [flattenArith] permission to use. *)
-> forall v1 v2, agree v1 v2
(* The valuations on the two sides agree on non-temporaries. *)
(* THEN we conclude existence of further values, such that *)
-> exists v c v2',
silent_cstep^* (v2, flattenArith tempCount dst e) (v, c)
(* The compiled program steps silently to an intermediate state. *)
/\ cstep (v, c) None (v2', Skip)
(* Next, it runs one final silent step (arithmetic never outputs). *)
/\ agree (v1 $+ (dst, interp e v1)) v2'
(* The place we end up agrees with the original lefthand valuation, with the
* destination updated with the requested value. *)
/\ v2' $? dst = Some (interp e v1)
(* The destination has had the right value set. (This isn't redundant with
* the last fact because the destination might be a temporary, in which case
* [agree] ignores it.) *)
/\ (forall n, n < tempCount -> dst <> tempVar n -> v2' $? tempVar n = v2 $? tempVar n)
(* We have not touched any temporaries both less than [tempCount] and not
* equal to the destination *).
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.
split.
equality.
simplify.
equality.
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
first_order.
eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
first_order.
eexists; exists (dst <- tempVar tempCount + tempVar (S tempCount)); eexists.
split.
apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount + tempVar (S tempCount))).
apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount + tempVar (S tempCount))).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H10 by eauto.
eauto.
(* Apologies for the copy-and-paste between the binary-operator cases! *)
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
first_order.
eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
first_order.
eexists; exists (dst <- tempVar tempCount - tempVar (S tempCount)); eexists.
split.
apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount - tempVar (S tempCount))).
apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount - tempVar (S tempCount))).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H10 by eauto.
eauto.
eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
first_order.
eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
first_order.
eexists; exists (dst <- tempVar tempCount * tempVar (S tempCount)); eexists.
split.
apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount * tempVar (S tempCount))).
apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount * tempVar (S tempCount))).
eauto 7 using trc_trans.
eauto 7 using trc_trans.
eauto 7 using trc_trans.
split.
eauto.
split.
simplify.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
eauto.
simplify.
propositional.
rewrite H9.
rewrite H10 by eauto.
rewrite H5.
erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
auto.
simplify.
rewrite H10 by eauto.
eauto.
Qed.
(* Now our reasoning can be fit within a general theorem about [step0]. Note
* how the conclusions use [cstep] instead of [step0], to accommodate steps
* within the structure of a term in the [Assign] case. *)
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; try eassumption.
first_order.
do 3 eexists.
split.
eassumption.
split.
eassumption.
erewrite <- interp_agree; eauto.
simplify.
eauto using noUnderscoreVar_tempVar.
Qed.
(* Now, some thrilling lemmas about underscores and plugging! *)
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 : core.
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 : core.
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 : core.
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 : core.
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 : core.
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 : core.
(* Finally, the main correctness theorem. *)
Lemma flatten_ok : forall v c,
noUnderscore c = true
-> (v, c) =| (v, flatten c).
Proof.
simplify.
(* Note that our simulation relation remembers lack of underscores, and it
* enforces mere agreement between valuations, rather than full equality. *)
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 4.
eapply flatten_ok' in H5; eauto 4.
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.