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Revising before class

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Adam Chlipala 2021-03-21 10:14:31 -04:00
parent 3048b59f34
commit 25441b3991
3 changed files with 44 additions and 45 deletions
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52
CompilerCorrectness.v
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@ -15,7 +15,7 @@ Set Implicit Arguments.
* albeit on a simpler language and with simpler compiler phases. We'll stick * 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 * 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 * illustrate the big ideas. Here's the object language that we'll use, which
* is _almost_ the same as from Chapter 7. *) * is _almost_ the same as from Chapter 8. *)
Inductive arith : Set := Inductive arith : Set :=
| Const (n : nat) | Const (n : nat)
@ -36,7 +36,7 @@ Inductive cmd :=
* interesting differences between the behaviors of different nonterminating * interesting differences between the behaviors of different nonterminating
* programs. A correct compiler should preserve these differences. *) * programs. A correct compiler should preserve these differences. *)
(* The next span of notations and definitions is the same as from Chapter 7. *) (* The next span of notations and definitions is the same as from Chapter 8. *)
Coercion Const : nat >-> arith. Coercion Const : nat >-> arith.
Coercion Var : var >-> arith. Coercion Var : var >-> arith.
@ -127,7 +127,7 @@ Inductive generate : valuation * cmd -> list (option nat) -> Prop :=
-> generate vc' ns -> generate vc' ns
-> generate vc (Some n :: ns). -> generate vc (Some n :: ns).
Hint Constructors plug step0 cstep generate : core. Local Hint Constructors plug step0 cstep generate : core.
(* Notice that [generate] is defined so that, for any two of a starting state's (* 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 * traces, one is a prefix of the other. The same wouldn't necessarily hold if
@ -179,8 +179,8 @@ Example month_boundaries_in_days :=
* because the program does not terminate, generating new output infinitely * because the program does not terminate, generating new output infinitely
* often. *) * often. *)
Hint Extern 1 (interp _ _ = _) => simplify; equality : core. Local Hint Extern 1 (interp _ _ = _) => simplify; equality : core.
Hint Extern 1 (interp _ _ <> _) => simplify; equality : core. Local Hint Extern 1 (interp _ _ <> _) => simplify; equality : core.
Theorem first_few_values : Theorem first_few_values :
generate ($0, month_boundaries_in_days) [Some 28; Some 56]. generate ($0, month_boundaries_in_days) [Some 28; Some 56].
@ -318,7 +318,7 @@ Proof.
equality. equality.
Qed. Qed.
Hint Resolve peel_cseq : core. Local Hint Resolve peel_cseq : core.
Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2 Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2
-> forall l vc1, step0 (v, c1) l vc1 -> forall l vc1, step0 (v, c1) l vc1
@ -500,7 +500,7 @@ Proof.
induct 1; simplify; eauto. induct 1; simplify; eauto.
Qed. Qed.
Hint Resolve plug_cfoldExprs1 : core. Local Hint Resolve plug_cfoldExprs1 : core.
(* The main correctness property! *) (* The main correctness property! *)
Theorem cfoldExprs_ok : forall v c, Theorem cfoldExprs_ok : forall v c,
@ -594,7 +594,7 @@ Proof.
invert H4. invert H4.
Qed. Qed.
Hint Resolve silent_generate_fwd silent_generate_bwd generate_Skip : core. Local 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 (* 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 * new optimization. The reason is that, because the optimized program skips
@ -767,9 +767,9 @@ Section simulation_skipping.
Qed. Qed.
End simulation_skipping. End simulation_skipping.
Hint Extern 1 (_ < _) => linear_arithmetic : core. Local Hint Extern 1 (_ < _) => linear_arithmetic : core.
Hint Extern 1 (_ >= _) => linear_arithmetic : core. Local Hint Extern 1 (_ >= _) => linear_arithmetic : core.
Hint Extern 1 (_ <> _) => linear_arithmetic : core. Local Hint Extern 1 (_ <> _) => linear_arithmetic : core.
(* We will need to do some bookkeeping of [n] values. This function is the (* 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. * trick, as we only need to skip steps based on removing [If]s from the code.
@ -817,7 +817,7 @@ Proof.
induct 1; simplify; eauto. induct 1; simplify; eauto.
Qed. Qed.
Hint Resolve plug_cfold1 : core. Local Hint Resolve plug_cfold1 : core.
Lemma plug_samefold : forall C c1 c1', Lemma plug_samefold : forall C c1 c1',
plug C c1 c1' plug C c1 c1'
@ -829,7 +829,7 @@ Proof.
f_equal; eauto. f_equal; eauto.
Qed. Qed.
Hint Resolve plug_samefold : core. Local Hint Resolve plug_samefold : core.
Lemma plug_countIfs : forall C c1 c1', Lemma plug_countIfs : forall C c1 c1',
plug C c1 c1' plug C c1 c1'
@ -841,13 +841,13 @@ Proof.
apply IHplug in H5; linear_arithmetic. apply IHplug in H5; linear_arithmetic.
Qed. Qed.
Hint Resolve plug_countIfs : core. Local Hint Resolve plug_countIfs : core.
Hint Extern 1 (interp ?e _ = _) => Local Hint Extern 1 (interp ?e _ = _) =>
match goal with match goal with
| [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H | [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H
end : core. end : core.
Hint Extern 1 (interp ?e _ <> _) => Local Hint Extern 1 (interp ?e _ <> _) =>
match goal with match goal with
| [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H | [ H : cfoldArith e = _ |- _ ] => rewrite <- cfoldArith_ok; rewrite H
end : core. end : core.
@ -1320,7 +1320,7 @@ Proof.
first_order. first_order.
Qed. Qed.
Hint Resolve agree_add agree_add_tempVar_fwd agree_add_tempVar_bwd agree_add_tempVar_bwd_prime agree_refl : core. Local 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. *) (* And here are two more unremarkable lemmas. *)
@ -1333,7 +1333,7 @@ Proof.
eauto 6. eauto 6.
Qed. Qed.
Hint Resolve silent_csteps_front : core. Local Hint Resolve silent_csteps_front : core.
Lemma tempVar_contra : forall n1 n2, Lemma tempVar_contra : forall n1 n2,
tempVar n1 = tempVar n2 tempVar n1 = tempVar n2
@ -1344,7 +1344,7 @@ Proof.
first_order. first_order.
Qed. Qed.
Hint Resolve tempVar_contra : core. Local Hint Resolve tempVar_contra : core.
Lemma self_prime_contra : forall s, Lemma self_prime_contra : forall s,
(s ++ "'")%string = s -> False. (s ++ "'")%string = s -> False.
@ -1352,7 +1352,7 @@ Proof.
induct s; simplify; equality. induct s; simplify; equality.
Qed. Qed.
Hint Resolve self_prime_contra : core. Local Hint Resolve self_prime_contra : core.
(* We've now proved all properties of [tempVar] that we need, so let's ask Coq (* 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. *) * not to reduce applications of it anymore, to keep goals simpler. *)
@ -1560,7 +1560,7 @@ Proof.
induct 1; bool; auto. induct 1; bool; auto.
Qed. Qed.
Hint Immediate noUnderscore_plug : core. Local Hint Immediate noUnderscore_plug : core.
Lemma silent_csteps_plug : forall C c1 c1', Lemma silent_csteps_plug : forall C c1 c1',
plug C c1 c1' plug C c1 c1'
@ -1571,7 +1571,7 @@ Proof.
induct 1; invert 1; eauto. induct 1; invert 1; eauto.
Qed. Qed.
Hint Resolve silent_csteps_plug : core. Local Hint Resolve silent_csteps_plug : core.
Fixpoint flattenContext (C : context) : context := Fixpoint flattenContext (C : context) : context :=
match C with match C with
@ -1585,7 +1585,7 @@ Proof.
induct 1; simplify; eauto. induct 1; simplify; eauto.
Qed. Qed.
Hint Resolve plug_flatten : core. Local Hint Resolve plug_flatten : core.
Lemma plug_total : forall c C, exists c', plug C c c'. Lemma plug_total : forall c C, exists c', plug C c c'.
Proof. Proof.
@ -1604,7 +1604,7 @@ Proof.
eauto. eauto.
Qed. Qed.
Hint Resolve plug_cstep : core. Local Hint Resolve plug_cstep : core.
Lemma step0_noUnderscore : forall v c l v' c', Lemma step0_noUnderscore : forall v c l v' c',
step0 (v, c) l (v', c') step0 (v, c) l (v', c')
@ -1616,7 +1616,7 @@ Proof.
reflexivity. reflexivity.
Qed. Qed.
Hint Resolve step0_noUnderscore : core. Local Hint Resolve step0_noUnderscore : core.
Fixpoint noUnderscoreContext (C : context) : bool := Fixpoint noUnderscoreContext (C : context) : bool :=
match C with match C with
@ -1643,7 +1643,7 @@ Proof.
rewrite H4, H3; reflexivity. rewrite H4, H3; reflexivity.
Qed. Qed.
Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd : core. Local Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd : core.
(* Finally, the main correctness theorem. *) (* Finally, the main correctness theorem. *)
Lemma flatten_ok : forall v c, Lemma flatten_ok : forall v c,

36
CompilerCorrectness_template.v
View file

@ -101,7 +101,7 @@ Inductive generate : valuation * cmd -> list (option nat) -> Prop :=
-> generate vc' ns -> generate vc' ns
-> generate vc (Some n :: ns). -> generate vc (Some n :: ns).
Hint Constructors plug step0 cstep generate : core. Local Hint Constructors plug step0 cstep generate : core.
Definition traceInclusion (vc1 vc2 : valuation * cmd) := Definition traceInclusion (vc1 vc2 : valuation * cmd) :=
forall ns, generate vc1 ns -> generate vc2 ns. forall ns, generate vc1 ns -> generate vc2 ns.
@ -131,8 +131,8 @@ Example month_boundaries_in_days :=
done done
done. done.
Hint Extern 1 (interp _ _ = _) => simplify; equality : core. Local Hint Extern 1 (interp _ _ = _) => simplify; equality : core.
Hint Extern 1 (interp _ _ <> _) => simplify; equality : core. Local Hint Extern 1 (interp _ _ <> _) => simplify; equality : core.
Theorem first_few_values : Theorem first_few_values :
generate ($0, month_boundaries_in_days) [Some 28; Some 56]. generate ($0, month_boundaries_in_days) [Some 28; Some 56].
@ -251,7 +251,7 @@ Proof.
equality. equality.
Qed. Qed.
Hint Resolve peel_cseq : core. Local Hint Resolve peel_cseq : core.
Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2 Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2
-> forall l vc1, step0 (v, c1) l vc1 -> forall l vc1, step0 (v, c1) l vc1
@ -439,7 +439,7 @@ Proof.
invert H4. invert H4.
Qed. Qed.
Hint Resolve silent_generate_fwd silent_generate_bwd generate_Skip : core. Local Hint Resolve silent_generate_fwd silent_generate_bwd generate_Skip : core.
Section simulation_skipping. Section simulation_skipping.
Variable R : nat -> valuation * cmd -> valuation * cmd -> Prop. Variable R : nat -> valuation * cmd -> valuation * cmd -> Prop.
@ -593,9 +593,9 @@ Section simulation_skipping.
Qed. Qed.
End simulation_skipping. End simulation_skipping.
Hint Extern 1 (_ < _) => linear_arithmetic : core. Local Hint Extern 1 (_ < _) => linear_arithmetic : core.
Hint Extern 1 (_ >= _) => linear_arithmetic : core. Local Hint Extern 1 (_ >= _) => linear_arithmetic : core.
Hint Extern 1 (_ <> _) => linear_arithmetic : core. Local Hint Extern 1 (_ <> _) => linear_arithmetic : core.
Lemma cfold_ok : forall v c, Lemma cfold_ok : forall v c,
(v, c) =| (v, cfold c). (v, c) =| (v, cfold c).
@ -1062,7 +1062,7 @@ Proof.
first_order. first_order.
Qed. Qed.
Hint Resolve agree_add agree_add_tempVar_fwd agree_add_tempVar_bwd agree_add_tempVar_bwd_prime agree_refl : core. Local Hint Resolve agree_add agree_add_tempVar_fwd agree_add_tempVar_bwd agree_add_tempVar_bwd_prime agree_refl : core.
Lemma silent_csteps_front : forall c v1 v2 c1 c2, Lemma silent_csteps_front : forall c v1 v2 c1 c2,
silent_cstep^* (v1, c1) (v2, c2) silent_cstep^* (v1, c1) (v2, c2)
@ -1073,7 +1073,7 @@ Proof.
eauto 6. eauto 6.
Qed. Qed.
Hint Resolve silent_csteps_front : core. Local Hint Resolve silent_csteps_front : core.
Lemma tempVar_contra : forall n1 n2, Lemma tempVar_contra : forall n1 n2,
tempVar n1 = tempVar n2 tempVar n1 = tempVar n2
@ -1084,7 +1084,7 @@ Proof.
first_order. first_order.
Qed. Qed.
Hint Resolve tempVar_contra : core. Local Hint Resolve tempVar_contra : core.
Lemma self_prime_contra : forall s, Lemma self_prime_contra : forall s,
(s ++ "'")%string = s -> False. (s ++ "'")%string = s -> False.
@ -1092,7 +1092,7 @@ Proof.
induct s; simplify; equality. induct s; simplify; equality.
Qed. Qed.
Hint Resolve self_prime_contra : core. Local Hint Resolve self_prime_contra : core.
Opaque tempVar. Opaque tempVar.
@ -1270,7 +1270,7 @@ Proof.
induct 1; bool; auto. induct 1; bool; auto.
Qed. Qed.
Hint Immediate noUnderscore_plug. Local Hint Immediate noUnderscore_plug.
Lemma silent_csteps_plug : forall C c1 c1', Lemma silent_csteps_plug : forall C c1 c1',
plug C c1 c1' plug C c1 c1'
@ -1281,7 +1281,7 @@ Proof.
induct 1; invert 1; eauto. induct 1; invert 1; eauto.
Qed. Qed.
Hint Resolve silent_csteps_plug. Local Hint Resolve silent_csteps_plug.
Fixpoint flattenContext (C : context) : context := Fixpoint flattenContext (C : context) : context :=
match C with match C with
@ -1295,7 +1295,7 @@ Proof.
induct 1; simplify; eauto. induct 1; simplify; eauto.
Qed. Qed.
Hint Resolve plug_flatten. Local Hint Resolve plug_flatten.
Lemma plug_total : forall c C, exists c', plug C c c'. Lemma plug_total : forall c C, exists c', plug C c c'.
Proof. Proof.
@ -1314,7 +1314,7 @@ Proof.
eauto. eauto.
Qed. Qed.
Hint Resolve plug_cstep. Local Hint Resolve plug_cstep.
Lemma step0_noUnderscore : forall v c l v' c', Lemma step0_noUnderscore : forall v c l v' c',
step0 (v, c) l (v', c') step0 (v, c) l (v', c')
@ -1326,7 +1326,7 @@ Proof.
reflexivity. reflexivity.
Qed. Qed.
Hint Resolve step0_noUnderscore. Local Hint Resolve step0_noUnderscore.
Fixpoint noUnderscoreContext (C : context) : bool := Fixpoint noUnderscoreContext (C : context) : bool :=
match C with match C with
@ -1353,7 +1353,7 @@ Proof.
rewrite H4, H3; reflexivity. rewrite H4, H3; reflexivity.
Qed. Qed.
Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd. Local Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd.
Lemma flatten_ok : forall v c, Lemma flatten_ok : forall v c,
noUnderscore c = true noUnderscore c = true

1
frap_book.tex
View file

@ -2758,7 +2758,6 @@ Therefore, a very regular kind of \emph{simulation relation} connects them.
\end{definition} \end{definition}
The crucial second condition can be drawn like this. The crucial second condition can be drawn like this.
\[ \[
\begin{tikzcd} \begin{tikzcd}
s_1 \arrow{r}{R} \arrow{d}{\forall \stackrel{\ell}{\to_{\mathsf{c}}}} & s_2 \arrow{d}{\exists \stackrel{\ell}{\to_{\mathsf{c}}}} \\ s_1 \arrow{r}{R} \arrow{d}{\forall \stackrel{\ell}{\to_{\mathsf{c}}}} & s_2 \arrow{d}{\exists \stackrel{\ell}{\to_{\mathsf{c}}}} \\