Update for new Connecting chapter, modulo adding the LaTeX content

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Adam Chlipala 2018-05-02 11:56:01 -04:00
parent df4016a2c3
commit 369edcdd79
5 changed files with 114 additions and 50 deletions

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@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 16: Concurrent Separation Logic * Chapter 17: Concurrent Separation Logic
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

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@ -6,7 +6,6 @@
Require Import Frap SepCancel ModelCheck Classes.Morphisms. Require Import Frap SepCancel ModelCheck Classes.Morphisms.
Require Import Arith.Div2 Eqdep. Require Import Arith.Div2 Eqdep.
(** * Some odds and ends from past chapters *) (** * Some odds and ends from past chapters *)
Ltac simp := repeat (simplify; subst; propositional; Ltac simp := repeat (simplify; subst; propositional;
@ -15,12 +14,31 @@ Ltac simp := repeat (simplify; subst; propositional;
end); try linear_arithmetic. end); try linear_arithmetic.
(** * Orientation *)
(* We've now done plenty of Coq proofs that apply to idealizations of real-world
* programming languages. What happens when we want to connect to real
* development ecosystems? The corresponding book chapter works through several
* dimensions of variation across approaches. The whole subject is an active
* area of research, and there aren't standard solutions that everyone agrees
* on. The rest of this code file develops one avant-garde approach. *)
(** * Bitvectors of known length *) (** * Bitvectors of known length *)
(* One way we can increase realism is ditching the natural numbers for
* bitvectors of fixed size, as we find in, e.g., registers of most computer
* processors. A simple dependent type definition gets the job done. *)
Inductive word : nat -> Set := Inductive word : nat -> Set :=
| WO : word O | WO : word O
| WS : bool -> forall {n}, word n -> word (S n). | WS : bool -> forall {n}, word n -> word (S n).
(* The index of a [word] tells us how many bits it takes up. *)
(* Next come a set of operation definitions, whose details we will gloss
* over. *)
Fixpoint wordToNat {sz} (w : word sz) : nat := Fixpoint wordToNat {sz} (w : word sz) : nat :=
match w with match w with
| WO => O | WO => O
@ -44,11 +62,15 @@ Fixpoint natToWord (sz n : nat) : word sz :=
Definition wzero {sz} := natToWord sz 0. Definition wzero {sz} := natToWord sz 0.
Definition wone {sz} := natToWord sz 1. Definition wone {sz} := natToWord sz 1.
Notation "^" := (natToWord _) (at level 0). Notation "^" := (natToWord _) (at level 0).
(* Note this notation that we do use later, for "casting" a natural number into
* a word. (We might "lose" bits if the input number is too large!) *)
Definition wplus {sz} (w1 w2 : word sz) : word sz := Definition wplus {sz} (w1 w2 : word sz) : word sz :=
natToWord sz (wordToNat w1 + wordToNat w2). natToWord sz (wordToNat w1 + wordToNat w2).
Infix "^+" := wplus (at level 50, left associativity). Infix "^+" := wplus (at level 50, left associativity).
(* And we will also use this notation in the main development, as a binary
* addition operator. The remaining definitions are safe to gloss over. *)
Definition whd {sz} (w : word (S sz)) : bool := Definition whd {sz} (w : word (S sz)) : bool :=
match w in word sz' return match sz' with match w in word sz' return match sz' with
@ -148,6 +170,9 @@ Proof.
equality. equality.
Qed. Qed.
(* Oh, but pay attention to this one: much of our development will be
* paramterized over what word size to consider. Any module implementing this
* type explains one particular choice. *)
Module Type BIT_WIDTH. Module Type BIT_WIDTH.
Parameter bit_width : nat. Parameter bit_width : nat.
Axiom bit_width_nonzero : bit_width > 0. Axiom bit_width_nonzero : bit_width > 0.
@ -156,6 +181,10 @@ End BIT_WIDTH.
(** * A modification of last chapter's language, to use words instead of naturals *) (** * A modification of last chapter's language, to use words instead of naturals *)
(* There actually isn't much to say about this language and its separation
* logic. We are almost just copying and pasting word operations for [nat]
* operations. Also, we drop failure and dynamic memory allocation, since they
* would just distract from the main point. *)
Module MixedEmbedded(Import BW : BIT_WIDTH). Module MixedEmbedded(Import BW : BIT_WIDTH).
Definition wrd := word bit_width. Definition wrd := word bit_width.
@ -180,9 +209,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
| Write (a v : wrd) : cmd unit | Write (a v : wrd) : cmd unit
| Loop {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) : cmd acc. | Loop {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) : cmd acc.
(* We're leaving out allocation and deallocation, to avoid distraction from
* the main point of the examples to follow. *)
Notation "x <- c1 ; c2" := (Bind c1 (fun x => c2)) (right associativity, at level 80). Notation "x <- c1 ; c2" := (Bind c1 (fun x => c2)) (right associativity, at level 80).
Notation "'for' x := i 'loop' c1 'done'" := (Loop i (fun x => c1)) (right associativity, at level 80). Notation "'for' x := i 'loop' c1 'done'" := (Loop i (fun x => c1)) (right associativity, at level 80).
@ -217,37 +243,22 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
Step := (step (A := result))^* Step := (step (A := result))^*
|}. |}.
(* Now let's get into the first distinctive feature of separation logic: an
* assertion language that takes advantage of *partiality* of heaps. We give our
* definitions inside a module, which will shortly be used as a parameter to
* another module (from the book library), to get some free automation for
* implications between these assertions. *)
Module Import S <: SEP. Module Import S <: SEP.
Definition hprop := heap -> Prop. Definition hprop := heap -> Prop.
(* A [hprop] is a regular old assertion over heaps. *)
(* Implication *)
Definition himp (p q : hprop) := forall h, p h -> q h. Definition himp (p q : hprop) := forall h, p h -> q h.
(* Equivalence *)
Definition heq (p q : hprop) := forall h, p h <-> q h. Definition heq (p q : hprop) := forall h, p h <-> q h.
(* Lifting a pure proposition: it must hold, and the heap must be empty. *)
Definition lift (P : Prop) : hprop := Definition lift (P : Prop) : hprop :=
fun h => P /\ h = $0. fun h => P /\ h = $0.
(* Separating conjunction, one of the two big ideas of separation logic.
* When does [star p q] apply to [h]? When [h] can be partitioned into two
* subheaps [h1] and [h2], respectively compatible with [p] and [q]. See book
* module [Map] for definitions of [split] and [disjoint]. *)
Definition star (p q : hprop) : hprop := Definition star (p q : hprop) : hprop :=
fun h => exists h1 h2, split h h1 h2 /\ disjoint h1 h2 /\ p h1 /\ q h2. fun h => exists h1 h2, split h h1 h2 /\ disjoint h1 h2 /\ p h1 /\ q h2.
(* Existential quantification *)
Definition exis {A} (p : A -> hprop) : hprop := Definition exis {A} (p : A -> hprop) : hprop :=
fun h => exists x, p x h. fun h => exists x, p x h.
(* Convenient notations *)
Notation "[| P |]" := (lift P) : sep_scope. Notation "[| P |]" := (lift P) : sep_scope.
Infix "*" := star : sep_scope. Infix "*" := star : sep_scope.
Notation "'exists' x .. y , p" := (exis (fun x => .. (exis (fun y => p)) ..)) : sep_scope. Notation "'exists' x .. y , p" := (exis (fun x => .. (exis (fun y => p)) ..)) : sep_scope.
@ -257,9 +268,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
Local Open Scope sep_scope. Local Open Scope sep_scope.
(* And now we prove some key algebraic properties, whose details aren't so
* important. The library automation uses these properties. *)
Lemma iff_two : forall A (P Q : A -> Prop), Lemma iff_two : forall A (P Q : A -> Prop),
(forall x, P x <-> Q x) (forall x, P x <-> Q x)
-> (forall x, P x -> Q x) /\ (forall x, Q x -> P x). -> (forall x, P x -> Q x) /\ (forall x, Q x -> P x).
@ -355,7 +363,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
End S. End S.
Export S. Export S.
(* Instantiate our big automation engine to these definitions. *)
Module Import Se := SepCancel.Make(S). Module Import Se := SepCancel.Make(S).
Export Se. Export Se.
@ -436,11 +443,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
reflexivity. reflexivity.
Qed. Qed.
(* Now, we carry out a moderately laborious soundness proof! It's safe to skip
* ahead to the text "Examples", but a few representative lemma highlights
* include [invert_Read], [preservation], [progress], and the main theorem
* [hoare_triple_sound]. *)
Lemma invert_Return : forall {result : Set} (r : result) P Q, Lemma invert_Return : forall {result : Set} (r : result) P Q,
hoare_triple P (Return r) Q hoare_triple P (Return r) Q
-> forall h, P h -> Q r h. -> forall h, P h -> Q r h.
@ -582,7 +584,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
cancel. cancel.
Qed. Qed.
(* Temporarily transparent again! *)
Transparent heq himp lift star exis ptsto. Transparent heq himp lift star exis ptsto.
Lemma preservation : forall {result} (c : cmd result) h c' h', Lemma preservation : forall {result} (c : cmd result) h c' h',
@ -766,7 +767,6 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
cases s; simp; eauto. cases s; simp; eauto.
Qed. Qed.
(* Fancy theorem to help us rewrite within preconditions and postconditions *)
Instance hoare_triple_morphism : forall A, Instance hoare_triple_morphism : forall A,
Proper (heq ==> eq ==> (eq ==> heq) ==> iff) (@hoare_triple A). Proper (heq ==> eq ==> (eq ==> heq) ==> iff) (@hoare_triple A).
Proof. Proof.
@ -791,7 +791,7 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
Qed. Qed.
(** * Examples *) (** * Examples, starting with reusable tactics *)
Global Opaque heq himp lift star exis ptsto. Global Opaque heq himp lift star exis ptsto.
@ -942,7 +942,13 @@ Module MixedEmbedded(Import BW : BIT_WIDTH).
End MixedEmbedded. End MixedEmbedded.
(** * A simple C-like language with just two data types *) (** * A simple C-like language, embedded deeply *)
(* In [DeepAndShallowEmbeddings], we saw how to extract programs from the last
* language to OCaml and run them with an interpreter. That interpreter needs
* to be trusted, and its performance isn't so great. It could be better to
* generate C-like syntax trees in Coq and output them directly. We will use
* this next language to that end. *)
Module DeeplyEmbedded(Import BW : BIT_WIDTH). Module DeeplyEmbedded(Import BW : BIT_WIDTH).
Definition wrd := word bit_width. Definition wrd := word bit_width.
@ -1017,8 +1023,13 @@ Module DeeplyEmbedded(Import BW : BIT_WIDTH).
Step := step Step := step
|}. |}.
(** ** Printing as C code *) (** ** Printing as C code *)
(* Here we have the pay-off: even within Coq, it is easy to print these syntax
* trees as normal C (concrete) syntax. The functions speak for
* themselves! *)
Fixpoint wordS {sz} (w : word sz) : string := Fixpoint wordS {sz} (w : word sz) : string :=
match w with match w with
| WO => "" | WO => ""
@ -1063,10 +1074,19 @@ End DeeplyEmbedded.
(** * Connecting the two *) (** * Connecting the two *)
(* Reasoning about the mixed-embedding language is much more pleasant than for
* the deep-embedding language. Let's implement a verified translation from the
* former to the latter. The translation will be an inductive judgment. *)
Module MixedToDeep(Import BW : BIT_WIDTH). Module MixedToDeep(Import BW : BIT_WIDTH).
Module Import DE := DeeplyEmbedded(BW). Module Import DE := DeeplyEmbedded(BW).
Module Import ME := MixedEmbedded(BW). Module Import ME := MixedEmbedded(BW).
(* Key insight: we translate with respect to a valuation [V], telling us the
* values of the variables in the deep-embedding world. When we hit a value
* of the mixed-embedding world, one translation option is to find a variable
* known to hold that value, outputting that variable as the translation! *)
Inductive translate_exp (V : valuation) : forall {A}, A -> exp -> Prop := Inductive translate_exp (V : valuation) : forall {A}, A -> exp -> Prop :=
| TrAdd : forall (v1 v2 : wrd) e1 e2, | TrAdd : forall (v1 v2 : wrd) e1 e2,
translate_exp V v1 e1 translate_exp V v1 e1
@ -1077,16 +1097,12 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
-> translate_exp V v (Var x) -> translate_exp V v (Var x)
| TrConst : forall v, | TrConst : forall v,
translate_exp V v (Const v). translate_exp V v (Const v).
(* Something subtle happens in this last case. We can turn any value into a
* constant of the deep embedding? Sounds like a cop-out! See a note below
* on why the cop-out doesn't apply. *)
Fixpoint couldWrite (x : var) (s : stmt) : bool := (* Things get pretty intricate from here on, including with a weird sort of
match s with * polymorphism over relations. We will only comment on the main points. *)
| Assign y _ => if x ==v y then true else false
| Skip
| DE.Write _ _ => false
| Seq s1 s2
| IfThenElse _ s1 s2 => couldWrite x s1 || couldWrite x s2
| WhileLoop _ s1 => couldWrite x s1
end.
Inductive translate_result (V : valuation) (v : wrd) : stmt -> Prop := Inductive translate_result (V : valuation) (v : wrd) : stmt -> Prop :=
| TrReturn : forall e, | TrReturn : forall e,
@ -1122,6 +1138,7 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
| TwoWords => wrd * wrd | TwoWords => wrd * wrd
end%type. end%type.
(* This is the main relation for translating commands. *)
Inductive translate Inductive translate
: forall {rt}, (valuation -> rtt rt -> stmt -> Prop) : forall {rt}, (valuation -> rtt rt -> stmt -> Prop)
-> valuation -> forall {A}, cmd A -> stmt -> Prop := -> valuation -> forall {A}, cmd A -> stmt -> Prop :=
@ -1134,11 +1151,21 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
-> x <> "acc" -> x <> "acc"
-> translate_exp V v e -> translate_exp V v e
-> (forall w, translate translate_return (V $+ (x, w)) (c w) s1) -> (forall w, translate translate_return (V $+ (x, w)) (c w) s1)
(* ^-- Note this crucial case for translating [Bind]. We require that
* the translation of the body be correct for any possible value of the
* mixed-embedding variable, so long as we guarantee that the value is
* also stashed in deep-embedding variable [x] before proceeding! *)
-> translate translate_return V (Bind (Return v) c) (Seq (Assign x e) s1) -> translate translate_return V (Bind (Return v) c) (Seq (Assign x e) s1)
| TrAssigned : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (v : wrd) (c : wrd -> cmd B) x s1, | TrAssigned : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (v : wrd) (c : wrd -> cmd B) x s1,
V $? x = Some v V $? x = Some v
-> translate translate_return V (c v) s1 -> translate translate_return V (c v) s1
-> translate translate_return V (Bind (Return v) c) (Seq Skip s1) -> translate translate_return V (Bind (Return v) c) (Seq Skip s1)
(* ^-- Note also "extra" rules like this one, which won't be used in
* translating a command in the first place. Instead, they are only used to
* "strengthen the induction hypothesis" in the simulation proof we use to
* show soundness of translation. In other words, execution of
* mixed-embedding programs generates intermediate states (e.g., with "extra"
* [Skip]s) that we still need to relate to the deep embedding. *)
| TrRead : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (a : wrd) (c : wrd -> cmd B) e x s1, | TrRead : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (a : wrd) (c : wrd -> cmd B) e x s1,
V $? x = None V $? x = None
-> x <> "i" -> x <> "i"
@ -1154,6 +1181,10 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
| TrAssignedUnit : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (c : unit -> cmd B) s1, | TrAssignedUnit : forall rt (translate_return : valuation -> rtt rt -> stmt -> Prop) V B (c : unit -> cmd B) s1,
translate translate_return V (c tt) s1 translate translate_return V (c tt) s1
-> translate translate_return V (Bind (Return tt) c) (Seq Skip s1) -> translate translate_return V (Bind (Return tt) c) (Seq Skip s1)
(* Next, note that the [Loop] rules only apply to a restricted pattern, to
* simplify the formalism. The next rule is the one used in compilation,
* while the rest are only used internally in the soundness proof. *)
| TrLoop : forall V (initA initB : wrd) body {A} (c : wrd * wrd -> cmd A) ea s1 s2, | TrLoop : forall V (initA initB : wrd) body {A} (c : wrd * wrd -> cmd A) ea s1 s2,
V $? "i" = None V $? "i" = None
-> V $? "acc" = None -> V $? "acc" = None
@ -1276,6 +1307,8 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
c) c)
(Seq (Seq s' (WhileLoop (Var "i") s1)) s2). (Seq (Seq s' (WhileLoop (Var "i") s1)) s2).
(* Here are tactics to compile programs automatically. *)
Ltac freshFor vm k := Ltac freshFor vm k :=
let rec keepTrying x := let rec keepTrying x :=
let H := fresh in let H := fresh in
@ -1307,6 +1340,8 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
eapply TrLoop; [ simplify; equality | simplify; equality | | intros | intros ] eapply TrLoop; [ simplify; equality | simplify; equality | | intros | intros ]
end. end.
(** ** Some examples of compiling programs *)
Example adder (a b c : wrd) := Example adder (a b c : wrd) :=
Bind (Return (a ^+ b)) (fun ab => Return (ab ^+ c)). Bind (Return (a ^+ b)) (fun ab => Return (ab ^+ c)).
@ -1393,6 +1428,22 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
Definition reverse_alt_compiled := Eval simpl in proj1_sig translate_reverse_alt. Definition reverse_alt_compiled := Eval simpl in proj1_sig translate_reverse_alt.
(** ** Soundness proof *)
(* We omit explanation of most of these details, which get rather hairy.
* Also, these proof scripts are not exactly modeling best practices in
* automation. Maybe some day the author will be motivated to clean them
* up. *)
(* We do point out here that one recurring motif throughout the lemmas is
* taking a translation run and applying it in a *larger* valuation than was
* used as input. Intuitively, it is OK to run with extra variables around,
* if we don't actually read them. This opportunity is important for
* translated loop bodies, which, after the first loop iteration, get run with
* their own past variable settings still in place, even though the body
* provably never reads its own past settings of temporary variables. *)
Lemma eval_translate : forall H V V' e v, Lemma eval_translate : forall H V V' e v,
eval H (V $++ V') e v eval H (V $++ V') e v
-> forall (v' : wrd), translate_exp V v' e -> forall (v' : wrd), translate_exp V v' e
@ -1435,6 +1486,8 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
simplify; equality. simplify; equality.
Qed. Qed.
(* Complex statement here, really dealing mostly with the whole idea of
* ignoring parts of valuations! *)
Lemma step_translate_loop : forall V (c : cmd (wrd * wrd)) s, Lemma step_translate_loop : forall V (c : cmd (wrd * wrd)) s,
translate (rt := TwoWords) translate_loop_body V c s translate (rt := TwoWords) translate_loop_body V c s
-> forall H H' V' V'' s', -> forall H H' V' V'' s',
@ -2615,9 +2668,9 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
eassumption. eassumption.
eapply lookup_None_dom. eapply lookup_None_dom.
rewrite lookup_join2 by (eapply lookup_None_dom; simplify; eauto). rewrite lookup_join2 by (eapply lookup_None_dom; simplify; eauto).
eassumption. eassumption.
Unshelve. Grab Existential Variables.
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
@ -2636,6 +2689,9 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
Qed. Qed.
(* This is the invariant for simulation! Note the crucial addition of extra
* variables in the valuation. We carefully chose the two languages to treat
* the heap identically, so we can require heap equality. *)
Inductive translated : forall {A}, DE.heap * valuation * stmt -> ME.heap * cmd A -> Prop := Inductive translated : forall {A}, DE.heap * valuation * stmt -> ME.heap * cmd A -> Prop :=
| Translated : forall A H V V' s (c : cmd A), | Translated : forall A H V V' s (c : cmd A),
translate (rt := OneWord) translate_result V c s translate (rt := OneWord) translate_result V c s
@ -3107,7 +3163,7 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
eexists. eexists.
repeat econstructor. repeat econstructor.
Unshelve. Grab Existential Variables.
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
@ -3135,6 +3191,8 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
exact (^0) || exact (Return (^0)). exact (^0) || exact (Return (^0)).
Qed. Qed.
(* The main theorem! Prove a Hoare triple in the high-level language;
* conclude safe execution in the low-level language. *)
Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s H, Theorem hoare_triple_sound : forall P (c : cmd wrd) Q V s H,
hoare_triple P c Q hoare_triple P c Q
-> P H -> P H
@ -3170,6 +3228,9 @@ Module MixedToDeep(Import BW : BIT_WIDTH).
eauto. eauto.
Qed. Qed.
(** ** Applying the main theorem to the earlier examples *)
Theorem adder_ok : forall a b c, Theorem adder_ok : forall a b c,
{{emp}} {{emp}}
adder a b c adder a b c
@ -3277,6 +3338,8 @@ End MixedToDeep.
(** * Getting concrete *) (** * Getting concrete *)
(* Let's generate C code for the concrete bitwidth of 32. *)
Module Bw32. Module Bw32.
Definition bit_width := 32. Definition bit_width := 32.
Theorem bit_width_nonzero : bit_width > 0. Theorem bit_width_nonzero : bit_width > 0.

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@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 17: Process Algebra and Behavioral Refinement * Chapter 18: Process Algebra and Behavioral Refinement
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

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@ -26,9 +26,10 @@ The main narrative, also present in the book PDF, presents standard program-proo
* Chapter 12: `HoareLogic.v` * Chapter 12: `HoareLogic.v`
* Chapter 13: `DeepAndShallowEmbeddings.v` * Chapter 13: `DeepAndShallowEmbeddings.v`
* Chapter 14: `SeparationLogic.v` * Chapter 14: `SeparationLogic.v`
* Chapter 15: `SharedMemory.v` * Chapter 15: `Connecting.v`
* Chapter 16: `ConcurrentSeparationLogic.v` * Chapter 16: `SharedMemory.v`
* Chapter 17: `MessagesAndRefinement.v` * Chapter 17: `ConcurrentSeparationLogic.v`
* Chapter 18: `MessagesAndRefinement.v`
There are also two supplementary files that are independent of the main narrative, for introducing programming with dependent types, a distinctive Coq feature that we neither use nor recommend for the problem sets, but which many students find interesting (and useful in other contexts). There are also two supplementary files that are independent of the main narrative, for introducing programming with dependent types, a distinctive Coq feature that we neither use nor recommend for the problem sets, but which many students find interesting (and useful in other contexts).
* `SubsetTypes.v`: a first introduction to dependent types by attaching predicates to normal types (used after `CompilerCorrectness.v` in the last course offering) * `SubsetTypes.v`: a first introduction to dependent types by attaching predicates to normal types (used after `CompilerCorrectness.v` in the last course offering)

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@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 15: Operational Semantics for Shared-Memory Concurrency * Chapter 16: Operational Semantics for Shared-Memory Concurrency
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)