HoareAsLogicHoare Logic as a Logic

The presentation of Hoare logic in chapter Hoare could be described as "model-theoretic": the proof rules for each of the constructors were presented as theorems about the evaluation behavior of programs, and proofs of program correctness (validity of Hoare triples) were constructed by combining these theorems directly in Coq.
Another way of presenting Hoare logic is to define a completely separate proof system -- a set of axioms and inference rules that talk about commands, Hoare triples, etc. -- and then say that a proof of a Hoare triple is a valid derivation in that logic. We can do this by giving an inductive definition of valid derivations in this new logic.
This chapter is optional. Before reading it, you'll want to read the ProofObjects chapter in Logical Foundations (Software Foundations, volume 1).
Set Warnings "-deprecated-hint-without-locality,-deprecated-hint-without-locality".
From PLF Require Import Maps.
From PLF Require Import Hoare.
Hint Constructors ceval : core.

Hoare Logic and Model Theory

In Hoare we introduced Hoare triples, which contain a precondition, command, and postcondition. For example (and for the moment deliberately avoiding the notation we previously introduced),
      Pre: X = 0
      Com: X := X + 1
      Post: X = 1
is a Hoare triple, as is
      Pre: X = 0
      Com: skip
      Post: X = 1
But there's an important difference between those two triples: the former expresses a truth about how Imp programs execute, whereas the latter does not.
To capture that difference, we introduced a definition hoare_triple that described when a triple expresses such a truth. Let's repeat that definition, but this time we'll call it valid:
Definition valid (P : Assertion) (c : com) (Q : Assertion) : Prop :=
   st st',
     st =[ c ]=> st'
     P st
     Q st'.
This notion of validity is based on the underlying model of how Imp programs execute. That model itself is based on states. So,
      Pre: X = 0
      Com: X := X + 1
      Post: X = 1
is valid, because starting from any state in which X is 0, and executing X := X + 1, we are guaranteed to reach a state in which X is 1. But,
      Pre: X = 0
      Com: skip
      Post: X = 1
is invalid, because starting from any state in which X is 0, we are guaranteed not to change the state, so X cannot be 1.
So far, we have punned between the syntax of a Hoare triple, written {{P}} c {{Q}}, and its validity, as expressed by valid. In essence, we have said that the semantic meaning of that syntax is the proposition returned by valid. This way of giving semantic meaning to something syntactic is part of the branch of mathematical logic known as model theory.
Our approach to Hoare logic through model theory led us to state proof rules in terms of that same state-based model, and to prove program correctness in it, too. But there is another approach, which is arguably more common in Hoare logic. We turn to it, next.

Hoare Logic and Proof Theory

Instead of using states and evaluation as the basis for reasoning, let's take the proof rules from Hoare as the basis. Those proof rules give us a set of axioms and inference rules that constitute a logic in their own right. We repeat them here:

             ---------------- (hoare_skip)
             {{P}} skip {{P}}

             ----------------------------- (hoare_asgn)
             {{Q [X > a]}} X := a {{Q}}

               {{P}} c1 {{Q}}
               {{Q}} c2 {{R}}
              ------------------ (hoare_seq)
              {{P}} c1; c2 {{R}}

              {{Pb}} c1 {{Q}}
              {{P ∧ ¬b}} c2 {{Q}}
      ------------------------------------ (hoare_if)
      {{P}} if b then c1 else c2 end {{Q}}

            {{Pb}} c {{P}}
      ----------------------------- (hoare_while)
      {{P} while b do c end {{P ∧ ¬b}}

                {{P'}} c {{Q'}}
                   P ->> P'
                   Q' ->> Q
         ----------------------------- (hoare_consequence)
                {{P}} c {{Q}}
Read the Hoare triples in those rules as devoid of any meaning other than what the rules give them. Forget about states and evaluations. They are just syntax that the rules tell us how to manipulate in legal ways.
Through this new lens, triple {{X = 0}} X := X + 1 {{X = 1}} is derivable, because we can derive a proof tree using the rules:

                    --------------------------- (hoare_asgn)
   X=0 ->> X+1=1 {{X+1=1}} X := X+1 {{X=1}}
   ------------------------------------------------------- (hoare_consequence)
                     {{X=0}} X := X+1 {{X=1}}
At each step we have either used one of the rules, or we have appealed to reasoning about assertions, which do not involve Hoare triples. (Note that we have left off the trivial part of hoare_consequence above, namely X=1 ->> X=1, only because of horizontal space contraints: it's hard to fit that many characters on a line and have the page still be readable. If you prefer, think of it as using hoare_consequence_pre instead.)
On the other hand, {{X = 0}} skip {{X = 1}} is not derivable, because there is no way to apply the rules to construct a proof tree with this triple at its root.
This approach gives meaning to triples not in terms of a model, but in terms of how they can be used to construct proof trees. It's a different way of giving semantic meaning to something syntactic, and it's part of the branch of mathematical logic known as proof theory.
Our goal for the rest of this chapter is to formalize Hoare logic using proof theory, and then to prove that the model-theoretic and proof-theoretic formalizations are consistent with one another.

Derivability

To formalize derivability of Hoare triples, we introduce inductive type derivable, which describes legal proof trees using the Hoare rules.
Inductive derivable : Assertion com Assertion Type :=
  | H_Skip : P,
      derivable P <{skip}> P
  | H_Asgn : Q V a,
      derivable (Q [V > a]) <{V := a}> Q
  | H_Seq : P c Q d R,
      derivable P c Q derivable Q d R derivable P <{c;d}> R
  | H_If : P Q b c1 c2,
    derivable (fun stP st bassn b st) c1 Q
    derivable (fun stP st ~(bassn b st)) c2 Q
    derivable P <{if b then c1 else c2 end}> Q
  | H_While : P b c,
    derivable (fun stP st bassn b st) c P
    derivable P <{while b do c end}> (fun stP st ¬ (bassn b st))
  | H_Consequence : (P Q P' Q' : Assertion) c,
    derivable P' c Q'
    ( st, P st P' st)
    ( st, Q' st Q st)
    derivable P c Q.
We don't need to include axioms corresponding to hoare_consequence_pre or hoare_consequence_post, because these can be proven easily from H_Consequence.
Lemma H_Consequence_pre : (P Q P': Assertion) c,
    derivable P' c Q
    ( st, P st P' st)
    derivable P c Q.
Proof. eauto using H_Consequence. Qed.

Lemma H_Consequence_post : (P Q Q' : Assertion) c,
    derivable P c Q'
    ( st, Q' st Q st)
    derivable P c Q.
Proof. eauto using H_Consequence. Qed.
As an example, let's construct a proof tree for
        {{(X=3) [X > X + 2] [X > X + 1]}}
      X := X + 1;
      X := X + 2
        {{X=3}}
Example sample_proof :
  derivable
    ((fun st:statest X = 3) [X > X + 2] [X > X + 1])
    <{ X := X + 1; X := X + 2}>
    (fun st:statest X = 3).
Proof.
  eapply H_Seq.
  - apply H_Asgn.
  - apply H_Asgn.
Qed.
You can see how the structure of the proof script mirrors the structure of the proof tree: at the root there is a use of the sequence rule; and at the leaves, the assignment rule.

Exercise: 3 stars, standard (provable_true_post)

Show that any Hoare triple whose postcondition is True is derivable. Proceed by induction on c.
Theorem provable_true_post : c P,
    derivable P c True.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 3 stars, standard (provable_false_pre)

Show that any Hoare triple whose precondition is False is derivable. Again, proceed by induction on c.
Theorem provable_false_pre : c Q,
    derivable False c Q.
Proof.
  (* FILL IN HERE *) Admitted.

Soundness and Completeness

We now have two approaches to formulating Hoare logic:
  • The model-theoretic approach uses valid to characterize when a Hoare triple holds in a model, which is based on states.
  • The proof-theoretic approach uses derivable to characterize when a Hoare triple is derivable as the end of a proof tree.
Do these two approaches agree? That is, are the valid Hoare triples exactly the derivable ones? This is a standard question investigated in mathematical logic. There are two pieces to answering it:
  • A logic is sound if everything that is derivable is valid.
  • A logic is complete if everything that is valid is derivable.
We can prove that Hoare logic is sound and complete.

Exercise: 3 stars, standard (hoare_sound)

Prove that if a Hoare triple is derivable, then it is valid. Nearly all the work for this was already done in Hoare as theorems hoare_skip, hoare_asgn, etc.; leverage those proofs. Proceed by induction on the derivation of the triple.
Theorem hoare_sound : P c Q,
  derivable P c Q valid P c Q.
Proof.
  (* FILL IN HERE *) Admitted.
The proof of completeness is more challenging. To carry out the proof, we need to invent some intermediate assertions using a technical device known as weakest preconditions (which are also discussed in Hoare2). Given a command c and a desired postcondition assertion Q, the weakest precondition wp c Q is an assertion P such that {{P}} c {{Q}} holds, and moreover, for any other assertion P', if {{P'}} c {{Q}} holds then P' ->> P.
Another way of stating that idea is that wp c Q is the following assertion:
Definition wp (c:com) (Q:Assertion) : Assertion :=
  fun s s', s =[ c ]=> s' Q s'.

Hint Unfold wp : core.
The following two theorems show that the two ways of thinking about wp are the same.
Theorem wp_is_precondition : c Q,
  {{wp c Q}} c {{Q}}.
Proof. auto. Qed.

Theorem wp_is_weakest : c Q P',
    {{P'}} c {{Q}}
    P' ->> (wp c Q).
Proof. eauto. Qed.
Weakest preconditions are useful because they enable us to identify assertions that otherwise would require clever thinking. The next two lemmas show that in action.

Exercise: 1 star, standard (wp_seq)

What if we have a sequence c1; c2, but not an intermediate assertion for what should hold in between c1 and c2? No problem. Prove that wp c2 Q suffices as such an assertion.
Lemma wp_seq : P Q c1 c2,
    derivable P c1 (wp c2 Q) derivable (wp c2 Q) c2 Q derivable P <{c1; c2}> Q.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars, standard (wp_invariant)

What if we have a while loop, but not an invariant for it? No problem. Prove that for any Q, assertion wp (while b do c end) Q is a loop invariant of while b do c end.
Lemma wp_invariant : b c Q,
    valid (wp <{while b do c end}> Q b) c (wp <{while b do c end}> Q).
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 4 stars, standard (hoare_complete)

Now we are ready to prove the completeness of Hoare logic. Finish the proof of the theorem below.
Hint: for the while case, use the invariant suggested by wp_invariant.
Acknowledgment: Our approach to this proof is inspired by:
https://www.ps.uni-saarland.de/courses/sem-ws11/script/Hoare.html
Theorem hoare_complete: P c Q,
  valid P c Q derivable P c Q.
Proof.
  unfold valid. intros P c. generalize dependent P.
  induction c; intros P Q HT.
  (* FILL IN HERE *) Admitted.

Postscript: Decidability

We might hope that Hoare logic would be decidable; that is, that there would be an (terminating) algorithm (a decision procedure) that can determine whether or not a given Hoare triple is valid or derivable. Sadly, such a decision procedure cannot exist.
Consider the triple {{True}} c {{False}}. This triple is valid if and only if c is non-terminating. So any algorithm that could determine validity of arbitrary triples could solve the Halting Problem.
Similarly, the triple {{True}} skip {{P}} is valid if and only if s, P s is valid, where P is an arbitrary assertion of Coq's logic. But this logic is far too powerful to be decidable.
(* 2021-08-11 15:11 *)