HoareAsLogicHoare Logic as a Logic

(* $Date: 2011-04-16 14:56:54 -0400 (Sat, 16 Apr 2011) $ *)

Require Export Hoare.

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.

Inductive hoare_proof : Assertion com Assertion Type :=
  | H_Skip : P,
      hoare_proof P (SKIP) P
  | H_Asgn : Q V a,
      hoare_proof (assn_sub V a Q) (V ::= a) Q
  | H_Seq : P c Q d R,
      hoare_proof P c Q hoare_proof Q d R hoare_proof P (c;d) R
  | H_If : P Q b c1 c2,
    hoare_proof (fun st => P st bassn b st) c1 Q
    hoare_proof (fun st => P st ~(bassn b st)) c2 Q
    hoare_proof P (IFB b THEN c1 ELSE c2 FI) Q
  | H_While : P b c,
    hoare_proof (fun st => P st bassn b st) c P
    hoare_proof P (WHILE b DO c END) (fun st => P st ~ (bassn b st))
  | H_Consequence : (P Q P' Q' : Assertion) c,
    hoare_proof P' c Q'
    (st, P st P' st)
    (st, Q' st Q st)
    hoare_proof P c Q
  | H_Consequence_pre : (P Q P' : Assertion) c,
    hoare_proof P' c Q
    (st, P st P' st)
    hoare_proof P c Q
  | H_Consequence_post : (P Q Q' : Assertion) c,
    hoare_proof P c Q'
    (st, Q' st Q st)
    hoare_proof P c Q.

Tactic Notation "hoare_proof_cases" tactic(first) ident(c) :=
  [ Case_aux c "H_Skip" | Case_aux c "H_Asgn" | Case_aux c "H_Seq"
  | Case_aux c "H_If" | Case_aux c "H_While" | Case_aux c "H_Consequence"
  | Case_aux c "H_Consequence_pre" | Case_aux c "H_Consequence_post" ].

For example, let's construct a proof object representing a derivation for the hoare triple
      {{assn_sub X (X+1) (assn_sub X (X+2) (X=3))}} X::=X+1; X::=X+2 {{X=3}}.
We can use Coq's tactics to help us construct the proof object.

Example sample_proof
             : hoare_proof
                 (assn_sub X (APlus (AId X) (ANum 1))
                   (assn_sub X (APlus (AId X) (ANum 2))
                     (fun st => st X = 3) ))
                 (X ::= APlus (AId X) (ANum 1); (X ::= APlus (AId X) (ANum 2)))
                 (fun st => st X = 3).
  apply H_Seq with (assn_sub X (APlus (AId X) (ANum 2))
                     (fun st => st X = 3)).
  apply H_Asgn. apply H_Asgn.

Print sample_proof.
    (assn_sub X (APlus (AId X) (ANum 1))
       (assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3)))
    (X ::= APlus (AId X) (ANum 1))
    (assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3))
    (X ::= APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3)
       (assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3))
       X (APlus (AId X) (ANum 1)))
    (H_Asgn (fun st : state => st X = VNat 3) X (APlus (AId X) (ANum 2)))

Exercise: 2 stars

Prove that such proof objects represent true claims.

Theorem hoare_proof_sound : P c Q,
  hoare_proof P c Q {{P}} c {{Q}}.
  (* FILL IN HERE *) Admitted.
We can also use Coq's reasoning facilities to prove metatheorems about Hoare Logic. For example, here are the analogs of two theorems we saw in chapter Hoare — this time expressed in terms of the syntax of Hoare Logic derivations (provability) rather than directly in terms of the semantics of Hoare triples.
The first one says that, for every P and c, the assertion {{P}} c {{True}} is provable in Hoare Logic. Note that the proof is more complex than the semantic proof in Hoare: we actually need to perform an induction over the structure of the command c.

Theorem H_Post_True_deriv:
  c P, hoare_proof P c (fun _ => True).
  intro c.
  com_cases (induction c) Case; intro P.
  Case "SKIP".
    eapply H_Consequence_pre.
    apply H_Skip.
    (* Proof of True *)
    intros. apply I.
  Case "::=".
    eapply H_Consequence_pre.
    apply H_Asgn.
    intros. apply I.
  Case ";".
    eapply H_Consequence_pre.
    eapply H_Seq.
    apply (IHc1 (fun _ => True)).
    apply IHc2.
    intros. apply I.
  Case "IFB".
    apply H_Consequence_pre with (fun _ => True).
    apply H_If.
    apply IHc1.
    apply IHc2.
    intros. apply I.
  Case "WHILE".
    eapply H_Consequence.
    eapply H_While.
    eapply IHc.
    intros; apply I.
    intros; apply I.

Similarly, we can show that {{False}} c {{Q}} is provable for any c and Q.

Lemma False_and_P_imp: P Q,
  False P Q.
  intros P Q [CONTRA HP].
  destruct CONTRA.

Tactic Notation "pre_false_helper" constr(CONSTR) :=
  eapply H_Consequence_pre;
    [eapply CONSTR | intros ? CONTRA; destruct CONTRA].

Theorem H_Pre_False_deriv:
  c Q, hoare_proof (fun _ => False) c Q.
  intros c.
  com_cases (induction c) Case; intro Q.
  Case "SKIP". pre_false_helper H_Skip.
  Case "::=". pre_false_helper H_Asgn.
  Case ";". pre_false_helper H_Seq. apply IHc1. apply IHc2.
  Case "IFB".
    apply H_If; eapply H_Consequence_pre.
    apply IHc1. intro. eapply False_and_P_imp.
    apply IHc2. intro. eapply False_and_P_imp.
  Case "WHILE".
    eapply H_Consequence_post.
    eapply H_While.
    eapply H_Consequence_pre.
      apply IHc.
      intro. eapply False_and_P_imp.
    intro. simpl. eapply False_and_P_imp.

This style of presentation gives a clearer picture of what it means to "give a proof in Hoare logic." However, it is not entirely satisfactory from the point of view of writing down such proofs in practice: it is quite verbose. The section of chapter Hoare on formalizing decorated programs shows how we can do even better.