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INTUITIONISTIC SEPARATION LOGIC. For reasoning about languages with explicit deallocation, one wants rules such as {p x}free(p){emp}, and one often thinks of an assertion Q as holding on a heaplet with a precise domain; this is called classical separation logic. For reasoning about languages with automatic garbage collection, there is no rule for free, and one often thinks of Q as holding on any heaplet with at least a certain domain; this is called intuitionistic separation logic. In the intuitionistic style, one has Q ∗ true = Q, which is not true in classical style; and emp is not a useful concept in the intuitionistic style (because it is equivalent to true).
SOUNDNESS OF HOARE LOGIC 31 Lemma floyd-assign: ∀ (P: h-assert) (x: var) (e: expr), (∀ ρ , P ρ → ∃ v, eval e v ρ ) → Hoare P (Cassign x e) (fun ρ ⇒ ∃ v, ∃ v’, eval (Evar x) v ρ ∧ subst x v’ (eval e v) ρ ∧ subst x v’ P ρ ). Compare this lemma to Floyd’s assignment rule shown on page 29. To prove it, we unfold the definition of Hoare and guard to obtain the proof goal, P : h-assert x : var e : expr H : ∀ ρ : stack, P ρ → ∃ v : val, eval e v ρ k : command H0 : guard (fun ρ : stack ⇒ ∃ v : val, ∃ v’ : option val, eval (Evar x) v ρ ∧ subst x v’ (eval e v) ρ ∧ subst x v’ P ρ ) k ρ : stack H1 : P ρ --------------------------------------(1/1) safe (State ρ m0 (Cseq (Cassign x e) k)) Now, we can use H ρ H1 to find v such that ρ ⊢ e ⇓ v.
That is, the chain of list cells x y is the representation in memory of σ. listrep σ (x, y) = (x = y ∧ σ = ε ∧ emp) ∨(x ̸= y ∧ ∃σ′ ∃h ∃t. next t ∗ listrep σ′ (t, y)) σ y. We will notate listrep σ (x, y) as x Now we are ready to prove the list-reversal program. next = w; w = v; v = t; } rev σ assert{w 0} As usual in Hoare logic, we need a loop invariant: ∃σ1 , σ2 . σ = rev(σ1 ) · σ2 ∧ v σ2 0∗w σ1 0 This separation-logic formula describes w v the picture in which the original sequence σ can be viewed as the concatenation of some σ1 (reversed) and some σ2 —we use · to denote se- 3.