1. Before the loop \(i = 0\) and \(n = 10\), since that means \(i \leq n\) is true we move on to checking it holds true after each loop iteration. What we can observe is that the loop condition is \(i < n\) and during the loop we have \(i = i + 1\), what this suggests is that clearly during the iteratons the invariant holds, additionally when \(i = 10\) we have that \(10 < 0 \to \bot \) hence the invariant also holds upon the termination of the loop, i.e. after the final iteration, thus it's indeed a valid loop invariant.
2. No, this is not a valid invariant, we can see that its very close to being right, but it doesnt account for the case after the final iteration and before we check the loop condition, i.e. where \(i = 10\), hence its not a valid loop invariant because it does not hold true after each iteration.
3. No, this is quite easy to see as it immediately fails even before we get into the loop as \(0 \geq 10 \to \bot \).

As the usual piece of little assistance let's take a look at the while rule

1. Here we can try to construct the proof tree first of all, instantiating our rule with the given invariant we have: Using the assignment rule we then want to prove the antecident here, so instantiating this with or postcondition \(Q\) we have in other words our precondition becomes \[ (x + 1) \leq n \] Therefore the use the assignment rule we want we have to demonstrate \[ (x \leq n \land x < n \to (x + 1) \leq n) \] since we have that \[ x \leq n \land x < n \equiv x < n \to (x + 1) \leq n \] we clearly have that
2. Yes, let's again instantiate our rule since our postcondition for the assignmnet is a tautology, i.e. always valid, it will hold under substitution thus making the assignment trivially valid hence working as a correct loop invariant.
3. Something we could observe in question (1) is that so this is a bit of a trick question in that with question 1 we are actually just using a postcondition weaking on this postcondition, hence this is naturally a valid postcondition.

For reference lets pull up the while rule again:

One not entirely unreasonable albiet simple choice is to try out the post condition \(x \geq 0\), its not too uncommon that a valid postcondition - or more often a variant of it - are in fact valid loop invariants. Testing this out it means we would have to prove the body:

To see if this is inductive we can try and find a counterexample to demonstrate its not, and indeed we can see that \(\sigma = \{y \mapsto -1000, x \mapsto 1, n \mapsto 2 \}\) would mean that \(P(\sigma )\) would hold, and we could indeed step to a state \(\sigma '\) though clearly we have that:

Since we can see here that the unconstrained \(y\) is causing is issues lets go with:

Since this ensures that \(y\) is correctly related to \(x\) it means that we indeed have that \[ \sigma \vDash I (x := y; y := x + 1, \sigma ) \to \sigma ' \to \sigma ' \vDash I \] as desired

There are a few ways we can answer this, starting in with the most pedantic let's construct the proof tree and then see if we can come up with a counter example, we consider the following hoare triple:

By the sequence and precondition strengthening rule we have that

Of note here we are deriving \(i = 1\) through the following implications:

Where we know that \(i = 1 \to \top \) so we can use precondition strengthening to use \(i = 1\) as our precondition since it implies something we've already syntactically demonstrated to be provable.

Clearly we have that the first assignment in the sequence holds by the naive assignment, same for the second assignment \(a[1] = 2\) since it precondition reduces to the tautology \(\top \), so the final rule is then derivable. But we can obviously see that it erroneously assumes that \(i\) represents some kind of distinct index but, we can see that in the second assignment its aliased by the constant index \(1\) which means that while we can derive the rule it is not semantically valid hence unsound.

To compare this with the correct rule we have that:

Now again let's try to derive the hoare triple we want to prove correct:

To derive we want to find some assertion \(R\), applying the correct assignment rule to the second assignment we define \(R\) as.

we can think of this as defining the function:

we can then do a case split on the equality \(i = 1\) using our defined function

1. If \(i = 1\) then we have \[ a'[i] = a[1] = 2 \to a'[i] = 3 \equiv 2 = 3 \equiv \bot \]
2. If \(i \ \mathrlap {\,/}{=}\ 1\) we have \[ a'[i] = a[i] \to a'[i] = 3 \equiv a[i] = 3 \]

Since it clearly must be the case that \(i \ \mathrlap {\,/}{=}\ 1\) it implies that our assertion \(R\) is equivalent to:

Now testing this rule on the first hoare triple:

By the assignment rule we must somehow be able to derive \(i = 1\) from the following:

substituting in our \(R\) we get

after a reduction we have

simplifying this we get

Which is where the crux of our issue lies, the precondition of our Hoare triple was \(i = 1\), but clearly here to derive our sequence of assignments we must have that i is not 1, hence this hoare triple is not derivable as its also clearly not valid.

A simpler approach at establishing the same idea is by just directly working backwards

so first we substitute \(a\) for the assignment \(a[1] := 2\), then we update it for the assignment \(a[i] := 3\), then our precondition would imply that:

We indeed cannot derive the triple with the precondition, as it should be trivially apparent that if \(i = 1\) after overwriting with \(a[1] = 2\) we clearly cannot have \(a[1] = 3\).

1. This formula is satisfiable. For example, if we have an array \(a\) such that \(a[3] = 2\), then the formula holds true.
2. This formula is valid. According to the Read-Over-Write axiom, when we write a value to an index in an array and then read from that same index, we get the value we just wrote. So regardless of the initial contents of array \(a\), after performing the write operation \(a \langle 3 \triangleleft 5 \rangle \), reading from index \(3\) will always yield \(5\) hence making it always true (valid).
3. This formula is unsatisfiable. According to the Read-Over-Write axiom, when we write a value to an index in an array and then read from that same index, we get the value we just wrote. Therefore, after performing the write operation \(a \langle 3 \triangleleft 5 \rangle \), reading from index \(3\) will always yield \(5\), making it impossible for it to equal \(3\).
4. This formula is satisfiable. For example, if we have an array \(a\) such that \(a[3] = 2\), then after performing the write operation \(a \langle 3 \triangleleft 5 \rangle \), reading from index \(3\) will yield \(5\). Thus, both parts of the conjunction can be true simultaneously though are not valid in all interpretations.

This is almost a sort of canonical example where the index replacement trick works really nicely. On a high level our postcondition asserts something about the array over the first \(n\) indices, hence if we want to have an invariant which shows correctness, we can simply reframe our postcondition as an assertion up to the \(i\)th position / iteration, in other words we simply replace \(n\) with \(i\):

Say we want to verify a hoare triple \(\{P\}\ s\ \{Q\}\), the weakest precondition approach is that we:

1. Start with our postcondition \(Q\) and, going backwards, we compute a formula \(\texttt {wp}(s, Q)\) called the weakest precondition of \(Q\) w.r.t the statement \(s\).
2. \(\texttt {wp}(s, Q)\) has the property that it's the weakest condition which guarantees that \(Q\) will hold after the termination of \(s\).

Therefore we can say that the triple is valid:

The basic rules can be recursively defined as follows:

• For assignments: \[ \texttt {wp}(x := e, Q) \triangleq Q[x \mapsto e] \]
• For composition \[ \texttt {wp}(s_1; s_2, Q) \triangleq \texttt {wp}(s_1, \texttt {wp}(s_2, Q)) \]
• For conditionals \[ \texttt {wp}(\texttt {if}\ b\ \texttt {then}\ s_1\ \texttt {else}\ s_2, Q) \triangleq (b \to \texttt {wp}(s_1, Q)) \land (\neg b \to \texttt {wp}(s_2, Q)) \]

As a reminder, a Hoare triple:

Is simply a kind of syntax sugar for the logical formula:

So for all source and target states, if we terminate on \(s\) then \(Q\) holds. Additionally, for different kinds of expressions we also have our constructions rules to ensure soundness, take for example assignment, our rule to only derive valid Hoare triples was:

If we assume \(e\) to be some variable expression in a context \(\sigma : \texttt {Vars} \to \mathbb {Z}\) which has to evaluate to a number before assignment then we can write the above rule more pedantically as:

The idea of backwards reasoning here is as follows:

1. First we look at the big step evaluation \((x := e, \sigma ) \Downarrow \sigma '\), we know that if this assignment succeeded, it implies there must have been a successful evaluation of the \(e\) term, we know this because the only way to have derived this step is by the following inference rule:
2. Using what we've learned from (1) we apply the notion of inversion, i.e. if we know a conclusion to be true, we can assert the premises of that conclusion must have also been true, this in a sense gives us the premises to use as new judgements.
3. Now looking at the expression \[ \forall n : \mathbb {Z},.\ (e, \sigma ) \Downarrow n \to Q[x \mapsto n] \] we can clearly derive the precondition \(Q[x \mapsto n]\) since we know the antecedent \(\forall n.\ (e, \sigma ) \Downarrow n\) is true, as it must have been true to derive the big step evaluation.
4. Since we have now demonstrated we can indeed provide sufficient reasoning to arrive at our conclusion \(Q(\sigma ')\) this finishes the proof.

The shorthand of the weakest precondition is nothing more than an expression of precisely this idea in a more concise fashion. Take for example \(\texttt {if conditions}\).

1. We again start from our post-condition then proceed with a case split or more accurately an inversion on the evaluation rule.
2. The inversion naturally gives us two branches, the true and false branch. This represents the conjunction here, then with each conjunct the antecedent of the implication is represented by the true or false guard condition \(b\), the consequent in this instance is simply a recursive call on the \(\texttt {then}\) and \(\texttt {else}\) branch bodies.

The main thing I'm trying to drive home here is that the weakest precondition idea is fundamentally just based on the concept of:

1. Seeing how a term must have been derived
2. Recursively going up the chain of any other sub-terms

So in a straightforward way, it's nothing more than chaining together all the individual proof rules for the various hoare triples. To show an example in lean using a some nice macros to create a simple syntax:

    example (n : Num) : 
    {{ ⟦ "x" ↦ n ⟧ }} 
      tm{ x := x + 1 } 
    {{ ⟦ "x" ↦ n + 1 ⟧ }} := by
    apply Hoare.assign'
    intro σ m hpre heval
    cases heval with
    | sum he1 he2 =>
      cases he1; cases he2
      simp [hpre]
  

The hoare triple we are proving here corresponds to this in the normal syntax:

We can see here to "prove" this hoare triple was valid we started by doing our standard case split on the assign - that's the \(\texttt {apply Hoare.assign'}\) - statement, then we were inside the addition, here we did a case split which gave us the lhs and rhs we were adding, after doing a case split on those two sides we could finally just simplify (and finalize) our proof using the fact that in \(\sigma \) \(x\) evaluates to \(n\).

1. In an abstract sense the statement is a sequence of an assignment and an if statement, starting with the if statement we get: \[ (x > 0) \to \texttt {wp}(z := 1, Q) \land (x \leq 0) \to \texttt {wp}(z := -1, Q) \] Both of the assignments resolve to substitutions of 1 and -1, so we get \[ (x > 0) \to (z > 0)[z \mapsto 1] \land (x \leq 0) \to (z > 0)[z \mapsto -1] \] With sequences we know it's a nested pattern, i.e. \(\texttt {wp}(s_1, \texttt {wp}(s_2, Q))\), the second argument we already have, we know that \(s_1\) here is just the assignment, so again we apply the subtitution \[ ((x > 0) \to Q[z \mapsto 1] \land (x \leq 0) \to Q[z \mapsto -1])[x \mapsto (y + 1)] \] which reduces to \[ \begin {align} ((y + 1 > 0) \to (1 > 0) &\land (y + 1 \leq 0) \to (-1 > 0)) \\ ((y + 1 > 0) \to \top &\land (y + 1 \leq 0) \to \bot ) \\ ((y + 1 > 0) \to \top &\land \neg (y + 1 \leq 0)) \\ ((y + 1 > 0) \to \top &\land (y + 1 > 0)) \\ ((y + 1 > 0) \to \top )& \\ (y + 1 > 0)& \\ \end {align} \]
2. To skip ahead a slight bit, so we have \[ (x > 0) \to (z \leq 0)[z \mapsto 1] \land (x \leq 0) \to (z \leq 0)[z \mapsto -1] \] then substituting for the intial assignment of \(x\) \[ (y + 1 > 0) \to (1 \leq 0) \land (y + 1 \leq 0) \to (-1 \leq 0) \] Here we just end up in the reverse case, i.e. \[ (y + 1 > 0) \to (\bot ) \land (y + 1 \leq 0) \to \top \] reduces now to \[ (y + 1 \leq 0) \]
3. The high level idea here is that we computed the weakest precondition for \(z > 0\) already, it was \[ (y + 1 > 0) \] since it's the weakest we can only admit other preconditions if they imply the weakest one, so if we then look at the implication \[ -1 > y \to (y + 1 > 0) \] we can trivially see that this implication does not hold, i.e. we cannot apply precondition strengthening here hence the rule does not allow us to prove the hoare triple.
4. Same reasoning as above, but now it works since \[ y > -1 \to y + 1 > 0 \] indeed makes sense

I do think it's worth telling this as a bit of a story, we start with the fundamental theorem for while loops:

So this is just the expanded rule for constructing valid or sound while hoare triples, now this rule assumes that the invariant \(I\) is the same as the precondition \(P\) which is naturally not always the case, so it could be nice to have a rule which accounts for constructing valid while loops with invariants and custom preconditions. Same with custom post-conditions \(Q\).

If you want to be super pedantic you could show how this is just a combination of consequence + while rule. Anyway then, moving on we have the rule for \(\texttt {wp}\) soundness.

Now depending on how you actually design your \(\texttt {wp}\) and \(\texttt {vc}\) code this rule can look slightly different, though I like this distinction, in this instance for the case of while loops \(\texttt {wp}(W[I], Q)\) simply returns the invariant \(I\), this means that again if we want to have a custom precondition for \(W\) we'd want to verify we would extend the rule as:

Thus for while loops it would become:

Where the verification conditions would unfold to what we discussed before.

A nice point to make here is that all of these rules are essentially derived from one another, which is to say that:

In a conceptual sense verification conditions i.e. what we need to prove to demonstrate a hoare triple is valid, are simply a convenience mechanism derived from the fundamental logic of how we construct a hoare triple.