# Hoare2Hoare Logic, Part II

Set Warnings "-notation-overridden,-parsing".

From Coq Require Import Strings.String.

From PLF Require Import Maps.

From Coq Require Import Bool.Bool.

From Coq Require Import Arith.Arith.

From Coq Require Import Arith.EqNat.

From Coq Require Import Arith.PeanoNat. Import Nat.

From Coq Require Import omega.Omega.

From PLF Require Import Hoare.

From PLF Require Import Imp.

# Decorated Programs

*compositional*: the structure of proofs exactly follows the structure of programs.

*decorated program*carries within it an argument for its own correctness.

WHILE ~(X = 0) DO

Z ::= Z - 1;;

X ::= X - 1

END

{{ X = m ∧ Z = p }}

WHILE ~(X = 0) DO

Z ::= Z - 1;;

X ::= X - 1

END

{{ Z = p - m }}

*parameters*m and p, which stand for fixed-but-arbitrary numbers. Formally, they are simply Coq variables of type nat. Here is a decorated version of the program, embodying a proof of this specification:

{{ X = m ∧ Z = p }} ->>

{{ Z - X = p - m }}

WHILE ~(X = 0) DO

{{ Z - X = p - m ∧ X ≠ 0 }} ->>

{{ (Z - 1) - (X - 1) = p - m }}

Z ::= Z - 1;;

{{ Z - (X - 1) = p - m }}

X ::= X - 1

{{ Z - X = p - m }}

END

{{ Z - X = p - m ∧ ¬(X ≠ 0) }} ->>

{{ Z = p - m }}

*locally consistent*with its nearby assertions in the following sense:

- SKIP is locally consistent if its precondition and
postcondition are the same:

{{ P }} SKIP {{ P }}

- The sequential composition of c
_{1}and c_{2}is locally consistent (with respect to assertions P and R) if c_{1}is locally consistent (with respect to P and Q) and c_{2}is locally consistent (with respect to Q and R):

{{ P }} c_{1};; {{ Q }} c_{2}{{ R }}

- An assignment is locally consistent if its precondition is the
appropriate substitution of its postcondition:

{{ P [X ⊢> a] }}

X ::= a

{{ P }}

- A conditional is locally consistent (with respect to assertions
P and Q) if the assertions at the top of its "then" and
"else" branches are exactly P ∧ b and P ∧ ¬b and if its
"then" branch is locally consistent (with respect to P ∧ b
and Q) and its "else" branch is locally consistent (with
respect to P ∧ ¬b and Q):

{{ P }}

TEST b THEN

{{ P ∧ b }}

c_{1}

{{ Q }}

ELSE

{{ P ∧ ¬b }}

c_{2}

{{ Q }}

FI

{{ Q }}

- A while loop with precondition P is locally consistent if its
postcondition is P ∧ ¬b, if the pre- and postconditions of
its body are exactly P ∧ b and P, and if its body is
locally consistent:

{{ P }}

WHILE b DO

{{ P ∧ b }}

c_{1}

{{ P }}

END

{{ P ∧ ¬b }}

- A pair of assertions separated by ->> is locally consistent if
the first implies the second:

{{ P }} ->>

{{ P' }}*only*place in a decorated program where checking whether decorations are correct is not fully mechanical and syntactic, but rather may involve logical and/or arithmetic reasoning.

*verifying*the correctness of a given proof. This verification involves checking that every single command is locally consistent with the accompanying assertions.

*finding*a proof for a given specification, we need to discover the right assertions. This can be done in an almost mechanical way, with the exception of finding loop invariants. In the remainder of this section we explain in detail how to construct decorations for several short programs, all of which are loop-free or have simple loop invariants. We defer a full discussion of finding loop invariants to the next section.

## Example: Swapping Using Addition and Subtraction

X ::= X + Y;;

Y ::= X - Y;;

X ::= X - Y

(1) {{ X = m ∧ Y = n }} ->>

(2) {{ (X + Y) - ((X + Y) - Y) = n ∧ (X + Y) - Y = m }}

X ::= X + Y;;

(3) {{ X - (X - Y) = n ∧ X - Y = m }}

Y ::= X - Y;;

(4) {{ X - Y = n ∧ Y = m }}

X ::= X - Y

(5) {{ X = n ∧ Y = m }}

- We begin with the undecorated program (the unnumbered lines).
- We add the specification -- i.e., the outer precondition (1)
and postcondition (5). In the precondition, we use parameters
m and n to remember the initial values of variables X
and Y so that we can refer to them in the postcondition (5).
- We work backwards, mechanically, starting from (5) and proceeding until we get to (2). At each step, we obtain the precondition of the assignment from its postcondition by substituting the assigned variable with the right-hand-side of the assignment. For instance, we obtain (4) by substituting X with X - Y in (5), and we obtain (3) by substituting Y with X - Y in (4).

(m + n) - ((m + n) - n) = n ∧ (m + n) - n = m

(m + n) - m = n ∧ m = m

n = n ∧ m = m

## Example: Simple Conditionals

(1) {{True}}

TEST X ≤ Y THEN

(2) {{True ∧ X ≤ Y}} ->>

(3) {{(Y - X) + X = Y ∨ (Y - X) + Y = X}}

Z ::= Y - X

(4) {{Z + X = Y ∨ Z + Y = X}}

ELSE

(5) {{True ∧ ~(X ≤ Y) }} ->>

(6) {{(X - Y) + X = Y ∨ (X - Y) + Y = X}}

Z ::= X - Y

(7) {{Z + X = Y ∨ Z + Y = X}}

FI

(8) {{Z + X = Y ∨ Z + Y = X}}

- We start with the outer precondition (1) and postcondition (8).
- We follow the format dictated by the hoare_if rule and copy the
postcondition (8) to (4) and (7). We conjoin the precondition (1)
with the guard of the conditional to obtain (2). We conjoin (1)
with the negated guard of the conditional to obtain (5).
- In order to use the assignment rule and obtain (3), we substitute Z by Y - X in (4). To obtain (6) we substitute Z by X - Y in (7).

*not*hold for arbitrary natural numbers n and m (for example, 3 - 5 + 5 = 5).

#### Exercise: 2 stars, standard (if_minus_plus_reloaded)

Fill in valid decorations for the following program:{{ True }}

TEST X ≤ Y THEN

{{ }} ->>

{{ }}

Z ::= Y - X

{{ }}

ELSE

{{ }} ->>

{{ }}

Y ::= X + Z

{{ }}

FI

{{ Y = X + Z }}

(* Do not modify the following line: *)

Definition manual_grade_for_decorations_in_if_minus_plus_reloaded : option (nat×string) := None.

☐

## Example: Reduce to Zero

(1) {{ True }}

WHILE ~(X = 0) DO

(2) {{ True ∧ X ≠ 0 }} ->>

(3) {{ True }}

X ::= X - 1

(4) {{ True }}

END

(5) {{ True ∧ ~(X ≠ 0) }} ->>

(6) {{ X = 0 }}

- Start with the outer precondition (1) and postcondition (6).
- Following the format dictated by the hoare_while rule, we copy
(1) to (4). We conjoin (1) with the guard to obtain (2). The guard
is a Boolean expression ~(X = 0), which for simplicity we
express in assertion (2) as X ≠ 0. We also conjoin (1) with the
negation of the guard to obtain (5).
- Because the outer postcondition (6) does not syntactically match (5),
we add an implication between them.
- Using the assignment rule with assertion (4), we trivially substitute
and obtain assertion (3).
- We add the implication between (2) and (3).

*not*unfold the definition of hoare_triple anywhere in this proof -- the idea is to use the Hoare rules as a self-contained logic for reasoning about programs.

Definition reduce_to_zero' : com :=

WHILE ~(X = 0) DO

X ::= X - 1

END.

Theorem reduce_to_zero_correct' :

{{True}}

reduce_to_zero'

{{X = 0}}.

Proof.

unfold reduce_to_zero'.

(* First we need to transform the postcondition so

that hoare_while will apply. *)

eapply hoare_consequence_post.

- apply hoare_while.

+ (* Loop body preserves invariant *)

(* Need to massage precondition before hoare_asgn applies *)

eapply hoare_consequence_pre.

× apply hoare_asgn.

× (* Proving trivial implication (2) ->> (3) *)

unfold assn_sub, "->>". simpl. intros. exact I.

- (* Invariant and negated guard imply postcondition *)

intros st [Inv GuardFalse].

unfold bassn in GuardFalse. simpl in GuardFalse.

rewrite not_true_iff_false in GuardFalse.

rewrite negb_false_iff in GuardFalse.

apply eqb_eq in GuardFalse.

apply GuardFalse.

Qed.

In Hoare we introduced a series of tactics named assn_auto
to automate proofs involving just assertions. We can try using
the most advanced of those tactics to streamline the previous proof:

Theorem reduce_to_zero_correct'' :

{{True}}

reduce_to_zero'

{{X = 0}}.

Proof.

unfold reduce_to_zero'.

eapply hoare_consequence_post.

- apply hoare_while.

+ eapply hoare_consequence_pre.

× apply hoare_asgn.

× assn_auto''.

- assn_auto''. (* doesn't succeed *)

Abort.

Let's introduce a much more sophisticated tactic that will
help with proving assertions throughout the rest of this chapter.
You don't need to understand the details of it. Briefly, it uses
split repeatedly to turn all the conjunctions into separate
subgoals, tries to use several theorems about booleans and
(in)equalities, then uses eauto and omega to finish off as
many subgoals as possible. What's left after verify does its
thing is just the "interesting parts" of checking that the
assertions correct --which might be nothing.

Ltac verify_assn :=

repeat split;

simpl; unfold assert_implies;

unfold ap in *; unfold ap

_{2}in *;

unfold bassn in *; unfold beval in *; unfold aeval in *;

unfold assn_sub; intros;

repeat (simpl in *;

rewrite t_update_eq ||

(try rewrite t_update_neq; [| (intro X; inversion X; fail)]));

simpl in *;

repeat match goal with [H : _ ∧ _ ⊢ _] ⇒ destruct H end;

repeat rewrite not_true_iff_false in *;

repeat rewrite not_false_iff_true in *;

repeat rewrite negb_true_iff in *;

repeat rewrite negb_false_iff in *;

repeat rewrite eqb_eq in *;

repeat rewrite eqb_neq in *;

repeat rewrite leb_iff in *;

repeat rewrite leb_iff_conv in *;

try subst;

simpl in *;

repeat

match goal with

[st : state ⊢ _] ⇒

match goal with

| [H : st _ = _ ⊢ _] ⇒ rewrite → H in *; clear H

| [H : _ = st _ ⊢ _] ⇒ rewrite <- H in *; clear H

end

end;

try eauto; try omega.

All that automation makes it easy to verify reduce_to_zero':

Theorem reduce_to_zero_correct''' :

{{True}}

reduce_to_zero'

{{X = 0}}.

Proof.

unfold reduce_to_zero'.

eapply hoare_consequence_post.

- apply hoare_while.

+ eapply hoare_consequence_pre.

× apply hoare_asgn.

× verify_assn.

- verify_assn.

Qed.

unfold reduce_to_zero'.

eapply hoare_consequence_post.

- apply hoare_while.

+ eapply hoare_consequence_pre.

× apply hoare_asgn.

× verify_assn.

- verify_assn.

Qed.

## Example: Division

X ::= m;;

Y ::= 0;;

WHILE n ≤ X DO

X ::= X - n;;

Y ::= Y + 1

END;

(1) {{ True }} ->>

(2) {{ n × 0 + m = m }}

X ::= m;;

(3) {{ n × 0 + X = m }}

Y ::= 0;;

(4) {{ n × Y + X = m }}

WHILE n ≤ X DO

(5) {{ n × Y + X = m ∧ n ≤ X }} ->>

(6) {{ n × (Y + 1) + (X - n) = m }}

X ::= X - n;;

(7) {{ n × (Y + 1) + X = m }}

Y ::= Y + 1

(8) {{ n × Y + X = m }}

END

(9) {{ n × Y + X = m ∧ ¬(n ≤ X) }} ->>

(10) {{ n × Y + X = m ∧ X < n }}

- (1) ->> (2): trivial, by algebra.
- (5) ->> (6): because n ≤ X, we are guaranteed that the subtraction in (6) does not get zero-truncated. We can therefore rewrite (6) as n × Y + n + X - n and cancel the ns, which results in the left conjunct of (5).
- (9) ->> (10): if ¬ (n ≤ X) then X < n. That's straightforward from high-school algebra.

# Finding Loop Invariants

- Strengthening the *loop invariant* means that you have a stronger
assumption to work with when trying to establish the
postcondition of the loop body, but it also means that the loop
body's postcondition is stronger and thus harder to prove.
- Strengthening the *induction hypothesis* means that you have a stronger assumption to work with when trying to complete the induction step of the proof, but it also means that the statement being proved inductively is stronger and thus harder to prove.

## Example: Slow Subtraction

{{ X = m ∧ Y = n }}

WHILE ~(X = 0) DO

Y ::= Y - 1;;

X ::= X - 1

END

{{ Y = n - m }}

*skeleton*for the proof by applying the rules for local consistency (working from the end of the program to the beginning, as usual, and without any thinking at all yet).

(1) {{ X = m ∧ Y = n }} ->> (a)

(2) {{ Inv }}

WHILE ~(X = 0) DO

(3) {{ Inv ∧ X ≠ 0 }} ->> (c)

(4) {{ Inv [X ⊢> X-1] [Y ⊢> Y-1] }}

Y ::= Y - 1;;

(5) {{ Inv [X ⊢> X-1] }}

X ::= X - 1

(6) {{ Inv }}

END

(7) {{ Inv ∧ ¬(X ≠ 0) }} ->> (b)

(8) {{ Y = n - m }}

- (a) it must be
*weak*enough to be implied by the loop's precondition, i.e., (1) must imply (2); - (b) it must be
*strong*enough to imply the program's postcondition, i.e., (7) must imply (8); - (c) it must be
*preserved*by each iteration of the loop (given that the loop guard evaluates to true), i.e., (3) must imply (4).

(1) {{ X = m ∧ Y = n }} ->> (a - OK)

(2) {{ True }}

WHILE ~(X = 0) DO

(3) {{ True ∧ X ≠ 0 }} ->> (c - OK)

(4) {{ True }}

Y ::= Y - 1;;

(5) {{ True }}

X ::= X - 1

(6) {{ True }}

END

(7) {{ True ∧ ~(X ≠ 0) }} ->> (b - WRONG!)

(8) {{ Y = n - m }}

*be*the postcondition. So let's return to our skeleton, instantiate Inv with Y = n - m, and check conditions (a) to (c) again.

(1) {{ X = m ∧ Y = n }} ->> (a - WRONG!)

(2) {{ Y = n - m }}

WHILE ~(X = 0) DO

(3) {{ Y = n - m ∧ X ≠ 0 }} ->> (c - WRONG!)

(4) {{ Y - 1 = n - m }}

Y ::= Y - 1;;

(5) {{ Y = n - m }}

X ::= X - 1

(6) {{ Y = n - m }}

END

(7) {{ Y = n - m ∧ ~(X ≠ 0) }} ->> (b - OK)

(8) {{ Y = n - m }}

(1) {{ X = m ∧ Y = n }} ->> (a - OK)

(2) {{ Y - X = n - m }}

WHILE ~(X = 0) DO

(3) {{ Y - X = n - m ∧ X ≠ 0 }} ->> (c - OK)

(4) {{ (Y - 1) - (X - 1) = n - m }}

Y ::= Y - 1;;

(5) {{ Y - (X - 1) = n - m }}

X ::= X - 1

(6) {{ Y - X = n - m }}

END

(7) {{ Y - X = n - m ∧ ~(X ≠ 0) }} ->> (b - OK)

(8) {{ Y = n - m }}

## Exercise: Slow Assignment

#### Exercise: 2 stars, standard (slow_assignment)

A roundabout way of assigning a number currently stored in X to the variable Y is to start Y at 0, then decrement X until it hits 0, incrementing Y at each step. Here is a program that implements this idea:{{ X = m }}

Y ::= 0;;

WHILE ~(X = 0) DO

X ::= X - 1;;

Y ::= Y + 1

END

{{ Y = m }}

(* FILL IN HERE *)

(* Do not modify the following line: *)

Definition manual_grade_for_decorations_in_slow_assignment : option (nat×string) := None.

☐

## Exercise: Slow Addition

#### Exercise: 3 stars, standard, optional (add_slowly_decoration)

The following program adds the variable X into the variable Z by repeatedly decrementing X and incrementing Z.WHILE ~(X = 0) DO

Z ::= Z + 1;;

X ::= X - 1

END

(* FILL IN HERE *)

☐

## Example: Parity

{{ X = m }}

WHILE 2 ≤ X DO

X ::= X - 2

END

{{ X = parity m }}

The postcondition does not hold at the beginning of the loop,
since m = parity m does not hold for an arbitrary m, so we
cannot use that as an invariant. To find an invariant that works,
let's think a bit about what this loop does. On each iteration it
decrements X by 2, which preserves the parity of X. So the
parity of X does not change, i.e., it is invariant. The initial
value of X is m, so the parity of X is always equal to the
parity of m. Using parity X = parity m as an invariant we
obtain the following decorated program:

{{ X = m }} ->> (a - OK)

{{ parity X = parity m }}

WHILE 2 ≤ X DO

{{ parity X = parity m ∧ 2 ≤ X }} ->> (c - OK)

{{ parity (X-2) = parity m }}

X ::= X - 2

{{ parity X = parity m }}

END

{{ parity X = parity m ∧ ~(2 ≤ X) }} ->> (b - OK)

{{ X = parity m }}
With this invariant, conditions (a), (b), and (c) are all
satisfied. For verifying (b), we observe that, when X < 2, we
have parity X = X (we can easily see this in the definition of
parity). For verifying (c), we observe that, when 2 ≤ X, we
have parity X = parity (X-2).
To formally state the invariant, you will need the ap operator
to apply parity to an Imp variable --e.g., ap parity X.
After using verify_assn, you will be left needing to prove some facts
about parity. The following lemmas will be helpful, as will
leb_complete and leb_correct.

{{ X = m }} ->> (a - OK)

{{ parity X = parity m }}

WHILE 2 ≤ X DO

{{ parity X = parity m ∧ 2 ≤ X }} ->> (c - OK)

{{ parity (X-2) = parity m }}

X ::= X - 2

{{ parity X = parity m }}

END

{{ parity X = parity m ∧ ~(2 ≤ X) }} ->> (b - OK)

{{ X = parity m }}

#### Exercise: 3 stars, standard, optional (parity_formal)

Translate the above informal decorated program into a formal proof in Coq. Your proof should use the Hoare logic rules and should not unfold hoare_triple. Refer to reduce_to_zero for an example.Lemma parity_ge_2 : ∀ x,

2 ≤ x →

parity (x - 2) = parity x.

Proof.

induction x; intros; simpl.

- reflexivity.

- destruct x.

+ omega.

+ inversion H; subst; simpl.

× reflexivity.

× rewrite sub_0_r. reflexivity.

Qed.

induction x; intros; simpl.

- reflexivity.

- destruct x.

+ omega.

+ inversion H; subst; simpl.

× reflexivity.

× rewrite sub_0_r. reflexivity.

Qed.

Lemma parity_lt_2 : ∀ x,

¬ 2 ≤ x →

parity x = x.

Proof.

induction x; intros; simpl.

- reflexivity.

- destruct x.

+ reflexivity.

+ omega.

Qed.

induction x; intros; simpl.

- reflexivity.

- destruct x.

+ reflexivity.

+ omega.

Qed.

Theorem parity_correct : ∀ (m:nat),

{{ X = m }}

WHILE 2 ≤ X DO

X ::= X - 2

END

{{ X = parity m }}.

Proof.

(* FILL IN HERE *) Admitted.

☐

## Example: Finding Square Roots

{{ X=m }}

Z ::= 0;;

WHILE (Z+1)*(Z+1) ≤ X DO

Z ::= Z+1

END

{{ Z×Z≤m ∧ m<(Z+1)*(Z+1) }}

(1) {{ X=m }} ->> (a - second conjunct of (2) WRONG!)

(2) {{ 0*0 ≤ m ∧ m<(0+1)*(0+1) }}

Z ::= 0;;

(3) {{ Z×Z ≤ m ∧ m<(Z+1)*(Z+1) }}

WHILE (Z+1)*(Z+1) ≤ X DO

(4) {{ Z×Z≤m ∧ (Z+1)*(Z+1)<=X }} ->> (c - WRONG!)

(5) {{ (Z+1)*(Z+1)<=m ∧ m<((Z+1)+1)*((Z+1)+1) }}

Z ::= Z+1

(6) {{ Z×Z≤m ∧ m<(Z+1)*(Z+1) }}

END

(7) {{ Z×Z≤m ∧ m<(Z+1)*(Z+1) ∧ ~((Z+1)*(Z+1)<=X) }} ->> (b - OK)

(8) {{ Z×Z≤m ∧ m<(Z+1)*(Z+1) }}

{{ X=m }} ->> (a - OK)

{{ X=m ∧ 0*0 ≤ m }}

Z ::= 0;

{{ X=m ∧ Z×Z ≤ m }}

WHILE (Z+1)*(Z+1) ≤ X DO

{{ X=m ∧ Z×Z≤m ∧ (Z+1)*(Z+1)<=X }} ->> (c - OK)

{{ X=m ∧ (Z+1)*(Z+1)<=m }}

Z ::= Z + 1

{{ X=m ∧ Z×Z≤m }}

END

{{ X=m ∧ Z×Z≤m ∧ ~((Z+1)*(Z+1)<=X) }} ->> (b - OK)

{{ Z×Z≤m ∧ m<(Z+1)*(Z+1) }}

## Example: Squaring

{{ X = m }}

Y ::= 0;;

Z ::= 0;;

WHILE ~(Y = X) DO

Z ::= Z + X;;

Y ::= Y + 1

END

{{ Z = m×m }}

{{ X = m }} ->> (a - WRONG)

{{ 0 = m×m ∧ X = m }}

Y ::= 0;;

{{ 0 = m×m ∧ X = m }}

Z ::= 0;;

{{ Z = m×m ∧ X = m }}

WHILE ~(Y = X) DO

{{ Z = m×m ∧ X = m ∧ Y ≠ X }} ->> (c - WRONG)

{{ Z+X = m×m ∧ X = m }}

Z ::= Z + X;;

{{ Z = m×m ∧ X = m }}

Y ::= Y + 1

{{ Z = m×m ∧ X = m }}

END

{{ Z = m×m ∧ X = m ∧ ~(Y ≠ X) }} ->> (b - OK)

{{ Z = m×m }}

{{ X = m }} ->> (a - OK)

{{ 0 = 0*m ∧ X = m }}

Y ::= 0;;

{{ 0 = Y×m ∧ X = m }}

Z ::= 0;;

{{ Z = Y×m ∧ X = m }}

WHILE ~(Y = X) DO

{{ Z = Y×m ∧ X = m ∧ Y ≠ X }} ->> (c - OK)

{{ Z+X = (Y+1)*m ∧ X = m }}

Z ::= Z + X;

{{ Z = (Y+1)*m ∧ X = m }}

Y ::= Y + 1

{{ Z = Y×m ∧ X = m }}

END

{{ Z = Y×m ∧ X = m ∧ ~(Y ≠ X) }} ->> (b - OK)

{{ Z = m×m }}

## Exercise: Factorial

#### Exercise: 3 stars, standard (factorial)

Recall that n! denotes the factorial of n (i.e., n! = 1*2*...*n). Here is an Imp program that calculates the factorial of the number initially stored in the variable X and puts it in the variable Y:{{ X = m }}

Y ::= 1 ;;

WHILE ~(X = 0)

DO

Y ::= Y × X ;;

X ::= X - 1

END

{{ Y = m! }}

{{ X = m }} ->>

{{ }}

Y ::= 1;;

{{ }}

WHILE ~(X = 0)

DO {{ }} ->>

{{ }}

Y ::= Y × X;;

{{ }}

X ::= X - 1

{{ }}

END

{{ }} ->>

{{ Y = m! }}

(* Do not modify the following line: *)

Definition manual_grade_for_decorations_in_factorial : option (nat×string) := None.

☐

## Exercise: Min

#### Exercise: 3 stars, standard (Min_Hoare)

Lemma lemma1 : ∀ x y,

(x<>0 ∧ y<>0) → min x y ≠ 0.

Lemma lemma2 : ∀ x y,

min (x-1) (y-1) = (min x y) - 1.

{{ X = a ∧ Y = b }} ->>

{{ }}

Z ::= 0;;

{{ }}

WHILE ~(X = 0) && ~(Y = 0) DO

{{ }} ->>

{{ }}

X := X - 1;;

{{ }}

Y := Y - 1;;

{{ }}

Z := Z + 1

{{ }}

END

{{ }} ->>

{{ Z = min a b }}

(* Do not modify the following line: *)

Definition manual_grade_for_decorations_in_Min_Hoare : option (nat×string) := None.

☐

#### Exercise: 3 stars, standard (two_loops)

Here is a very inefficient way of adding 3 numbers:X ::= 0;;

Y ::= 0;;

Z ::= c;;

WHILE ~(X = a) DO

X ::= X + 1;;

Z ::= Z + 1

END;;

WHILE ~(Y = b) DO

Y ::= Y + 1;;

Z ::= Z + 1

END

{{ True }} ->>

{{ }}

X ::= 0;;

{{ }}

Y ::= 0;;

{{ }}

Z ::= c;;

{{ }}

WHILE ~(X = a) DO

{{ }} ->>

{{ }}

X ::= X + 1;;

{{ }}

Z ::= Z + 1

{{ }}

END;;

{{ }} ->>

{{ }}

WHILE ~(Y = b) DO

{{ }} ->>

{{ }}

Y ::= Y + 1;;

{{ }}

Z ::= Z + 1

{{ }}

END

{{ }} ->>

{{ Z = a + b + c }}

(* Do not modify the following line: *)

Definition manual_grade_for_decorations_in_two_loops : option (nat×string) := None.

☐

## Exercise: Power Series

#### Exercise: 4 stars, standard, optional (dpow2_down)

Here is a program that computes the series: 1 + 2 + 2^2 + ... + 2^m = 2^(m+1) - 1X ::= 0;;

Y ::= 1;;

Z ::= 1;;

WHILE ~(X = m) DO

Z ::= 2 × Z;;

Y ::= Y + Z;;

X ::= X + 1

END

(* FILL IN HERE *)

☐

# Weakest Preconditions (Optional)

{{ False }} X ::= Y + 1 {{ X ≤ 5 }}

*not*very interesting: although it is perfectly valid Hoare triple, it tells us nothing useful. Since the precondition isn't satisfied by any state, it doesn't describe any situations where we can use the command X ::= Y + 1 to achieve the postcondition X ≤ 5.

{{ Y ≤ 4 ∧ Z = 0 }} X ::= Y + 1 {{ X ≤ 5 }}

{{ Y ≤ 4 }} X ::= Y + 1 {{ X ≤ 5 }}

*weakest precondition*of command X ::= Y + 1 for postcondition X ≤ 5.

*weakest precondition*of command X ::= Y + 1 with respect to postcondition X ≤ 5. Think of

*weakest*here as meaning "easiest to satisfy": a weakest precondition is one that as many states as possible can satisfy.

- P is a precondition, that is, {{P}} c {{Q}}; and
- P is at least as weak as all other preconditions, that is, if {{P'}} c {{Q}} then P' ->> P.

#### Exercise: 1 star, standard, optional (wp)

What are weakest preconditions of the following commands for the following postconditions?1) {{ ? }} SKIP {{ X = 5 }}

2) {{ ? }} X ::= Y + Z {{ X = 5 }}

3) {{ ? }} X ::= Y {{ X = Y }}

4) {{ ? }}

TEST X = 0 THEN Y ::= Z + 1 ELSE Y ::= W + 2 FI

{{ Y = 5 }}

5) {{ ? }}

X ::= 5

{{ X = 0 }}

6) {{ ? }}

WHILE true DO X ::= 0 END

{{ X = 0 }}

(* FILL IN HERE *)

☐

☐

#### Exercise: 3 stars, advanced, optional (is_wp_formal)

Prove formally, using the definition of hoare_triple, that Y ≤ 4 is indeed a weakest precondition of X ::= Y + 1 with respect to postcondition X ≤ 5.#### Exercise: 2 stars, advanced, optional (hoare_asgn_weakest)

Show that the precondition in the rule hoare_asgn is in fact the weakest precondition.Theorem hoare_asgn_weakest : ∀ Q X a,

is_wp (Q [X ⊢> a]) (X ::= a) Q.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 2 stars, advanced, optional (hoare_havoc_weakest)

Show that your havoc_pre function from the himp_hoare exercise in the Hoare chapter returns a weakest precondition.
Module Himp2.

Import Himp.

Lemma hoare_havoc_weakest : ∀ (P Q : Assertion) (X : string),

{{ P }} HAVOC X {{ Q }} →

P ->> havoc_pre X Q.

Proof.

(* FILL IN HERE *) Admitted.

☐

Import Himp.

Lemma hoare_havoc_weakest : ∀ (P Q : Assertion) (X : string),

{{ P }} HAVOC X {{ Q }} →

P ->> havoc_pre X Q.

Proof.

(* FILL IN HERE *) Admitted.

☐

# Formal Decorated Programs (Advanced)

## Syntax

*decorated commands*, or dcoms.

{{P}} ({{P}} SKIP {{P}}) ;; ({{P}} SKIP {{P}}) {{P}},

- Command SKIP is decorated only with its postcondition, as
SKIP {{ Q }}.
- Sequence d
_{1};; d_{2}contains no additional decoration. Inside d_{2}there will be a postcondition; that serves as the postcondition of d_{1};; d_{2}. Inside d_{1}there will also be a postcondition; it additionally serves as the precondition for d_{2}. - Assignment X ::= a is decorated only with its postcondition,
as X ::= a {{ Q }}.
- If statement TEST b THEN d
_{1}ELSE d_{2}is decorated with a postcondition for the entire statement, as well as preconditions for each branch, as TEST b THEN {{ P_{1}}} d_{1}ELSE {{ P_{2}}} d_{2}FI {{ Q }}. - While loop WHILE b DO d END is decorated with its
postcondition and a precondition for the body, as
WHILE b DO {{ P }} d END {{ Q }}. The postcondition inside
d serves as the loop invariant.
- Implications ->> are added as decorations for a precondition as ->> {{ P }} d, or for a postcondition as d ->> {{ Q }}. The former is waiting for another precondition to eventually be supplied, e.g., {{ P'}} ->> {{ P }} d, and the latter relies on the postcondition already embedded in d.

Inductive dcom : Type :=

| DCSkip (Q : Assertion)

(* SKIP {{ Q }} *)

| DCSeq (d

_{1}d

_{2}: dcom)

(* d

_{1};; d

_{2}*)

| DCAsgn (X : string) (a : aexp) (Q : Assertion)

(* X := a {{ Q }} *)

| DCIf (b : bexp) (P

_{1}: Assertion) (d

_{1}: dcom)

(P

_{2}: Assertion) (d

_{2}: dcom) (Q : Assertion)

(* TEST b THEN {{ P

_{1}}} d

_{1}ELSE {{ P

_{2}}} d

_{2}FI {{ Q }} *)

| DCWhile (b : bexp) (P : Assertion) (d : dcom) (Q : Assertion)

(* WHILE b DO {{ P }} d END {{ Q }} *)

| DCPre (P : Assertion) (d : dcom)

(* ->> {{ P }} d *)

| DCPost (d : dcom) (Q : Assertion)

(* d ->> {{ Q }} *)

.

DCPre is used to provide the weakened precondition from
the rule of consequence. To provide the initial precondition
at the very top of the program, we use Decorated:

To avoid clashing with the existing Notation definitions for
ordinary commands, we introduce these notations in a special
scope called dcom_scope, and we Open this scope for the
remainder of the file.

Notation "'SKIP' {{ P }}"

:= (DCSkip P)

(at level 10) : dcom_scope.

Notation "l '::=' a {{ P }}"

:= (DCAsgn l a P)

(at level 60, a at next level) : dcom_scope.

Notation "'WHILE' b 'DO' {{ Pbody }} d 'END' {{ Ppost }}"

:= (DCWhile b Pbody d Ppost)

(at level 80, right associativity) : dcom_scope.

Notation "'TEST' b 'THEN' {{ P }} d 'ELSE' {{ P' }} d' 'FI' {{ Q }}"

:= (DCIf b P d P' d' Q)

(at level 80, right associativity) : dcom_scope.

Notation "'->>' {{ P }} d"

:= (DCPre P d)

(at level 90, right associativity) : dcom_scope.

Notation "d '->>' {{ P }}"

:= (DCPost d P)

(at level 80, right associativity) : dcom_scope.

Notation " d ;; d' "

:= (DCSeq d d')

(at level 80, right associativity) : dcom_scope.

Notation "{{ P }} d"

:= (Decorated P d)

(at level 90) : dcom_scope.

Delimit Scope dcom_scope with dcom.

Open Scope dcom_scope.

Example dec0 :=

SKIP {{ True }}.

Example dec1 :=

WHILE true DO {{ True }} SKIP {{ True }} END

{{ True }}.

:= (DCSkip P)

(at level 10) : dcom_scope.

Notation "l '::=' a {{ P }}"

:= (DCAsgn l a P)

(at level 60, a at next level) : dcom_scope.

Notation "'WHILE' b 'DO' {{ Pbody }} d 'END' {{ Ppost }}"

:= (DCWhile b Pbody d Ppost)

(at level 80, right associativity) : dcom_scope.

Notation "'TEST' b 'THEN' {{ P }} d 'ELSE' {{ P' }} d' 'FI' {{ Q }}"

:= (DCIf b P d P' d' Q)

(at level 80, right associativity) : dcom_scope.

Notation "'->>' {{ P }} d"

:= (DCPre P d)

(at level 90, right associativity) : dcom_scope.

Notation "d '->>' {{ P }}"

:= (DCPost d P)

(at level 80, right associativity) : dcom_scope.

Notation " d ;; d' "

:= (DCSeq d d')

(at level 80, right associativity) : dcom_scope.

Notation "{{ P }} d"

:= (Decorated P d)

(at level 90) : dcom_scope.

Delimit Scope dcom_scope with dcom.

Open Scope dcom_scope.

Example dec0 :=

SKIP {{ True }}.

Example dec1 :=

WHILE true DO {{ True }} SKIP {{ True }} END

{{ True }}.

Recall that you can Set Printing All to see how all that
notation is desugared.

Set Printing All.

Print dec1.

Unset Printing All.

Example dec_while : decorated :=

{{ True }}

WHILE ~(X = 0)

DO

{{ True ∧ (X ≠ 0) }}

X ::= X - 1

{{ True }}

END

{{ True ∧ X = 0}} ->>

{{ X = 0 }}.

Print dec1.

Unset Printing All.

Example dec_while : decorated :=

{{ True }}

WHILE ~(X = 0)

DO

{{ True ∧ (X ≠ 0) }}

X ::= X - 1

{{ True }}

END

{{ True ∧ X = 0}} ->>

{{ X = 0 }}.

It is easy to go from a dcom to a com by erasing all
annotations.

Fixpoint extract (d : dcom) : com :=

match d with

| DCSkip _ ⇒ SKIP

| DCSeq d

_{1}d

_{2}⇒ (extract d

_{1};; extract d

_{2})

| DCAsgn X a _ ⇒ X ::= a

| DCIf b _ d

_{1}_ d

_{2}_ ⇒ TEST b THEN extract d

_{1}ELSE extract d

_{2}FI

| DCWhile b _ d _ ⇒ WHILE b DO extract d END

| DCPre _ d ⇒ extract d

| DCPost d _ ⇒ extract d

end.

Definition extract_dec (dec : decorated) : com :=

match dec with

| Decorated P d ⇒ extract d

end.

Example extract_while_ex :

extract_dec dec_while = (WHILE ¬ X = 0 DO X ::= X - 1 END)%imp.

Proof.

unfold dec_while.

reflexivity.

Qed.

It is straightforward to extract the precondition and
postcondition from a decorated program.

Fixpoint post (d : dcom) : Assertion :=

match d with

| DCSkip P ⇒ P

| DCSeq d

_{1}d

_{2}⇒ post d

_{2}

| DCAsgn X a Q ⇒ Q

| DCIf _ _ d

_{1}_ d

_{2}Q ⇒ Q

| DCWhile b Pbody c Ppost ⇒ Ppost

| DCPre _ d ⇒ post d

| DCPost c Q ⇒ Q

end.

Definition pre_dec (dec : decorated) : Assertion :=

match dec with

| Decorated P d ⇒ P

end.

Definition post_dec (dec : decorated) : Assertion :=

match dec with

| Decorated P d ⇒ post d

end.

Example pre_dec_while : pre_dec dec_while = True.

Proof. reflexivity. Qed.

Example post_dec_while : post_dec dec_while = (X = 0)%assertion.

Proof. reflexivity. Qed.

We can express what it means for a decorated program to be
correct as follows:

Definition dec_correct (dec : decorated) :=

{{pre_dec dec}} (extract_dec dec) {{post_dec dec}}.

Example dec_while_triple_correct :

dec_correct dec_while

= {{ True }}

(WHILE ~(X = 0) DO X ::= X - 1 END)%imp

{{ X = 0 }}.

Proof. reflexivity. Qed.

To check whether this Hoare triple is
The function verification_conditions takes a dcom d together
with a precondition P and returns a

*valid*, we need a way to extract the "proof obligations" from a decorated program. These obligations are often called*verification conditions*, because they are the facts that must be verified to see that the decorations are logically consistent and thus constitute a proof of correctness.## Extracting Verification Conditions

*proposition*that, if it can be proved, implies that the triple {{P}} (extract d) {{post d}} is valid. It does this by walking over d and generating a big conjunction that includes- all the local consistency checks
- many uses of ->> to bridge the gap between (i) assertions found inside decorated commands and (ii) assertions used by the local consistency checks. These uses correspond applications of the consequence rule.

Fixpoint verification_conditions (P : Assertion) (d : dcom) : Prop :=

match d with

| DCSkip Q ⇒

(P ->> Q)

| DCSeq d

_{1}d

_{2}⇒

verification_conditions P d

_{1}

∧ verification_conditions (post d

_{1}) d

_{2}

| DCAsgn X a Q ⇒

(P ->> Q [X ⊢> a])

| DCIf b P

_{1}d

_{1}P

_{2}d

_{2}Q ⇒

((P ∧ b) ->> P

_{1})%assertion

∧ ((P ∧ ¬ b) ->> P

_{2})%assertion

∧ (post d

_{1}->> Q) ∧ (post d

_{2}->> Q)

∧ verification_conditions P

_{1}d

_{1}

∧ verification_conditions P

_{2}d

_{2}

| DCWhile b Pbody d Ppost ⇒

(* post d is the loop invariant and the initial

precondition *)

(P ->> post d)

∧ ((post d ∧ b) ->> Pbody)%assertion

∧ ((post d ∧ ¬ b) ->> Ppost)%assertion

∧ verification_conditions Pbody d

| DCPre P' d ⇒

(P ->> P') ∧ verification_conditions P' d

| DCPost d Q ⇒

verification_conditions P d ∧ (post d ->> Q)

end.

And now the key theorem, stating that verification_conditions
does its job correctly. Not surprisingly, we need to use each of
the Hoare Logic rules at some point in the proof.

Theorem verification_correct : ∀ d P,

verification_conditions P d → {{P}} (extract d) {{post d}}.

Proof.

induction d; intros; simpl in ×.

- (* Skip *)

eapply hoare_consequence_pre.

+ apply hoare_skip.

+ assumption.

- (* Seq *)

destruct H as [H

eapply hoare_seq.

+ apply IHd2. apply H

+ apply IHd1. apply H

- (* Asgn *)

eapply hoare_consequence_pre.

+ apply hoare_asgn.

+ assumption.

- (* If *)

destruct H as [HPre1 [HPre2 [Hd

apply IHd1 in HThen. clear IHd1.

apply IHd2 in HElse. clear IHd2.

apply hoare_if.

+ eapply hoare_consequence; eauto.

+ eapply hoare_consequence; eauto.

- (* While *)

destruct H as [Hpre [Hbody1 [Hpost1 Hd]]].

eapply hoare_consequence; eauto.

apply hoare_while.

eapply hoare_consequence_pre; eauto.

- (* Pre *)

destruct H as [HP Hd].

eapply hoare_consequence_pre; eauto.

- (* Post *)

destruct H as [Hd HQ].

eapply hoare_consequence_post; eauto.

Qed.

induction d; intros; simpl in ×.

- (* Skip *)

eapply hoare_consequence_pre.

+ apply hoare_skip.

+ assumption.

- (* Seq *)

destruct H as [H

_{1}H_{2}].eapply hoare_seq.

+ apply IHd2. apply H

_{2}.+ apply IHd1. apply H

_{1}.- (* Asgn *)

eapply hoare_consequence_pre.

+ apply hoare_asgn.

+ assumption.

- (* If *)

destruct H as [HPre1 [HPre2 [Hd

_{1}[Hd_{2}[HThen HElse]]]]].apply IHd1 in HThen. clear IHd1.

apply IHd2 in HElse. clear IHd2.

apply hoare_if.

+ eapply hoare_consequence; eauto.

+ eapply hoare_consequence; eauto.

- (* While *)

destruct H as [Hpre [Hbody1 [Hpost1 Hd]]].

eapply hoare_consequence; eauto.

apply hoare_while.

eapply hoare_consequence_pre; eauto.

- (* Pre *)

destruct H as [HP Hd].

eapply hoare_consequence_pre; eauto.

- (* Post *)

destruct H as [Hd HQ].

eapply hoare_consequence_post; eauto.

Qed.

Now that all the pieces are in place, we can verify an entire program.

Definition verification_conditions_dec (dec : decorated) : Prop :=

match dec with

| Decorated P d ⇒ verification_conditions P d

end.

Corollary verification_correct_dec : ∀ dec,

verification_conditions_dec dec → dec_correct dec.

Proof.

intros [P d]. apply verification_correct.

Qed.

The propositions generated by verification_conditions are fairly
big, and they contain many conjuncts that are essentially trivial.
Our verify_assn can often take care of them.

Example vc_dec_while :

verification_conditions_dec dec_while =

((((fun _ : state ⇒ True) ->> (fun _ : state ⇒ True)) ∧

((fun st : state ⇒ True ∧ negb (st X =? 0) = true) ->>

(fun st : state ⇒ True ∧ st X ≠ 0)) ∧

((fun st : state ⇒ True ∧ negb (st X =? 0) ≠ true) ->>

(fun st : state ⇒ True ∧ st X = 0)) ∧

(fun st : state ⇒ True ∧ st X ≠ 0) ->> (fun _ : state ⇒ True) [X ⊢> X - 1]) ∧

(fun st : state ⇒ True ∧ st X = 0) ->> (fun st : state ⇒ st X = 0)).

Proof. verify_assn. Qed.

## Automation

Ltac verify :=

intros;

apply verification_correct;

verify_assn.

Theorem dec_while_correct :

dec_correct dec_while.

Proof. verify. Qed.

intros;

apply verification_correct;

verify_assn.

Theorem dec_while_correct :

dec_correct dec_while.

Proof. verify. Qed.

Let's use that automation to verify formal decorated programs
corresponding to informal ones we have seen.

### Slow Subtraction

Example subtract_slowly_dec (m : nat) (p : nat) : decorated :=

{{ X = m ∧ Z = p }} ->>

{{ Z - X = p - m }}

WHILE ~(X = 0)

DO {{ Z - X = p - m ∧ X ≠ 0 }} ->>

{{ (Z - 1) - (X - 1) = p - m }}

Z ::= Z - 1

{{ Z - (X - 1) = p - m }} ;;

X ::= X - 1

{{ Z - X = p - m }}

END

{{ Z - X = p - m ∧ X = 0 }} ->>

{{ Z = p - m }}.

Theorem subtract_slowly_dec_correct : ∀ m p,

dec_correct (subtract_slowly_dec m p).

Proof. verify. (* this grinds for a bit! *) Qed.

Definition swap : com :=

X ::= X + Y;;

Y ::= X - Y;;

X ::= X - Y.

Definition swap_dec (m n:nat) : decorated :=

{{ X = m ∧ Y = n}} ->>

{{ (X + Y) - ((X + Y) - Y) = n

∧ (X + Y) - Y = m }}

X ::= X + Y

{{ X - (X - Y) = n ∧ X - Y = m }};;

Y ::= X - Y

{{ X - Y = n ∧ Y = m }};;

X ::= X - Y

{{ X = n ∧ Y = m}}.

Theorem swap_correct : ∀ m n,

dec_correct (swap_dec m n).

Proof. verify. Qed.

Definition div_mod_dec (a b : nat) : decorated :=

{{ True }} ->>

{{ b × 0 + a = a }}

X ::= a

{{ b × 0 + X = a }};;

Y ::= 0

{{ b × Y + X = a }};;

WHILE b ≤ X DO

{{ b × Y + X = a ∧ b ≤ X }} ->>

{{ b × (Y + 1) + (X - b) = a }}

X ::= X - b

{{ b × (Y + 1) + X = a }};;

Y ::= Y + 1

{{ b × Y + X = a }}

END

{{ b × Y + X = a ∧ ~(b ≤ X) }} ->>

{{ b × Y + X = a ∧ (X < b) }}.

Theorem div_mod_dec_correct : ∀ a b,

dec_correct (div_mod_dec a b).

Proof.

verify.

(* verify_assn didn't quite get all the way there *)

rewrite mult_plus_distr_l. omega.

Qed.

There are actually several ways to phrase the loop invariant for
this program. Here is one natural one, which leads to a rather
long proof:

Inductive ev : nat → Prop :=

| ev_0 : ev O

| ev_SS : ∀ n : nat, ev n → ev (S (S n)).

Definition find_parity_dec (m:nat) : decorated :=

{{ X = m }} ->>

{{ X ≤ m ∧ ap ev (m - X) }}

WHILE 2 ≤ X DO

{{ (X ≤ m ∧ ap ev (m - X)) ∧ 2 ≤ X }} ->>

{{ X - 2 ≤ m ∧ ap ev (m - (X - 2)) }}

X ::= X - 2

{{ X ≤ m ∧ ap ev (m - X) }}

END

{{ (X ≤ m ∧ ap ev (m - X)) ∧ X < 2 }} ->>

{{ X = 0 ↔ ev m }}.

Lemma l

_{1}: ∀ m n p,

p ≤ n →

n ≤ m →

m - (n - p) = m - n + p.

Proof. intros. omega. Qed.

Lemma l

_{2}: ∀ m,

ev m →

ev (m + 2).

Proof. intros. rewrite plus_comm. simpl. constructor. assumption. Qed.

Lemma l

_{3}' : ∀ m,

ev m →

¬ev (S m).

Proof. induction m; intros H

_{1}H

_{2}. inversion H

_{2}. apply IHm.

inversion H

_{2}; subst; assumption. assumption. Qed.

Lemma l

_{3}: ∀ m,

1 ≤ m →

ev m →

ev (m - 1) →

False.

Proof. intros. apply l

_{2}in H

_{1}.

assert (G : m - 1 + 2 = S m). clear H

_{0}H

_{1}. omega.

rewrite G in H

_{1}. apply l

_{3}' in H

_{0}. apply H

_{0}. assumption. Qed.

Theorem find_parity_correct : ∀ m,

dec_correct (find_parity_dec m).

Proof.

intro m. verify;

(* simplification too aggressive ... reverting a bit *)

fold (2 <=? (st X)) in *;

try rewrite leb_iff in *;

try rewrite leb_iff_conv in *; eauto; try omega.

- (* invariant holds initially *)

rewrite minus_diag. constructor.

- (* invariant preserved *)

rewrite l

_{1}; try assumption.

apply l

_{2}; assumption.

- (* invariant strong enough to imply conclusion

(-> direction) *)

rewrite <- minus_n_O in H

_{2}. assumption.

- (* invariant strong enough to imply conclusion

(<- direction) *)

destruct (st X) as [| [| n]].

(* by H

_{1}X can only be 0 or 1 *)

+ (* st X = 0 *)

reflexivity.

+ (* st X = 1 *)

apply l

_{3}in H; try assumption. inversion H.

+ (* st X = 2 *)

clear H

_{0}H

_{2}. (* omega confused otherwise *)

omega.

Qed.

Here is a more intuitive way of writing the invariant:

Definition find_parity_dec' (m:nat) : decorated :=

{{ X = m }} ->>

{{ ap ev X ↔ ev m }}

WHILE 2 ≤ X DO

{{ (ap ev X ↔ ev m) ∧ 2 ≤ X }} ->>

{{ ap ev (X - 2) ↔ ev m }}

X ::= X - 2

{{ ap ev X ↔ ev m }}

END

{{ (ap ev X ↔ ev m) ∧ ~(2 ≤ X) }} ->>

{{ X=0 ↔ ev m }}.

Lemma l

_{4}: ∀ m,

2 ≤ m →

(ev (m - 2) ↔ ev m).

Proof.

induction m; intros. split; intro; constructor.

destruct m. inversion H. inversion H

_{1}. simpl in ×.

rewrite <- minus_n_O in ×. split; intro.

constructor. assumption.

inversion H

_{0}. assumption.

Qed.

Theorem find_parity_correct' : ∀ m,

dec_correct (find_parity_dec' m).

Proof.

intros m. verify;

(* simplification too aggressive ... reverting a bit *)

fold (2 <=? (st X)) in *;

try rewrite leb_iff in *;

try rewrite leb_iff_conv in *; intuition; eauto; try omega.

- (* invariant preserved (part 1) *)

rewrite l

_{4}in H

_{0}; eauto.

- (* invariant preserved (part 2) *)

rewrite l

_{4}; eauto.

- (* invariant strong enough to imply conclusion

(-> direction) *)

apply H

_{0}. constructor.

- (* invariant strong enough to imply conclusion

(<- direction) *)

destruct (st X) as [| [| n]]. (* by H

_{1}X can only be 0 or 1 *)

+ (* st X = 0 *)

reflexivity.

+ (* st X = 1 *)

inversion H.

+ (* st X = 2 *)

clear H

_{0}H H

_{3}. (* omega confused otherwise *)

omega.

Qed.

Definition sqrt_dec (m:nat) : decorated :=

{{ X = m }} ->>

{{ X = m ∧ 0×0 ≤ m }}

Z ::= 0

{{ X = m ∧ Z×Z ≤ m }};;

WHILE (Z+1)*(Z+1) ≤ X DO

{{ (X = m ∧ Z×Z≤m)

∧ (Z + 1)*(Z + 1) ≤ X }} ->>

{{ X = m ∧ (Z+1)*(Z+1)≤m }}

Z ::= Z + 1

{{ X = m ∧ Z×Z≤m }}

END

{{ (X = m ∧ Z×Z≤m)

∧ ~((Z + 1)*(Z + 1) ≤ X) }} ->>

{{ Z×Z≤m ∧ m<(Z+1)*(Z+1) }}.

Theorem sqrt_correct : ∀ m,

dec_correct (sqrt_dec m).

Proof. verify. Qed.

### Squaring

Definition square_dec (m : nat) : decorated :=

{{ X = m }}

Y ::= X

{{ X = m ∧ Y = m }};;

Z ::= 0

{{ X = m ∧ Y = m ∧ Z = 0}} ->>

{{ Z + X × Y = m × m }};;

WHILE ¬ (Y = 0) DO

{{ Z + X × Y = m × m ∧ Y ≠ 0 }} ->>

{{ (Z + X) + X × (Y - 1) = m × m }}

Z ::= Z + X

{{ Z + X × (Y - 1) = m × m }};;

Y ::= Y - 1

{{ Z + X × Y = m × m }}

END

{{ Z + X × Y = m × m ∧ Y = 0 }} ->>

{{ Z = m × m }}.

Theorem square_dec_correct : ∀ m,

dec_correct (square_dec m).

Proof.

verify.

- (* invariant preserved *)

destruct (st Y) as [| y'].

+ exfalso. auto.

+ simpl. rewrite <- minus_n_O.

assert (G : ∀ n m, n × S m = n + n × m). {

clear. intros. induction n. reflexivity. simpl.

rewrite IHn. omega. }

rewrite <- H. rewrite G. omega.

Qed.

Definition square_dec' (n : nat) : decorated :=

{{ True }}

X ::= n

{{ X = n }};;

Y ::= X

{{ X = n ∧ Y = n }};;

Z ::= 0

{{ X = n ∧ Y = n ∧ Z = 0 }} ->>

{{ Z = X × (X - Y)

∧ X = n ∧ Y ≤ X }};;

WHILE ~(Y = 0) DO

{{ (Z = X × (X - Y)

∧ X = n ∧ Y ≤ X)

∧ Y ≠ 0 }}

Z ::= Z + X

{{ Z = X × (X - (Y - 1))

∧ X = n ∧ Y ≤ X }};;

Y ::= Y - 1

{{ Z = X × (X - Y)

∧ X = n ∧ Y ≤ X }}

END

{{ (Z = X × (X - Y)

∧ X = n ∧ Y ≤ X)

∧ Y = 0 }} ->>

{{ Z = n × n }}.

Theorem square_dec'_correct : ∀ (n:nat),

dec_correct (square_dec' n).

Proof.

verify.

- (* invariant holds initially *)

rewrite minus_diag. omega.

- (* invariant preserved *) subst.

rewrite mult_minus_distr_l.

repeat rewrite mult_minus_distr_l. rewrite mult_1_r.

assert (G : ∀ n m p,

m ≤ n → p ≤ m → n - (m - p) = n - m + p).

intros. omega.

rewrite G. reflexivity. apply mult_le_compat_l. assumption.

destruct (st Y).

+ exfalso. auto.

+ clear. rewrite mult_succ_r. rewrite plus_comm.

apply le_plus_l.

- (* invariant + negation of guard imply

desired postcondition *)

rewrite <- minus_n_O. reflexivity.

Qed.

Definition square_simpler_dec (m : nat) : decorated :=

{{ X = m }} ->>

{{ 0 = 0×m ∧ X = m }}

Y ::= 0

{{ 0 = Y×m ∧ X = m }};;

Z ::= 0

{{ Z = Y×m ∧ X = m }};;

WHILE ~(Y = X) DO

{{ (Z = Y×m ∧ X = m)

∧ Y ≠ X }} ->>

{{ Z + X = (Y + 1)*m ∧ X = m }}

Z ::= Z + X

{{ Z = (Y + 1)*m ∧ X = m }};;

Y ::= Y + 1

{{ Z = Y×m ∧ X = m }}

END

{{ (Z = Y×m ∧ X = m) ∧ Y = X }} ->>

{{ Z = m×m }}.

Theorem square_simpler_dec_correct : ∀ m,

dec_correct (square_simpler_dec m).

Proof.

verify.

rewrite mult_plus_distr_r. omega.

Qed.

Fixpoint pow2 n :=

match n with

| 0 ⇒ 1

| S n' ⇒ 2 × (pow2 n')

end.

Definition dpow2_down (n : nat) :=

{{ True }} ->>

{{ 1 = (pow2 (0 + 1))-1 ∧ 1 = pow2 0 }}

X ::= 0

{{ 1 = (pow2 (0 + 1))-1 ∧ 1 = ap pow2 X }};;

Y ::= 1

{{ Y = (ap pow2 (X + 1))-1 ∧ 1 = ap pow2 X}};;

Z ::= 1

{{ Y = (ap pow2 (X + 1))-1 ∧ Z = ap pow2 X }};;

WHILE ~(X = n) DO

{{ (Y = (ap pow2 (X + 1))-1 ∧ Z = ap pow2 X)

∧ X ≠ n }} ->>

{{ Y + 2 × Z = (ap pow2 (X + 2))-1

∧ 2 × Z = ap pow2 (X + 1) }}

Z ::= 2 × Z

{{ Y + Z = (ap pow2 (X + 2))-1

∧ Z = ap pow2 (X + 1) }};;

Y ::= Y + Z

{{ Y = (ap pow2 (X + 2))-1

∧ Z = ap pow2 (X + 1) }};;

X ::= X + 1

{{ Y = (ap pow2 (X + 1))-1

∧ Z = ap pow2 X }}

END

{{ (Y = (ap pow2 (X + 1))-1 ∧ Z = ap pow2 X)

∧ X = n }} ->>

{{ Y = pow2 (n+1) - 1 }}.

Lemma pow2_plus_1 : ∀ n,

pow2 (n+1) = pow2 n + pow2 n.

Proof. induction n; simpl. reflexivity. omega. Qed.

Lemma pow2_le_1 : ∀ n, pow2 n ≥ 1.

Proof. induction n. simpl. constructor. simpl. omega. Qed.

Theorem dpow2_down_correct : ∀ n,

dec_correct (dpow2_down n).

Proof.

intros m. verify.

- (* 1 *)

rewrite pow2_plus_1. rewrite <- H

_{0}. reflexivity.

- (* 2 *)

rewrite <- plus_n_O.

rewrite <- pow2_plus_1. remember (st X) as n.

replace (pow2 (n + 1) - 1 + pow2 (n + 1))

with (pow2 (n + 1) + pow2 (n + 1) - 1) by omega.

rewrite <- pow2_plus_1.

replace (n + 1 + 1) with (n + 2) by omega.

reflexivity.

- (* 3 *)

rewrite <- plus_n_O. rewrite <- pow2_plus_1.

reflexivity.

- (* 4 *)

replace (st X + 1 + 1) with (st X + 2) by omega.

reflexivity.

Qed.

## Further Exercises

#### Exercise: 3 stars, advanced (slow_assignment_dec)

Transform the informal decorated program your wrote for slow_assignment into a formal decorated program. If all goes well, the only change you will need to make is to move semicolons, which go after the postcondition of an assignment in a formal decorated program. For example,{{ X = m ∧ 0 = 0 }}

Y ::= 0;;

{{ X = m ∧ Y = 0 }}

becomes

{{ X = m ∧ 0 = 0 }}

Y ::= 0

{{ X = m ∧ Y = 0 }} ;;

Example slow_assignment_dec (m : nat) : decorated

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Now prove the correctness of your decorated program. If all goes well,
you will need only verify.

Theorem slow_assignment_dec_correct : ∀ m,

dec_correct (slow_assignment_dec m).

Proof. (* FILL IN HERE *) Admitted.

(* Do not modify the following line: *)

Definition manual_grade_for_check_defn_of_slow_assignment_dec : option (nat×string) := None.

☐

#### Exercise: 4 stars, advanced (factorial_dec)

The factorial function is defined recursively in the Coq standard library in a way that is equivalent to the following:Fixpoint fact (n : nat) : nat :=

match n with

| O ⇒ 1

| S n' ⇒ n × (fact n')

end.

Using your solutions to factorial and slow_assignment_dec as a
guide, write a formal decorated program factorial_dec that
implements the factorial function. Hint: recall the use of ap in
assertions to apply a function to an Imp variable.
Then state a theorem named factorial_dec_correct that says
factorial_dec is correct, and prove the theorem. If all goes
well, verify will leave you with just two subgoals, each of
which requires establishing some mathematical property of fact,
rather than proving anything about your program. Hint: if those
two subgoals become tedious to prove, give some though to how
you could restate your assertions such that the mathematical
operations are more amenable to manipulation in Coq. For example,
recall that 1 + ... is easier to work with than ... + 1.

(* FILL IN HERE *)

(* Do not modify the following line: *)

Definition manual_grade_for_factorial_dec : option (nat×string) := None.

☐

#### Exercise: 2 stars, advanced, optional (fib_eqn)

The Fibonacci function is usually written like this:Fixpoint fib n :=

match n with

| 0 ⇒ 1

| 1 ⇒ 1

| _ ⇒ fib (pred n) + fib (pred (pred n))

end.

Fixpoint fib n :=

match n with

| 0 ⇒ 1

| S n' ⇒ match n' with

| 0 ⇒ 1

| S n'' ⇒ fib n' + fib n''

end

end.

Prove that fib satisfies the following equation. You will need this
as a lemma in the next exercise.

Lemma fib_eqn : ∀ n,

n > 0 →

fib n + fib (pred n) = fib (1 + n).

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 4 stars, advanced, optional (fib)

The following Imp program leaves the value of fib n in the variable Y when it terminates:X ::= 1;;

Y ::= 1;;

Z ::= 1;;

WHILE ~(X = 1 + n) DO

T ::= Z;;

Z ::= Z + Y;;

Y ::= T;;

X ::= 1 + X

END

{{ True }} dfib {{ Y = fib n }}

Definition T : string := "T".

Definition dfib (n : nat) : decorated

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem dfib_correct : ∀ n,

dec_correct (dfib n).

(* FILL IN HERE *) Admitted.

☐

Close Scope dcom_scope.

#### Exercise: 5 stars, advanced, optional (improve_dcom)

The formal decorated programs defined in this section are intended to look as similar as possible to the informal ones defined earlier in the chapter. If we drop this requirement, we can eliminate almost all annotations, just requiring final postconditions and loop invariants to be provided explicitly. Do this -- i.e., define a new version of dcom with as few annotations as possible and adapt the rest of the formal development leading up to the verification_correct theorem.(* FILL IN HERE *)

☐

#### Exercise: 4 stars, advanced, optional (implement_dcom)

Adapt the parser for Imp presented in chapter ImpParser of*Logical Foundations*to parse decorated commands (either ours or, even better, the ones you defined in the previous exercise).

(* FILL IN HERE *)

☐

(* 30 Apr 2020 *)