(** * Perm: Basic Techniques for Permutations and Ordering *)
(** Consider these algorithms and data structures:
- sort a sequence of numbers;
- finite maps from numbers to (arbitrary-type) data
- finite maps from any ordered type to (arbitrary-type) data
- priority queues: finding/deleting the highest number in a set
To prove the correctness of such programs, we need to reason
about less-than comparisons (for example, on integers) and about
"these two sets/sequences have the same contents". In this
chapter, we introduce some techniques for reasoning about:
- less-than comparisons on natural numbers
- permutations (rearrangements of lists)
Then, in later chapters, we'll apply these proof techniques
to reasoning about algorithms and data structures.
*)
From Coq Require Import Strings.String.
Require Export Coq.Bool.Bool.
Require Export Coq.Arith.Arith.
Require Export Coq.Arith.EqNat.
Require Export Coq.omega.Omega.
Require Export Coq.Lists.List.
Export ListNotations.
Require Export Permutation.
(* ################################################################# *)
(** * The Less-Than Order on the Natural Numbers *)
(** These [Check] and [Locate] commands remind us about
_Propositional_ and the _Boolean_ less-than operators
in the Coq standard library. *)
Check Nat.lt. (* : nat -> nat -> Prop *)
Check lt. (* : nat -> nat -> Prop *)
Goal Nat.lt = lt. Proof. reflexivity. Qed. (* They are the same *)
Check Nat.ltb. (* : nat -> nat -> bool *)
Locate "_ < _". (* "x < y" := lt x y *)
Locate " n < m *)
(** For some reason, the Coq library has [ =? b" := (Nat.leb b a)
(at level 70, only parsing) : nat_scope.
Notation "a >? b" := (Nat.ltb b a)
(at level 70, only parsing) : nat_scope.
(* ================================================================= *)
(** ** Relating [Prop] to [bool] *)
(** The [reflect] relation connects a [Proposition] to a [Boolean]. *)
Print reflect.
(*
Inductive reflect (P : Prop) : bool -> Set :=
| ReflectT : P -> reflect P true
| ReflectF : ~ P -> reflect P false
*)
(** That is, [reflect P b] means that [P<->True] if and only if [b=true].
The way to use [reflect] is, for each of your operators, make a
lemma like these next three:
*)
Lemma eqb_reflect : forall x y, reflect (x = y) (x =? y).
Proof.
intros x y.
apply iff_reflect. symmetry. apply Nat.eqb_eq.
Qed.
Lemma ltb_reflect : forall x y, reflect (x < y) (x
~ (k - 3 <= j) ->
k > i.
Proof.
intros.
(** Now, there's a hard way to prove this, and an easy way.
Here's the hard way. *)
(* try to remember the name of the lemma about negation and [<=] *)
Search (~ _ <= _ -> _).
apply not_le in H0.
(* try to remember the name of the transitivity lemma about [>] *)
Search (_ > _ -> _ > _ -> _ > _).
apply gt_trans with j.
apply gt_trans with (k-3).
(* _OBVIOUSLY_, [k] is greater than [k-3]. But _OOPS_,
this is not actually true, because we are talking about
natural numbers with "bogus subtraction." *)
Abort.
Theorem bogus_subtraction: ~ (forall k:nat, k > k - 3).
Proof.
(* [intro] introduces exactly one thing, like [intros ?] *)
intro.
(* [specialize] applies a hypothesis to an argument *)
specialize (H O).
simpl in H. inversion H.
Qed.
(** With bogus subtraction, this omega_example1 theorem even True?
Yes it is; let's try again, the hard way, to find the proof. *)
Theorem omega_example1:
forall i j k,
i < j ->
~ (k - 3 <= j) ->
k > i.
Proof. (* try again! *)
intros.
apply not_le in H0.
unfold gt in H0.
unfold gt.
(* try to remember the name ... *)
Search (_ < _ -> _ <= _ -> _ < _).
apply lt_le_trans with j.
apply H.
apply le_trans with (k-3).
Search (_ < _ -> _ <= _).
apply lt_le_weak.
auto.
apply le_minus.
Qed. (* Oof! That was exhausting and tedious. *)
(** And here's the easy way. *)
Theorem omega_example2:
forall i j k,
i < j ->
~ (k - 3 <= j) ->
k > i.
Proof.
intros.
omega.
Qed.
(** Here we have used the [omega] tactic, made available by importing
[Coq.omega.Omega] as we have done above. Omega is an algorithm
for integer linear programming, invented in 1991 by William Pugh.
Because ILP is NP-complete, we might expect that this algorithm is
exponential-time in the worst case, and indeed that's true: if you
have [N] equations, it could take [2^N] time. But in the typical
cases that result from reasoning about programs, omega is much
faster than that. Coq's [omega] tactic is an implementation of
this algorithm that generates a machine-checkable Coq proof. It
"understands" the types Z and nat, and these operators: [<] [=] [>] [<=]
[>=] [+] [-] [~], as well as multiplication by small integer
literals (such as 0,1,2,3...) and some uses of [\/] and [/\].
Omega does _not_ understand other operators. It treats things
like [a*b] and [f x y] as if they were variables. That is, it can
prove [f x y > a*b -> f x y + 3 >= a*b], in the same way it would
prove [u > v -> u+3 >= v].
Now let's consider a silly little program: swap the first two
elements of a list, if they are out of order. *)
Definition maybe_swap (al: list nat) : list nat :=
match al with
| a :: b :: ar => if a >? b then b::a::ar else a::b::ar
| _ => al
end.
Example maybe_swap_123:
maybe_swap [1; 2; 3] = [1; 2; 3].
Proof. reflexivity. Qed.
Example maybe_swap_321:
maybe_swap [3; 2; 1] = [2; 3; 1].
Proof. reflexivity. Qed.
(** In this program, we wrote [a>?b] instead of [a>b]. Why is that? *)
Check (1>2). (* : Prop *)
Check (1>?2). (* : bool *)
(** We cannot compute with elements of [Prop]: we need some kind of
constructible (and pattern-matchable) value. For that we use
[bool]. *)
Locate ">?". (* a >? b := ltb b a *)
(** The name [ltb] stands for "less-than boolean." *)
Print Nat.ltb.
(* = fun n m : nat => S n <=? m : nat -> nat -> bool *)
Locate ">=?".
(** Instead of defining an operator [Nat.geb], the standard library just
defines the notation for greater-or-equal-boolean as a
less-or-equal-boolean with the arguments swapped. *)
Locate leb.
Print leb.
Print Nat.leb. (* The computation to compare natural numbers. *)
(** Here's a theorem: [maybe_swap] is idempotent -- that is, applying it
twice gives the same result as applying it once. *)
Theorem maybe_swap_idempotent:
forall al, maybe_swap (maybe_swap al) = maybe_swap al.
Proof.
intros.
destruct al as [ | a al].
simpl.
reflexivity.
destruct al as [ | b al].
simpl.
reflexivity.
simpl.
(** What do we do here? We must proceed by case analysis on
whether a>b. *)
destruct (b b]; not upon comparisons yielding bool, such as [a>?b].
We need to convert these comparisons to [Prop], so that we can use
[omega].
Actually, we don't "need" to. Instead, we could reason directly
about these operations in [bool]. But that would be even more
tedious than the [omega_example1] proof. Therefore: let's set up
some machinery so that we can use [omega] on boolean tests. *)
Abort.
(** Let's try again, a new way: *)
Theorem maybe_swap_idempotent:
forall al, maybe_swap (maybe_swap al) = maybe_swap al.
Proof.
intros.
destruct al as [ | a al].
simpl.
reflexivity.
destruct al as [ | b al].
simpl.
reflexivity.
simpl.
(** This is where we left off before. Now, watch: *)
destruct (ltb_reflect b a). (* THIS LINE *)
(* Notice that [b= b *)
simpl.
bdestruct (b list A -> Prop *)
(** We say "list [al] is a permutation of list [bl]",
written [Permutation al bl], if the elements of [al] can be
reordered (without insertions or deletions) to get the list [bl]. *)
Print Permutation.
(*
Inductive Permutation {A : Type} : list A -> list A -> Prop :=
perm_nil : Permutation [] []
| perm_skip : forall (x : A) (l l' : list A),
Permutation l l' ->
Permutation (x :: l) (x :: l')
| perm_swap : forall (x y : A) (l : list A),
Permutation (y :: x :: l) (x :: y :: l)
| perm_trans : forall l l' l'' : list A,
Permutation l l' ->
Permutation l' l'' ->
Permutation l l''.
*)
(** You might wonder, "is that really the right definition?" And
indeed, it's important that we get a right definition, because
[Permutation] is going to be used in the specification of
correctness of our searching and sorting algorithms. If we have
the wrong specification, then all our proofs of "correctness" will
be useless.
It's not obvious that this is indeed the right specification of
permutations. (It happens to be true, but it's not obvious!) In
order to gain confidence that we have the right specification, we
should use this specification to prove some properties that we
think permutations ought to have. *)
(** **** Exercise: 2 stars, standard, optional (Permutation_properties)
Think of some properties of the [Permutation] relation and write
them down informally in English, or a mix of Coq and English.
Here are four to get you started:
- 1. If [Permutation al bl], then [length al = length bl].
- 2. If [Permutation al bl], then [Permutation bl al].
- 3. [[1;1]] is NOT a permutation of [[1;2]].
- 4. [[1;2;3;4]] IS a permutation of [[3;4;2;1]].
YOUR ASSIGNMENT: Add three more properties. Write them here: *)
(** Now, let's examine all the theorems in the Coq library about
permutations: *)
Search Permutation. (* Browse through the results of this query! *)
(** Which of the properties that you wrote down above have already
been proved as theorems by the Coq library developers? Answer
here:
[] *)
(** Let's use the permutation rules in the library to prove the
following theorem. *)
Example butterfly: forall b u t e r f l y : nat,
Permutation ([b;u;t;t;e;r]++[f;l;y]) ([f;l;u;t;t;e;r]++[b;y]).
Proof.
intros.
(* Just to illustrate a method, let's group [u;t;t;e;r] together: *)
change [b;u;t;t;e;r] with ([b]++[u;t;t;e;r]).
change [f;l;u;t;t;e;r] with ([f;l]++[u;t;t;e;r]).
remember [u;t;t;e;r] as utter.
clear Hequtter.
(* Next, let's cancel [utter] from both sides. In order to do that,
we need to bring [utter] to the beginning of each list. *)
Check app_assoc.
rewrite <- app_assoc.
rewrite <- app_assoc.
Check perm_trans.
apply perm_trans with (utter ++ [f;l;y] ++ [b]).
rewrite (app_assoc utter [f;l;y]).
Check Permutation_app_comm.
apply Permutation_app_comm.
eapply perm_trans.
2: apply Permutation_app_comm.
rewrite <- app_assoc.
Search (Permutation (_++_) (_++_)).
apply Permutation_app_head.
(* Now that [utter] is utterly removed from the goal, let's cancel [f;l]. *)
eapply perm_trans.
2: apply Permutation_app_comm.
simpl.
Check perm_skip.
apply perm_skip.
apply perm_skip.
Search (Permutation (_::_) (_::_)).
apply perm_swap.
Qed.
(** That example illustrates a general method for proving
permutations involving cons [::] and append [++].
You identify some portion appearing in both sides;
you bring that portion to the front on each side using
lemmas such as [Permutation_app_comm] and [perm_swap],
with generous use of [perm_trans]. Then, you use
[perm_skip] to cancel a single element, or [Permutation_app_head]
to cancel an append-chunk. *)
(** **** Exercise: 3 stars, standard (permut_example)
Use the permutation rules in the library (see the [Search],
above) to prove the following theorem. These [Check] commands
are a hint about useful lemmas; you won't necessarily need
all of them. *)
Check perm_skip.
Check perm_trans.
Check Permutation_refl.
Check Permutation_app_comm.
Check app_assoc.
Check app_nil_r.
Example permut_example: forall (a b: list nat),
Permutation (5::6::a++b) ((5::b)++(6::a++[])).
Proof.
(* Hint: after you cancel the [5], bring the [6] to the front. *)
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, standard (not_a_permutation)
Prove that [[1;1]] is not a permutation of [[1;2]].
Hints are given as [Check] commands. *)
Check Permutation_cons_inv.
Check Permutation_length_1_inv.
Example not_a_permutation:
~ Permutation [1;1] [1;2].
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Back to [maybe_swap]. We prove that it doesn't lose or gain
any elements, only reorders them. *)
Theorem maybe_swap_perm: forall al,
Permutation al (maybe_swap al).
Proof.
(* WORK IN CLASS *) Admitted.
(** Now let us specify functional correctness of [maybe_swap]:
it rearranges the elements in such a way that the first is
less than or equal to the second. *)
Definition first_le_second (al: list nat) : Prop :=
match al with
| a::b::_ => a <= b
| _ => True
end.
Theorem maybe_swap_correct: forall al,
Permutation al (maybe_swap al)
/\ first_le_second (maybe_swap al).
Proof.
intros.
split.
apply maybe_swap_perm.
(* WORK IN CLASS *) Admitted.
End Exploration1.
(* ################################################################# *)
(** * Summary: Comparisons and Permutations *)
(** To prove correctness of algorithms for sorting and searching,
we'll reason about comparisons and permutations using the tools
developed in this chapter. The [maybe_swap] program is a tiny
little example of a sorting program. The proof style in
[maybe_swap_correct] will be applied (at a larger scale) in
the next few chapters. *)
(** **** Exercise: 2 stars, standard (Forall_perm)
To close, a useful utility tactic and lemma. First, the tactic.
Coq's [inversion H] tactic is so good at extracting information
from the hypothesis [H] that [H] becomes completely redundant,
and one might as well [clear] it from the goal. Then, since the
[inversion] typically creates some equality facts, why not then
[subst] ? This motivates the following useful tactic, [inv]. *)
Ltac inv H := inversion H; clear H; subst.
(** Second, the lemma. You will find [inv] useful in proving it.
Prove the lemma by induction. You will need to decide whether
to induct on [al], [bl], [Permutation al bl], or [Forall f al]. *)
Theorem Forall_perm: forall {A} (f: A -> Prop) al bl,
Permutation al bl ->
Forall f al -> Forall f bl.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)