Peano Arithmetic

Section Arith.

# The natural numbers

Guiseppe Peano defined the natural numbers as given by 0 : nat and if n is a natural number then S n : nat is a natural number called the successor of n. Given this we can construct all the natural numbers, e.g.
• 1 = S 0
• 2 = S 1 = S (S 0)
• 3 = S 2 = S (S (S 0))
Moreover these are all natural numbers (we say they are defined inductively). Peano went on to represent the fundamental properties of the natural numbers using axioms. Some of the axioms express general properties of equality, which we have already seen. But the following three are specific to the natural numbers. Indeed, they are provable propositions in Coq:
• Axiom 7 : 0 is not the successor of any number. forall n:nat, S n <> 0
• Axiom 8 : If two numbers have the same successor, then they are equal. forall m n:nat, S m = S n -> m = n
• Axiom 9 : If any property holds for 0, and is closed under successor, then it holds for all natural numbers (principle of induction). forall P : nat -> Prop, P 0 -> forall( m : nat, P m -> P (S m)) -> forall n : nat, P n
For illustration we are going to prove these principles:

Lemma peano7 : forall n:nat, S n <> 0.
intro n.
intro h.

This is basically the same problem as proving true <> false, we could apply the same technique here. To avoid repetetion we just use the discriminate tactic.

discriminate h.
Qed.

To prove the next axiom, it is useful to define the inverse to S, the predecessor function pred. We arbitrarily decide that the predecessor of 0 is 0.

Definition pred (n : nat) : nat :=
match n with
| 0 => 0
| S n => n
end.

Lemma peano8 : forall m n:nat, S m = S n -> m = n.
intros m n h.

By folding with pred we can change the current goal so that we can apply our hypothesis.

fold (pred (S m)).
rewrite h.

And now we just have to unfold. simpl would have done the job too.

unfold pred.
reflexivity.
Qed.

The 8th axiom says that the successor function is injective. Can we prove the other direction too? forall m n:nat, m = n -> S m = S n Does this tell us anything new about the successor function?

The proof of the induction axiom is rather boring. It just uses a tactic which is called induction...

Lemma peano9 : forall P : nat -> Prop, P 0
-> (forall m : nat, P m -> P (S m))
-> forall n : nat, P n.
intros P h0 hS n.
induction n.
exact h0.
apply hS.
exact IHn.
Qed.

Peano defined the operations addition and multiplication. These are actually examples of functions defined by primitive recursion a general scheme which can be used to define many other functions. A function is definable by primitive recursion if we can give a case for 0 and reduce the computation for the value at S n to the value at n. In Coq we have to use the keyword fixpoint instead of definition and we have to indicate on which argument we want to do primitive recursion.

The idea is that we can define addition like this:
• to add 0 to a number is just this number,
• to add one more that n to a number is one more than adding n to the number.

Fixpoint add (m n : nat) {struct m} : nat :=
match m with
| 0 => n
| S m => S (add m n)
end.

Eval compute in (add 2 3).

In the Coq library addition is defined using the usual infix notation +.

To define multiplication we use primitive recursion again. This time the idea is the following.
• multiplying 0 with a number is just 0.
• multiplying one more than n with a number is obtained by adding the number to multiplying n with the number.

Fixpoint mult (m n : nat) {struct m} : nat :=
match m with
| 0 => 0
| S m => add n (mult m n)
end.

Eval compute in (mult 2 3).

In the Coq library addition is defined using the usual infix notation + and * with the usual rules of precedence. From now on we shall use the library versions which are defined exactly in the same way as we have defined add and mult

# Algebraic properties

Addition and multiplication satisfy a number of important equations:
• 0 is a neutral element for addition 0 + m = m and m + 0 = m
• Addition is associative. m + (n + l) = (m + l) + n
• Addition is commutative. m + n = n + m
• 1 is a neutral element for multiplication 1 * m = m and m * 1 = m
• Multiplication is associative. m * (n * l) = (m * n) * l
• Multiplication is commutative. m * n = n * m
• 0 is a null for multiplication. m * 0 = 0 and 0 * m = 0
• Addition distributes over multiplication. m * (n + l) = m * n + m * l and (m + n) * l = m * l + n * l
In the language of universal algebra, we say that
• (+,0) is a commutative monoid, because 0 is neutral, + is associative and commutative.
• ( *,1) is a commutative monoid, because 1 is neutral, * is associative and commutative.
• (+,0,*,1) is a commutative semiring because (+,0) and ( *,1) are commutative monoids and 0 is a zero for multiplication and addition distributes over multiplication.
We are going to prove that (+,0) is a commutative monoid and leave the remaining properties as an exercise.

Lemma plus_O_n : forall n:nat, n = 0 + n.
This property is very easy to prove. Can you see why?
intro n.
reflexivity.
Qed.

Lemma plus_n_O : forall n:nat, n = n + 0.
intro n.

This one cannot be proven by reflexivity. So we have to use induction.

induction n.

n = 0 This is easy.

simpl.
reflexivity.

We can simplify S n + 0 using the definition of +

simpl.
rewrite<- IHn.
reflexivity.
Qed.

Lemma plus_assoc : forall (l m n:nat),l + (m + n) = (l + m) + n.
intros l m n.

There seems to be quite a choice what to do induction over: l,m,n but only one of them works. Why?

induction l.
simpl.
reflexivity.
simpl.
rewrite IHl.
reflexivity.
Qed.

To prove commutativity we first prove a lemma we know already that 0 + m = m = m + 0 but what about S m + n = S (m + n) = m + S n ?

Lemma plus_n_Sm : forall n m : nat, S (m + n) = m + S n.
intros.
induction m.
simpl.
reflexivity.
simpl.
rewrite IHm.
reflexivity.
Qed.

We are now ready to prove commutativity.

Lemma plus_comm : forall n m:nat, n + m = m + n.
intros.
induction n.
simpl.
apply plus_n_O.
simpl.
rewrite IHn.
apply plus_n_Sm.
Qed.

# Ordering the numbers

We define the relation <= on natural numbers by saying that m <= n holds if there is a number k such that m = k + n.

Definition leq (m n : nat) : Prop :=
exists k : nat, n = k + m.

Notation "m <= n" := (leq m n).

We verify some basic properties of <=:
• <= is reflexive. forall n:nat, n <= n
• <= is transitive. forall l m n:nat, l <= m -> m <= n -> l <= n
• <= is antisymmetric. forall l m : nat, l <= m -> m <= l -> m = l
Any relation which is reflexive, transitive and antisymmetric is a partial order. Here the word partial is used to differentiate <= from a total order like <. We verify the first two properties in Coq, but leave antisymmetry as an exercise.

Lemma le_refl: forall n:nat,n <= n.
intro n.
exists 0.
reflexivity.
Qed.

Lemma le_trans : forall (l m n : nat), l <= m -> m <= n -> l <= n.
intros l m n lm mn.
destruct lm as [k klm].
destruct mn as [j jmn].
exists (j+k).
rewrite<- plus_assoc.
rewrite<- klm.
rewrite<- jmn.
reflexivity.
Qed.

# Decidable properties

We say a predicate is P : A -> Prop decidable if we can define a boolean function decP : A -> bool which agrees with the predicate, i.e. forall a:A, P a <-> decP a = true. This also extends to relations in the obvious way.

We show below that equality on natural numbers is decidable. Do you know any undecidable predicates? Is equality always decidable?

First we define the decision procedure. In the case of equality this is quite obvious: we inspect both parameters, if they start with different constructors (i.e. 0 vs S) they are certainly not equal. If they are both 0 they are equal, and if they both start with S then we recursively compare the arguments.

Fixpoint eqnat (m n : nat) {struct m} : bool :=
match m with
| 0 => match n with
| 0 => true
| S n' => false
end
| S m' => match n with
| 0 => false
| S n' => eqnat m' n'
end
end.

Now we show both direction seperately. The -> direction just boils down to showing that eqnat is reflexive. Why?

Lemma eqnat_refl : forall m : nat, eqnat m m = true.
intro m.
induction m.
reflexivity.
simpl.
exact IHm.
Qed.

The other direction is more interesting and requires a double induction over m and n.

Lemma eqnat_compl : forall m n : nat, eqnat m n = true -> m = n.
intro m.
Here it would have been a mistake to do intros m n. Why?
m = 0
induction m.
intro n.
induction n.
n = 0
intro h.
reflexivity.
n = S n'
intro h.
simpl in h.
discriminate h.
m = S m'
intro n.
induction n.
n = 0
intro h.
discriminate h.
n = S n'
intro h.
assert (h' : m = n).
apply IHm.
exact h.
rewrite h'.
reflexivity.
Qed.

Finally, we can prove the theorem that equality for natural numbers is decidable.

Theorem eqnat_dec : forall m n : nat, m = n <-> eqnat m n = true.
intros m n.
split.
intro h.
rewrite h.
apply eqnat_refl.
apply eqnat_compl.
Qed.

End Arith.