Library Coq.Numbers.NatInt.NZDomain
In this file, we investigate the shape of domains satisfying
the NZDomainSig interface. In particular, we define a
translation from Peano numbers nat into NZ.
First, a section about iterating a function.
Section Iter.
Variable A : Type.
Fixpoint iter (f:A->A)(n:nat) : A -> A := fun a =>
match n with
| O => a
| S n => f (iter f n a)
end.
Infix "^" := iter.
Lemma iter_alt : forall f n m, (f^(Datatypes.S n)) m = (f^n) (f m).
Lemma iter_plus : forall f n n' m, (f^(n+n')) m = (f^n) ((f^n') m).
Lemma iter_plus_bis : forall f n n' m, (f^(n+n')) m = (f^n') ((f^n) m).
Global Instance iter_wd (R:relation A) : Proper ((R==>R)==>eq==>R==>R) iter.
End Iter.
Implicit Arguments iter [A].
Local Infix "^" := iter.
Module NZDomainProp (Import NZ:NZDomainSig').
We prove that any points in NZ have a common descendant by succ
Definition common_descendant n m := exists k, exists l, (S^k) n == (S^l) m.
Instance common_descendant_wd : Proper (eq==>eq==>iff) common_descendant.
Instance common_descendant_equiv : Equivalence common_descendant.
Lemma common_descendant_with_0 : forall n, common_descendant n 0.
Lemma common_descendant_always : forall n m, common_descendant n m.
Thanks to succ being injective, we can then deduce that for any two
points, one is an iterated successor of the other.
Generalized version of pred_succ when iterating
From a given point, all others are iterated successors
or iterated predecessors.
In particular, all points are either iterated successors of 0
or iterated predecessors of 0 (or both).
Definition initial n := forall m, n ~= S m.
Lemma initial_alt : forall n, initial n <-> S (P n) ~= n.
Lemma initial_alt2 : forall n, initial n <-> ~exists m, n == S m.
First case: let's assume such an initial point exists
(i.e. S isn't surjective)...
... then we have unicity of this initial point.
... then all other points are descendant of it.
NB : We would like to have pred n == n for the initial element,
but nothing forces that. For instance we can have -3 as initial point,
and P(-3) = 2. A bit odd indeed, but legal according to NZDomainSig.
We can hence have n == (P^k) m without exists k', m == (S^k') n.
We need decidability of eq (or classical reasoning) for this:
Section SuccPred.
Hypothesis eq_decidable : forall n m, n==m \/ n~=m.
Lemma succ_pred_approx : forall n, ~initial n -> S (P n) == n.
End SuccPred.
End InitialExists.
Second case : let's suppose now S surjective, i.e. no initial point.
Section InitialDontExists.
Hypothesis succ_onto : forall n, exists m, n == S m.
Lemma succ_onto_gives_succ_pred : forall n, S (P n) == n.
Lemma succ_onto_pred_injective : forall n m, P n == P m -> n == m.
End InitialDontExists.
To summarize:
S is always injective, P is always surjective (thanks to pred_succ).
I) If S is not surjective, we have an initial point, which is unique. This bottom is below zero: we have N shifted (or not) to the left. P cannot be injective: P init = P (S (P init)). (P init) can be arbitrary.
II) If S is surjective, we have forall n, S (P n) = n, S and P are bijective and reciprocal.
IIa) if exists k<>O, 0 == S^k 0, then we have a cyclic structure Z/nZ IIb) otherwise, we have Z
S is always injective, P is always surjective (thanks to pred_succ).
I) If S is not surjective, we have an initial point, which is unique. This bottom is below zero: we have N shifted (or not) to the left. P cannot be injective: P init = P (S (P init)). (P init) can be arbitrary.
II) If S is surjective, we have forall n, S (P n) = n, S and P are bijective and reciprocal.
IIa) if exists k<>O, 0 == S^k 0, then we have a cyclic structure Z/nZ IIb) otherwise, we have Z
It is weaker than bi_induction. For instance it cannot prove that
we can go from one point by many S or many P, but only by many
S mixed with many P. Think of a model with two copies of N:
0, 1=S 0, 2=S 1, ... 0', 1'=S 0', 2'=S 1', ...
and P 0 = 0' and P 0' = 0.
0, 1=S 0, 2=S 1, ... 0', 1'=S 0', 2'=S 1', ...
and P 0 = 0' and P 0' = 0.
Lemma bi_induction_pred :
forall A : t -> Prop, Proper (eq==>iff) A ->
A 0 -> (forall n, A n -> A (S n)) -> (forall n, A n -> A (P n)) ->
forall n, A n.
Lemma central_induction_pred :
forall A : t -> Prop, Proper (eq==>iff) A -> forall n0,
A n0 -> (forall n, A n -> A (S n)) -> (forall n, A n -> A (P n)) ->
forall n, A n.
End NZDomainProp.
We now focus on the translation from nat into NZ.
First, relationship with 0, succ, pred.
Module NZOfNat (Import NZ:NZDomainSig').
Definition ofnat (n : nat) : t := (S^n) 0.
Notation "[ n ]" := (ofnat n) (at level 7) : ofnat.
Local Open Scope ofnat.
Lemma ofnat_zero : [O] == 0.
Lemma ofnat_succ : forall n, [Datatypes.S n] == succ [n].
Lemma ofnat_pred : forall n, n<>O -> [Peano.pred n] == P [n].
Since P 0 can be anything in NZ (either -1, 0, or even other
numbers, we cannot state previous lemma for n=O.
If we require in addition a strict order on NZ, we can prove that
ofnat is injective, and hence that NZ is infinite
(i.e. we ban Z/nZ models)
Module NZOfNatOrd (Import NZ:NZOrdSig').
Include NZOfNat NZ.
Include NZOrderPropFunct NZ.
Local Open Scope ofnat.
Theorem ofnat_S_gt_0 :
forall n : nat, 0 < [Datatypes.S n].
Theorem ofnat_S_neq_0 :
forall n : nat, 0 ~= [Datatypes.S n].
Lemma ofnat_injective : forall n m, [n]==[m] -> n = m.
Lemma ofnat_eq : forall n m, [n]==[m] <-> n = m.
Lemma ofnat_lt : forall n m : nat, [n]<[m] <-> (n<m)%nat.
Lemma ofnat_le : forall n m : nat, [n]<=[m] <-> (n<=m)%nat.
End NZOfNatOrd.
For basic operations, we can prove correspondance with
their counterpart in nat.
Module NZOfNatOps (Import NZ:NZAxiomsSig').
Include NZOfNat NZ.
Local Open Scope ofnat.
Lemma ofnat_add_l : forall n m, [n]+m == (S^n) m.
Lemma ofnat_add : forall n m, [n+m] == [n]+[m].
Lemma ofnat_mul : forall n m, [n*m] == [n]*[m].
Lemma ofnat_sub_r : forall n m, n-[m] == (P^m) n.
Lemma ofnat_sub : forall n m, m<=n -> [n-m] == [n]-[m].
End NZOfNatOps.