Require Import ZArith Nnat NAxioms NDiv NSig.

Module NTypeIsNAxioms (Import N : NType').

Hint Rewrite
 spec_0 spec_succ spec_add spec_mul spec_pred spec_sub
 spec_div spec_modulo spec_gcd spec_compare spec_eq_bool
 spec_max spec_min spec_power_pos spec_power
 : nsimpl.
Ltac nsimpl := autorewrite with nsimpl.
Ltac ncongruence := unfold eq; repeat red; intros; nsimpl; congruence.
Ltac zify := unfold eq, lt, le in *; nsimpl.

Local Obligation Tactic := ncongruence.

Instance eq_equiv : Equivalence eq.

Program Instance succ_wd : Proper (eq==>eq) succ.
Program Instance pred_wd : Proper (eq==>eq) pred.
Program Instance add_wd : Proper (eq==>eq==>eq) add.
Program Instance sub_wd : Proper (eq==>eq==>eq) sub.
Program Instance mul_wd : Proper (eq==>eq==>eq) mul.

Theorem pred_succ : forall n, pred (succ n) == n.

Definition N_of_Z z := of_N (Zabs_N z).

Section Induction.

Variable A : N.t -> Prop.
Hypothesis A_wd : Proper (eq==>iff) A.
Hypothesis A0 : A 0.
Hypothesis AS : forall n, A n <-> A (succ n).

Let B (z : Z) := A (N_of_Z z).

Lemma B0 : B 0.

Lemma BS : forall z : Z, (0 <= z)%Z -> B z -> B (z + 1).

Lemma B_holds : forall z : Z, (0 <= z)%Z -> B z.

Theorem bi_induction : forall n, A n.

End Induction.

Theorem add_0_l : forall n, 0 + n == n.

Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m).

Theorem sub_0_r : forall n, n - 0 == n.

Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m).

Theorem mul_0_l : forall n, 0 * n == 0.

Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m.

Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).

Definition eqb := eq_bool.

Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y.

Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare.

Instance lt_wd : Proper (eq ==> eq ==> iff) lt.

Theorem lt_eq_cases : forall n m, n <= m <-> n < m \/ n == m.

Theorem lt_irrefl : forall n, ~ n < n.

Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m.

Theorem min_l : forall n m, n <= m -> min n m == n.

Theorem min_r : forall n m, m <= n -> min n m == m.

Theorem max_l : forall n m, m <= n -> max n m == n.

Theorem max_r : forall n m, n <= m -> max n m == m.

Theorem pred_0 : pred 0 == 0.

Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.

Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b).

Theorem mod_upper_bound : forall a b, ~b==0 -> modulo a b < b.

Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) :=
  Nrect (fun _ => A) a (fun n a => f (N.of_N n) a) (N.to_N n).
Implicit Arguments recursion [A].

Instance recursion_wd (A : Type) (Aeq : relation A) :
 Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) (@recursion A).

Theorem recursion_0 :
  forall (A : Type) (a : A) (f : N.t -> A -> A), recursion a f 0 = a.

Theorem recursion_succ :
  forall (A : Type) (Aeq : relation A) (a : A) (f : N.t -> A -> A),
    Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
      forall n, Aeq (recursion a f (succ n)) (f n (recursion a f n)).

End NTypeIsNAxioms.

Module NType_NAxioms (N : NType)
 <: NAxiomsSig <: NDivSig <: HasCompare N <: HasEqBool N <: HasMinMax N
 := N <+ NTypeIsNAxioms.