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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
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(*         *     GNU Lesser General Public License Version 2.1          *)
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Local Open Scope Z_scope.
(** Our purpose is to write an induction shema for {0,1,2,...} similar to the [nat] schema (Theorem [Natlike_rec]). For that the following implications will be used : << ∀n:nat, Q n == ∀n:nat, P (Z.of_nat n) ===> ∀x:Z, x <= 0 -> P x  /\ || ||  (Q O) ∧ (∀n:nat, Q n -> Q (S n)) <=== (P 0) ∧ (∀x:Z, P x -> P (Z.succ x))  <=== (Z.of_nat (S n) = Z.succ (Z.of_nat n))  <=== Z_of_nat_complete >> Then the  diagram will be closed and the theorem proved. *)

Lemma Z_of_nat_complete (x : Z) :
0 <= x -> exists n : nat, x = Z.of_nat n.