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Require Import Coq.Setoids.Setoid Coq.PArith.BinPos Coq.PArith.Pnat ZArith_base.
Section P_of_nat.
Variables (n: nat) (E: n <> O).
Lemma P_of_nat: positive.
apply P_of_succ_nat.
destruct n as [|p].
exfalso. apply E. reflexivity.
exact p.
Defined.
Lemma P_of_nat_correct: nat_of_P P_of_nat = n.
unfold P_of_nat.
destruct n. exfalso. intuition.
apply nat_of_P_o_P_of_succ_nat_eq_succ.
Qed.
End P_of_nat.
Lemma nat_of_P_inj_iff (p q : positive): nat_of_P p = nat_of_P q <-> p = q.
Proof with auto.
split; intro. apply nat_of_P_inj... subst...
Qed.
Lemma nat_of_P_nonzero (p: positive): nat_of_P p <> 0.
Proof.
intro H.
apply Nat.lt_irrefl with 0.
rewrite <- H at 2.
apply lt_O_nat_of_P.
Qed.
#[global]
Hint Immediate nat_of_P_nonzero.
Lemma Plt_lt (p q: positive): Pos.lt p q <-> (nat_of_P p < nat_of_P q).
Proof.
split. apply nat_of_P_lt_Lt_compare_morphism.
apply nat_of_P_lt_Lt_compare_complement_morphism.
Qed.
Lemma Ple_le (p q: positive): Pos.le p q <-> le (nat_of_P p) (nat_of_P q).
Proof.
rewrite Pos.le_lteq, Plt_lt, Nat.lt_eq_cases, nat_of_P_inj_iff.
reflexivity.
Qed.
Lemma Ple_refl p: Pos.le p p.
Proof. intros. apply Pos.le_lteq. firstorder. Qed.
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