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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import NZAxioms.
Module Type NZBaseProp (Import NZ : NZDomainSig').
Include BackportEq NZ NZ. (** eq_refl, eq_sym, eq_trans *)
Lemma eq_sym_iff : forall x y, x==y <-> y==x.
Proof.
intros; split; symmetry; auto.
Qed.
(* TODO: how register ~= (which is just a notation) as a Symmetric relation,
hence allowing "symmetry" tac ? *)
Theorem neq_sym : forall n m, n ~= m -> m ~= n.
Proof.
intros n m H1 H2; symmetry in H2; false_hyp H2 H1.
Qed.
Theorem eq_stepl : forall x y z, x == y -> x == z -> z == y.
Proof.
intros x y z H1 H2; now rewrite <- H1.
Qed.
Declare Left Step eq_stepl.
(* The right step lemma is just the transitivity of eq *)
Declare Right Step (@Equivalence_Transitive _ _ eq_equiv).
Theorem succ_inj : forall n1 n2, S n1 == S n2 -> n1 == n2.
Proof.
intros n1 n2 H.
apply pred_wd in H. now do 2 rewrite pred_succ in H.
Qed.
(* The following theorem is useful as an equivalence for proving
bidirectional induction steps *)
Theorem succ_inj_wd : forall n1 n2, S n1 == S n2 <-> n1 == n2.
Proof.
intros; split.
apply succ_inj.
intros. now f_equiv.
Qed.
Theorem succ_inj_wd_neg : forall n m, S n ~= S m <-> n ~= m.
Proof.
intros; now rewrite succ_inj_wd.
Qed.
(* We cannot prove that the predecessor is injective, nor that it is
left-inverse to the successor at this point *)
Section CentralInduction.
Variable A : t -> Prop.
Hypothesis A_wd : Proper (eq==>iff) A.
Theorem central_induction :
forall z, A z ->
(forall n, A n <-> A (S n)) ->
forall n, A n.
Proof.
intros z Base Step; revert Base; pattern z; apply bi_induction.
solve_proper.
intro; now apply bi_induction.
intro; pose proof (Step n); tauto.
Qed.
End CentralInduction.
Tactic Notation "nzinduct" ident(n) :=
induction_maker n ltac:(apply bi_induction).
Tactic Notation "nzinduct" ident(n) constr(u) :=
induction_maker n ltac:(apply central_induction with (z := u)).
End NZBaseProp.
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