Turn out Coq 8.5 already comes with a module to get lia without axioms: Lia

theories/psatz_axiomfree.v

-(** This file is a hack that lets us use Psatz without importing all sorts of
-    axioms about real numbers.  It has been created by copying the file
-    Psatz.v from the Coq distribution, removing everything defined after the lia
-    tactic, and removing the two lines importing RMicromega and Rdefinitions.
-    The original license header follows. *)
-
-(************************************************************************)
-(*  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        *)
-(************************************************************************)
-(*                                                                      *)
-(* Micromega: A reflexive tactic using the Positivstellensatz           *)
-(*                                                                      *)
-(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
-(*                                                                      *)
-(************************************************************************)
-
-From Coq Require Import ZMicromega.
-From Coq Require Import QMicromega.
-From Coq Require Import QArith.
-From Coq Require Import ZArith.
-From Coq Require Import RingMicromega.
-From Coq Require Import VarMap.
-From Coq Require Tauto.
-Declare ML Module "micromega_plugin".
-
-Ltac preprocess := 
-    zify ; unfold Z.succ in * ; unfold Z.pred in *.
-
-Ltac lia :=
-  preprocess;
-  xlia ;
-  abstract (
-  intros __wit __varmap __ff ;
-    change (Tauto.eval_f (Zeval_formula (@find Z Z0 __varmap)) __ff) ;
-      apply (ZTautoChecker_sound __ff __wit); vm_cast_no_check (eq_refl true)).

theories/tactics.v

@@ -3,7 +3,7 @@
 (** This file collects general purpose tactics that are used throughout
 the development. *)
 From Coq Require Import Omega.
-From stdpp Require Export psatz_axiomfree.
+From Coq Require Export Lia.
 From stdpp Require Export decidable.
 
 Lemma f_equal_dep {A B} (f g : ∀ x : A, B x) x : f = g → f x = g x.