Experiment for a MathComp lra tactic

This is simple and already works but has major limitations:
- only works for R (from Coq stdlib)
- pulls axioms from real numbers in Coq stdlib every time it's used

We may like this to work for any realFieldType, maybe we should
write a new MathComp frontend to the lra.

theories/mclra.v

@@ -0,0 +1,73 @@
+Require Import Rdefinitions Raxioms RIneq.
+Require Import Lra.
+From mathcomp Require Import ssreflect ssrbool eqtype ssralg ssrint ssrnum.
+Require Import Rstruct.
+
+Delimit Scope R_scope with Re.
+
+Local Open Scope ring_scope.
+
+(* Adapted from http://github.com/pi8027/formalized-postscript/blob/master/stdlib_ext.v *)
+
+Ltac mclra_hyps2R :=
+  match goal with
+  | H : context [andb _ _] |- _ => let H0 := fresh in case/andP: H => H H0
+  | H : context [orb _ _] |- _ => case/orP: H => H
+  | H : context [?L <= ?R] |- _ => move/RleP: H => H
+  | H : context [?L < ?R] |- _ => move/RltP : H => H
+  | H : context [?L == ?R] |- _ => move/eqP : H => H
+  | H : context [?L + ?R] |- _ => rewrite -RplusE in H
+  | H : context [?L * ?R] |- _ => rewrite -RmultE in H
+  | H : context [?L - ?R] |- _ => rewrite -RminusE in H
+  | H : context [- ?R] |- _ => rewrite -RoppE in H
+  | H : context [?x%:R] |- _ =>
+    rewrite -INRE INR_IZR_INZ [IZR (Z.of_nat _)]/= in H
+  | H : context [?x%:~R] |- _ =>
+    rewrite -pmulrn -INRE INR_IZR_INZ [IZR (Z.of_nat _)]/= in H
+  end.
+
+Ltac mclra_goal2R :=
+  match goal with
+  | |- is_true (andb _ _) => apply/andP; split
+  | |- is_true (orb _ _) => try apply/orP
+  | |- is_true (_ <= _) => try apply/RleP
+  | |- is_true (_ < _) => try apply/RltP
+  | |- is_true (_ == _) => try apply/eqP
+  | |- context [?L + ?R] => rewrite -RplusE
+  | |- context [?L * ?R] => rewrite -RmultE
+  | |- context [?L - ?R] => rewrite -RminusE
+  | |- context [- ?R] => rewrite -RoppE
+  | |- context [?x%:R] => rewrite -INRE INR_IZR_INZ [IZR (Z.of_nat _)]/=
+  | |- context [?x%:~R] =>
+    rewrite -pmulrn -INRE INR_IZR_INZ [IZR (Z.of_nat _)]/=
+  end.
+
+Ltac mclra := repeat mclra_hyps2R; repeat mclra_goal2R; lra.
+
+(* A few tests/examples *)
+
+Goal forall x y : R,
+  x + 2%:R * y <= 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
+Proof.
+move=> x y lineq_xy lineq'_xy.
+mclra.
+Qed.
+
+Goal forall x y : R,
+  x + 2%:R * y <= 3%:R -> 2%:R * x + y <= 3%:R -> x + y > 2%:R -> False.
+Proof.
+move=> x y lineq_xy lineq_xy' lineq_xy''.
+mclra.
+Qed.
+
+Goal forall x : R, x <= -11%:~R -> x <= 12%:R.
+Proof.
+move=> x lineq_xy.
+mclra.
+Qed.
+
+Goal forall (n : int) (x : R), x <= n%:~R -> x + x <= n%:~R + n%:~R.
+Proof.
+move=> n x lineq_xy.
+mclra.
+Qed.