Define constructive real numbers as Cauchy sequences of rational numbers. Redefine classical real numbers as a quotient of those constructive real numbers.

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diff --git a/doc/changelog/10-standard-library/10445-constructive-reals.rst b/doc/changelog/10-standard-library/10445-constructive-reals.rst
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@@ -0,0 +1,30 @@
+- New module `Reals.ConstructiveCauchyReals` defines constructive real numbers
+  by Cauchy sequences of rational numbers. Classical real numbers are now defined
+  as a quotient of these constructive real numbers, which significantly reduces
+  the number of axioms needed (see `Reals.Rdefinitions` and `Reals.Raxioms`),
+  while preserving backward compatibility.
+
+  Futhermore, the new axioms for classical real numbers include the limited
+  principle of omniscience (`sig_forall_dec`), which is a logical principle
+  instead of an ad hoc property of the real numbers. In a future pull request
+  we propose to deduce the other axiom, `completeness`, from the logical
+  principle `sig_not_dec` (see `Reals.Rlogic`), which is the excluded middle
+  of negations in sort `Type`.
+
+  If we want the axioms to be completely logical, we could also replace the
+  quotient axioms by functional extensionality, which gives the correct
+  equality to the equivalences classes represented as `bool`-valued functions
+  (also using Hedberg to get the unicity of proofs that those functions
+  are indeed equivalence classes). However that last part is not
+  completely backward compatible : it assumes funext, which is more
+  powerful than ad hoc quotient axioms.
+
+  Future work includes the development of constructive analysis, by
+  moving the constructive lemmas in new modules (see `Reals.RIneq_constr`),
+  and by interfacing tighter with libraries C-CoRN and math-classes.
+  Since the constructive definitions are weaker, this will require
+  to adapt the definitions in some places. We also need to work on
+  efficient extractions of constructive real numbers, so that we can
+  compute them in practice.
+
+  See `#10445 <https://github.com/coq/coq/pull/10445>`_, by Vincent Semeria.
diff --git a/doc/stdlib/index-list.html.template b/doc/stdlib/index-list.html.template
index a561de1d0c..eb7c982d49 100644
--- a/doc/stdlib/index-list.html.template
+++ b/doc/stdlib/index-list.html.template
@@ -516,7 +516,9 @@ through the <tt>Require Import</tt> command.</p>
   </dt>
   <dd>
     theories/Reals/Rdefinitions.v
+    theories/Reals/ConstructiveCauchyReals.v
     theories/Reals/Raxioms.v
+    theories/Reals/RIneq_constr.v
     theories/Reals/RIneq.v
     theories/Reals/DiscrR.v
     theories/Reals/ROrderedType.v
@@ -561,6 +563,7 @@ through the <tt>Require Import</tt> command.</p>
     theories/Reals/Ranalysis5.v
     theories/Reals/Ranalysis_reg.v
     theories/Reals/Rcomplete.v
+    theories/Reals/Rcomplete_constr.v
     theories/Reals/RiemannInt.v
     theories/Reals/RiemannInt_SF.v
     theories/Reals/Rpow_def.v