stdlib/array: make array.Sorted compatible with strict orders

examples/mergesort_array/why3session.xml

 <?xml version="1.0" encoding="UTF-8"?>
 <!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
 "http://why3.lri.fr/why3session.dtd">
-<why3session shape_version="5">
+<why3session shape_version="6">
 <prover id="0" name="Alt-Ergo" version="2.0.0" timelimit="11" steplimit="0" memlimit="1000"/>
 <prover id="3" name="CVC3" version="2.4.1" timelimit="11" steplimit="0" memlimit="1000"/>
 <prover id="4" name="CVC4" version="1.4" timelimit="5" steplimit="0" memlimit="1000"/>
@@ -196,7 +196,7 @@
  <goal name="VC mergesort_rec" expl="VC for mergesort_rec" proved="true">
  <transf name="split_vc" proved="true" >
   <goal name="VC mergesort_rec.0" expl="postcondition" proved="true">
-  <proof prover="0" timelimit="5"><result status="valid" time="0.01" steps="11"/></proof>
+  <proof prover="0" timelimit="5"><result status="valid" time="0.01" steps="7"/></proof>
   </goal>
   <goal name="VC mergesort_rec.1" expl="postcondition" proved="true">
   <proof prover="0" timelimit="5"><result status="valid" time="0.01" steps="13"/></proof>
@@ -229,7 +229,7 @@
   <proof prover="0" timelimit="5"><result status="valid" time="0.01" steps="18"/></proof>
   </goal>
   <goal name="VC mergesort_rec.11" expl="precondition" proved="true">
-  <proof prover="0" timelimit="5"><result status="valid" time="0.15" steps="184"/></proof>
+  <proof prover="0" timelimit="5"><result status="valid" time="0.27" steps="308"/></proof>
   </goal>
   <goal name="VC mergesort_rec.12" expl="precondition" proved="true">
   <proof prover="0" timelimit="5"><result status="valid" time="0.01" steps="22"/></proof>
@@ -283,7 +283,7 @@
   <proof prover="0"><result status="valid" time="0.03" steps="53"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.11" expl="precondition" proved="true">
-  <proof prover="0"><result status="valid" time="0.01" steps="50"/></proof>
+  <proof prover="0"><result status="valid" time="0.01" steps="59"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.12" expl="precondition" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="21"/></proof>
@@ -305,7 +305,7 @@
   <proof prover="0"><result status="valid" time="0.01" steps="22"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.17" expl="assertion" proved="true">
-  <proof prover="0"><result status="valid" time="0.75" steps="490"/></proof>
+  <proof prover="0"><result status="valid" time="0.75" steps="506"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.18" expl="assertion" proved="true">
   <transf name="split_goal_right" proved="true" >
@@ -319,7 +319,7 @@
    <proof prover="0"><result status="valid" time="0.03" steps="42"/></proof>
    </goal>
    <goal name="VC bottom_up_mergesort.18.3" expl="assertion" proved="true">
-   <proof prover="0"><result status="valid" time="0.03" steps="103"/></proof>
+   <proof prover="0"><result status="valid" time="0.03" steps="143"/></proof>
    </goal>
    <goal name="VC bottom_up_mergesort.18.4" expl="assertion" proved="true">
    <proof prover="0"><result status="valid" time="0.01" steps="27"/></proof>
@@ -339,13 +339,13 @@
   <proof prover="0"><result status="valid" time="0.01" steps="57"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.22" expl="loop invariant preservation" proved="true">
-  <proof prover="0"><result status="valid" time="0.03" steps="80"/></proof>
+  <proof prover="0"><result status="valid" time="0.03" steps="72"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.23" expl="loop invariant preservation" proved="true">
   <proof prover="0"><result status="valid" time="0.02" steps="71"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.24" expl="assertion" proved="true">
-  <proof prover="0"><result status="valid" time="0.11" steps="177"/></proof>
+  <proof prover="0"><result status="valid" time="0.11" steps="181"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.25" expl="loop variant decrease" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="14"/></proof>
@@ -363,7 +363,7 @@
   <proof prover="0"><result status="valid" time="0.01" steps="23"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.30" expl="postcondition" proved="true">
-  <proof prover="0"><result status="valid" time="0.01" steps="32"/></proof>
+  <proof prover="0"><result status="valid" time="0.01" steps="31"/></proof>
   </goal>
   <goal name="VC bottom_up_mergesort.31" expl="postcondition" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="9"/></proof>
@@ -373,12 +373,12 @@
 </theory>
 <theory name="NaturalMergesort" proved="true">
  <goal name="VC find_run" expl="VC for find_run" proved="true">
- <proof prover="0"><result status="valid" time="0.13" steps="178"/></proof>
+ <proof prover="0"><result status="valid" time="0.13" steps="167"/></proof>
  </goal>
  <goal name="VC natural_mergesort" expl="VC for natural_mergesort" proved="true">
  <transf name="split_goal_right" proved="true" >
   <goal name="VC natural_mergesort.0" expl="postcondition" proved="true">
-  <proof prover="0"><result status="valid" time="0.01" steps="7"/></proof>
+  <proof prover="0"><result status="valid" time="0.01" steps="3"/></proof>
   </goal>
   <goal name="VC natural_mergesort.1" expl="postcondition" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="5"/></proof>
@@ -479,7 +479,7 @@
   <proof prover="0"><result status="valid" time="0.01" steps="37"/></proof>
   </goal>
   <goal name="VC natural_mergesort.30" expl="loop invariant preservation" proved="true">
-  <proof prover="0"><result status="valid" time="0.02" steps="102"/></proof>
+  <proof prover="0"><result status="valid" time="0.02" steps="128"/></proof>
   </goal>
   <goal name="VC natural_mergesort.31" expl="loop invariant preservation" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="39"/></proof>
@@ -510,7 +510,7 @@
   <proof prover="0"><result status="valid" time="0.00" steps="1"/></proof>
   </goal>
   <goal name="VC naturalrec.1" expl="postcondition" proved="true">
-  <proof prover="0"><result status="valid" time="0.01" steps="11"/></proof>
+  <proof prover="0"><result status="valid" time="0.01" steps="7"/></proof>
   </goal>
   <goal name="VC naturalrec.2" expl="postcondition" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="13"/></proof>
@@ -565,7 +565,7 @@
   <proof prover="0"><result status="valid" time="0.01" steps="47"/></proof>
   </goal>
   <goal name="VC naturalrec.18" expl="precondition" proved="true">
-  <proof prover="0"><result status="valid" time="0.02" steps="107"/></proof>
+  <proof prover="0"><result status="valid" time="0.02" steps="136"/></proof>
   </goal>
   <goal name="VC naturalrec.19" expl="precondition" proved="true">
   <proof prover="0"><result status="valid" time="0.01" steps="30"/></proof>

examples/skew_heaps/why3session.xml

 <?xml version="1.0" encoding="UTF-8"?>
 <!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
 "http://why3.lri.fr/why3session.dtd">
-<why3session shape_version="5">
+<why3session shape_version="6">
 <prover id="2" name="Alt-Ergo" version="2.0.0" timelimit="5" steplimit="0" memlimit="1000"/>
 <prover id="3" name="CVC4" version="1.5" timelimit="5" steplimit="0" memlimit="1000"/>
 <file proved="true">

stdlib/array.mlw

@@ -168,12 +168,12 @@ module Sorted
   predicate le elt elt
 
   predicate sorted_sub (a: array elt) (l u: int) =
-    forall i1 i2 : int. l <= i1 <= i2 < u -> le a[i1] a[i2]
+    forall i1 i2 : int. l <= i1 < i2 < u -> le a[i1] a[i2]
   (** `sorted_sub a l u` is true whenever the array segment `a(l..u-1)`
       is sorted w.r.t order relation `le` *)
 
   predicate sorted (a: array elt) =
-    forall i1 i2 : int. 0 <= i1 <= i2 < length a -> le a[i1] a[i2]
+    forall i1 i2 : int. 0 <= i1 < i2 < length a -> le a[i1] a[i2]
   (** `sorted a` is true whenever the array `a` is sorted w.r.t `le` *)
 
 end