diff --git a/examples/in_progress/binomial_heap.mlw b/examples/in_progress/binomial_heap.mlw
index aa39d30f4b42b1fafb410e724c2ec264f952529b..1d41078d770701295cdeeaf0100d1b33053d7588 100644
--- a/examples/in_progress/binomial_heap.mlw
+++ b/examples/in_progress/binomial_heap.mlw
@@ -19,6 +19,7 @@ module BinomialHeap
   type tree = {
     elem: elt;
     children: list tree;
+    rank: int;
   }
 
   (* [e] is no greater than the roots of the trees in [l] *)
@@ -41,14 +42,14 @@ module BinomialHeap
   lemma heaps_reverse:
     forall h. heaps h -> heaps (reverse h)
 
-  function rank (t: tree) : int =
-    length t.children
+  (* function rank (t: tree) : int = *)
+  (*   length t.children *)
 
   function link (t1 t2: tree) : tree =
     if le t1.elem t2.elem then
-      { t1 with children = Cons t2 t1.children }
+      { elem = t1.elem; rank = t1.rank + 1; children = Cons t2 t1.children }
     else
-      { t2 with children = Cons t1 t2.children }
+      { elem = t2.elem; rank = t2.rank + 1; children = Cons t1 t2.children }
 
   function occ (x: elt) (l: list tree) : int =
     match l with
@@ -86,7 +87,8 @@ module BinomialHeap
     end
 
   predicate binomial_tree (t: tree) =
-    has_order (rank t) t.children
+    t.rank = length t.children &&
+    has_order t.rank t.children
 
   (* a binomial heap is a list of trees *)
   type heap = list tree
@@ -96,7 +98,7 @@ module BinomialHeap
   predicate inv (m: int) (h: heap) =
     match h with
     | Nil -> true
-    | Cons t r -> let k = rank t in m <= k && binomial_tree t && inv (k + 1) r
+    | Cons t r -> let k = t.rank in m <= k && binomial_tree t && inv (k + 1) r
     end
 
   lemma inv_reverse:
@@ -166,7 +168,7 @@ module BinomialHeap
     ensures  { occ x result = occ x h + 1 }
     ensures  { forall e. e <> x -> occ e result = occ e h }
     =
-    add_tree { elem = x; children = Nil } h
+    add_tree { elem = x; rank = 0; children = Nil } h
 
   let rec merge (ghost k: int) (h1 h2: heap)
     requires { heaps h1 }
diff --git a/examples/in_progress/binomial_heap/why3session.xml b/examples/in_progress/binomial_heap/why3session.xml
index 6ec0e36e6709632a21163b872f105893a609e790..2cc028f0208d4cc9d00eee4c60b5d879b65051e9 100644
--- a/examples/in_progress/binomial_heap/why3session.xml
+++ b/examples/in_progress/binomial_heap/why3session.xml
@@ -6,28 +6,30 @@
 <prover id="1" name="Alt-Ergo" version="1.01" timelimit="6" steplimit="0" memlimit="1000"/>
 <prover id="2" name="Eprover" version="1.8-001" timelimit="6" steplimit="0" memlimit="1000"/>
 <file name="../binomial_heap.mlw" expanded="true">
-<theory name="BinomialHeap" sum="129ba3e361754e4b52388d22424142ee" expanded="true">
+<theory name="BinomialHeap" sum="0b4cf8ddb9d3580c11d55bf59ef6dc13" expanded="true">
  <goal name="le_roots_trans">
- <proof prover="0" obsolete="true"><result status="unknown" time="0.01"/></proof>
+ <proof prover="0"><result status="unknown" time="0.01"/></proof>
  <proof prover="1"><result status="unknown" time="0.02"/></proof>
  </goal>
  <goal name="heaps_reverse">
- <proof prover="0" obsolete="true"><result status="timeout" time="6.98"/></proof>
+ <proof prover="0"><result status="timeout" time="6.98"/></proof>
  <proof prover="1"><result status="unknown" time="0.03"/></proof>
  </goal>
  <goal name="occ_nonneg">
- <proof prover="0" obsolete="true"><result status="timeout" time="6.01"/></proof>
+ <proof prover="0"><result status="timeout" time="6.01"/></proof>
  <proof prover="1"><result status="unknown" time="0.03"/></proof>
  </goal>
  <goal name="occ_reverse">
- <proof prover="0" obsolete="true"><result status="timeout" time="6.01"/></proof>
+ <proof prover="0"><result status="timeout" time="6.01"/></proof>
  <proof prover="1"><result status="unknown" time="0.06"/></proof>
  </goal>
  <goal name="heaps_mem">
- <proof prover="0" obsolete="true"><result status="timeout" time="6.02"/></proof>
+ <proof prover="0"><result status="timeout" time="6.02"/></proof>
  <proof prover="1"><result status="unknown" time="0.05"/></proof>
  </goal>
- <goal name="inv_reverse" expanded="true">
+ <goal name="inv_reverse">
+ <proof prover="0" obsolete="true"><result status="timeout" time="6.02"/></proof>
+ <proof prover="1" obsolete="true"><result status="unknown" time="0.07"/></proof>
  </goal>
  <goal name="WP_parameter empty" expl="VC for empty">
  <proof prover="1"><result status="valid" time="0.01" steps="4"/></proof>
@@ -50,16 +52,16 @@
   <proof prover="1"><result status="valid" time="0.00" steps="11"/></proof>
   </goal>
   <goal name="WP_parameter get_min.5" expl="5. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="23"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="24"/></proof>
   </goal>
   <goal name="WP_parameter get_min.6" expl="6. postcondition">
   <proof prover="1"><result status="valid" time="0.01" steps="27"/></proof>
   </goal>
   <goal name="WP_parameter get_min.7" expl="7. postcondition">
-  <proof prover="1"><result status="valid" time="0.00" steps="36"/></proof>
+  <proof prover="0"><result status="valid" time="0.06"/></proof>
   </goal>
   <goal name="WP_parameter get_min.8" expl="8. postcondition">
-  <proof prover="0"><result status="valid" time="0.06"/></proof>
+  <proof prover="0"><result status="valid" time="0.07"/></proof>
   </goal>
   <goal name="WP_parameter get_min.9" expl="9. precondition">
   <proof prover="2"><result status="valid" time="0.01"/></proof>
@@ -78,60 +80,60 @@
   <proof prover="1"><result status="valid" time="0.01" steps="6"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.2" expl="2. postcondition">
-  <proof prover="0"><result status="valid" time="0.01"/></proof>
-  <proof prover="1"><result status="valid" time="0.00" steps="11"/></proof>
+  <proof prover="1" timelimit="21"><result status="valid" time="0.01" steps="12"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.3" expl="3. postcondition">
-  <proof prover="1"><result status="valid" time="0.00" steps="8"/></proof>
+  <proof prover="1"><result status="valid" time="0.00" steps="9"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.4" expl="4. postcondition">
-  <proof prover="0"><result status="valid" time="0.02"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="57"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.5" expl="5. postcondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="42"/></proof>
+  <proof prover="1" timelimit="21"><result status="valid" time="0.01" steps="42"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.6" expl="6. postcondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="26"/></proof>
+  <proof prover="0"><result status="valid" time="0.14"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.7" expl="7. assertion">
   <transf name="compute_in_goal">
    <goal name="WP_parameter add_tree.7.1" expl="1. assertion">
-   <proof prover="1"><result status="valid" time="0.05" steps="145"/></proof>
+   <proof prover="1"><result status="valid" time="0.05" steps="52"/></proof>
    </goal>
   </transf>
   </goal>
   <goal name="WP_parameter add_tree.8" expl="8. variant decrease">
-  <proof prover="1"><result status="valid" time="0.01" steps="13"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="14"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.9" expl="9. precondition">
-  <proof prover="0"><result status="valid" time="0.14"/></proof>
+  <proof prover="0"><result status="valid" time="0.23"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.10" expl="10. precondition">
   <transf name="compute_in_goal">
    <goal name="WP_parameter add_tree.10.1" expl="1. precondition">
-   <proof prover="1"><result status="valid" time="0.52" steps="1189"/></proof>
+   <proof prover="1"><result status="valid" time="0.21" steps="1046"/></proof>
    </goal>
   </transf>
   </goal>
   <goal name="WP_parameter add_tree.11" expl="11. precondition">
-  <proof prover="1"><result status="valid" time="0.02" steps="64"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="70"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.12" expl="12. precondition">
   <proof prover="0"><result status="valid" time="0.07"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.13" expl="13. postcondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="14"/></proof>
+  <proof prover="0"><result status="valid" time="0.02"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.14" expl="14. postcondition">
-  <proof prover="0"><result status="valid" time="0.13"/></proof>
+  <proof prover="0"><result status="valid" time="0.01"/></proof>
   </goal>
   <goal name="WP_parameter add_tree.15" expl="15. postcondition">
-  <proof prover="0"><result status="valid" time="0.14"/></proof>
+  <proof prover="0"><result status="valid" time="0.10"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="WP_parameter add" expl="VC for add">
  <proof prover="0"><result status="valid" time="0.04"/></proof>
+ <proof prover="1"><result status="valid" time="0.02" steps="157"/></proof>
  </goal>
  <goal name="WP_parameter merge" expl="VC for merge">
  <transf name="split_goal_wp">
@@ -163,6 +165,7 @@
   <transf name="compute_in_goal">
    <goal name="WP_parameter merge.9.1" expl="1. postcondition">
    <proof prover="0"><result status="valid" time="0.02"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="51"/></proof>
    </goal>
   </transf>
   </goal>
@@ -170,25 +173,26 @@
   <proof prover="1"><result status="valid" time="0.01" steps="13"/></proof>
   </goal>
   <goal name="WP_parameter merge.11" expl="11. precondition">
-  <proof prover="1"><result status="valid" time="0.02" steps="62"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="57"/></proof>
   </goal>
   <goal name="WP_parameter merge.12" expl="12. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="26"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="22"/></proof>
   </goal>
   <goal name="WP_parameter merge.13" expl="13. precondition">
   <proof prover="1"><result status="valid" time="0.01" steps="8"/></proof>
   </goal>
   <goal name="WP_parameter merge.14" expl="14. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="50"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="41"/></proof>
   </goal>
   <goal name="WP_parameter merge.15" expl="15. postcondition">
   <proof prover="0"><result status="valid" time="0.03"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="121"/></proof>
   </goal>
   <goal name="WP_parameter merge.16" expl="16. postcondition">
-  <proof prover="1"><result status="valid" time="0.02" steps="63"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="51"/></proof>
   </goal>
   <goal name="WP_parameter merge.17" expl="17. postcondition">
-  <proof prover="1"><result status="valid" time="0.07" steps="161"/></proof>
+  <proof prover="1"><result status="valid" time="0.07" steps="159"/></proof>
   </goal>
   <goal name="WP_parameter merge.18" expl="18. variant decrease">
   <proof prover="1"><result status="valid" time="0.01" steps="18"/></proof>
@@ -197,19 +201,24 @@
   <proof prover="1"><result status="valid" time="0.01" steps="9"/></proof>
   </goal>
   <goal name="WP_parameter merge.20" expl="20. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="51"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="42"/></proof>
   </goal>
   <goal name="WP_parameter merge.21" expl="21. precondition">
-  <proof prover="1"><result status="valid" time="0.00" steps="60"/></proof>
+  <transf name="compute_in_goal">
+   <goal name="WP_parameter merge.21.1" expl="1. precondition">
+   <proof prover="1"><result status="valid" time="0.03" steps="451"/></proof>
+   </goal>
+  </transf>
   </goal>
   <goal name="WP_parameter merge.22" expl="22. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="27"/></proof>
+  <proof prover="1"><result status="valid" time="0.00" steps="23"/></proof>
   </goal>
   <goal name="WP_parameter merge.23" expl="23. postcondition">
   <proof prover="0"><result status="valid" time="0.02"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="122"/></proof>
   </goal>
   <goal name="WP_parameter merge.24" expl="24. postcondition">
-  <proof prover="1"><result status="valid" time="0.02" steps="64"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="52"/></proof>
   </goal>
   <goal name="WP_parameter merge.25" expl="25. postcondition">
   <proof prover="1"><result status="valid" time="0.06" steps="149"/></proof>
@@ -218,25 +227,22 @@
   <proof prover="1"><result status="valid" time="0.01" steps="14"/></proof>
   </goal>
   <goal name="WP_parameter merge.27" expl="27. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="65"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="61"/></proof>
   </goal>
   <goal name="WP_parameter merge.28" expl="28. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="27"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="23"/></proof>
   </goal>
   <goal name="WP_parameter merge.29" expl="29. precondition">
-  <transf name="compute_in_goal">
-   <goal name="WP_parameter merge.29.1" expl="1. precondition">
-   <proof prover="1"><result status="valid" time="0.03" steps="450"/></proof>
-   </goal>
-  </transf>
+  <proof prover="1"><result status="valid" time="0.00" steps="61"/></proof>
   </goal>
   <goal name="WP_parameter merge.30" expl="30. precondition">
-  <proof prover="1"><result status="valid" time="0.00" steps="27"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="23"/></proof>
   </goal>
   <goal name="WP_parameter merge.31" expl="31. precondition">
   <transf name="compute_in_goal">
    <goal name="WP_parameter merge.31.1" expl="1. precondition">
-   <proof prover="0"><result status="valid" time="0.65"/></proof>
+   <proof prover="0"><result status="valid" time="0.16"/></proof>
+   <proof prover="1"><result status="valid" time="2.62" steps="9363"/></proof>
    </goal>
   </transf>
   </goal>
@@ -245,7 +251,7 @@
    <goal name="WP_parameter merge.32.1" expl="1. precondition">
    <transf name="compute_in_goal">
     <goal name="WP_parameter merge.32.1.1" expl="1. precondition">
-    <proof prover="1"><result status="valid" time="0.04" steps="245"/></proof>
+    <proof prover="1"><result status="valid" time="0.04" steps="339"/></proof>
     </goal>
    </transf>
    </goal>
@@ -256,15 +262,18 @@
   </goal>
   <goal name="WP_parameter merge.34" expl="34. precondition">
   <proof prover="0"><result status="valid" time="0.08"/></proof>
+  <proof prover="1" obsolete="true"><result status="timeout" time="5.99"/></proof>
   </goal>
   <goal name="WP_parameter merge.35" expl="35. postcondition">
   <proof prover="1"><result status="valid" time="0.01" steps="20"/></proof>
   </goal>
   <goal name="WP_parameter merge.36" expl="36. postcondition">
   <proof prover="0"><result status="valid" time="0.10"/></proof>
+  <proof prover="1" obsolete="true"><result status="timeout" time="5.99"/></proof>
   </goal>
   <goal name="WP_parameter merge.37" expl="37. postcondition">
   <proof prover="0"><result status="valid" time="0.11"/></proof>
+  <proof prover="1" obsolete="true"><result status="timeout" time="5.99"/></proof>
   </goal>
  </transf>
  </goal>
@@ -274,20 +283,20 @@
   <proof prover="1"><result status="valid" time="0.01" steps="4"/></proof>
   </goal>
   <goal name="WP_parameter extract_min_tree.2" expl="2. postcondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="43"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="42"/></proof>
   </goal>
   <goal name="WP_parameter extract_min_tree.3" expl="3. variant decrease">
-  <proof prover="1"><result status="valid" time="0.00" steps="16"/></proof>
+  <proof prover="1"><result status="valid" time="0.00" steps="17"/></proof>
   </goal>
   <goal name="WP_parameter extract_min_tree.4" expl="4. precondition">
   <transf name="compute_in_goal">
    <goal name="WP_parameter extract_min_tree.4.1" expl="1. precondition">
-   <proof prover="1"><result status="valid" time="0.02" steps="221"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="243"/></proof>
    </goal>
   </transf>
   </goal>
   <goal name="WP_parameter extract_min_tree.5" expl="5. precondition">
-  <proof prover="1"><result status="valid" time="0.02" steps="46"/></proof>
+  <proof prover="1"><result status="valid" time="0.02" steps="24"/></proof>
   </goal>
   <goal name="WP_parameter extract_min_tree.6" expl="6. precondition">
   <proof prover="1"><result status="valid" time="0.01" steps="7"/></proof>
@@ -296,40 +305,44 @@
   <transf name="split_goal_wp">
    <goal name="WP_parameter extract_min_tree.7.1" expl="1. VC for extract_min_tree">
    <proof prover="0"><result status="valid" time="0.03"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="107"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.2" expl="2. VC for extract_min_tree">
    <proof prover="1"><result status="valid" time="0.00" steps="16"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.3" expl="3. VC for extract_min_tree">
-   <proof prover="1"><result status="valid" time="0.01" steps="111"/></proof>
+   <proof prover="1"><result status="valid" time="0.01" steps="108"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.4" expl="4. VC for extract_min_tree">
-   <proof prover="1"><result status="valid" time="0.01" steps="69"/></proof>
+   <proof prover="1"><result status="valid" time="0.01" steps="65"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.5" expl="5. VC for extract_min_tree">
-   <proof prover="1"><result status="valid" time="0.01" steps="48"/></proof>
+   <proof prover="1"><result status="valid" time="0.01" steps="49"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.6" expl="6. VC for extract_min_tree">
-   <proof prover="0"><result status="valid" time="0.02"/></proof>
+   <proof prover="0"><result status="valid" time="0.06"/></proof>
+   <proof prover="1"><result status="valid" time="0.09" steps="299"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.7" expl="7. VC for extract_min_tree">
    <proof prover="1"><result status="valid" time="0.01" steps="15"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.8" expl="8. VC for extract_min_tree">
    <proof prover="0"><result status="valid" time="0.03"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="160"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.9" expl="9. VC for extract_min_tree">
-   <proof prover="1"><result status="valid" time="0.01" steps="79"/></proof>
+   <proof prover="1"><result status="valid" time="0.01" steps="74"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.10" expl="10. VC for extract_min_tree">
-   <proof prover="1"><result status="valid" time="0.01" steps="52"/></proof>
+   <proof prover="1"><result status="valid" time="0.01" steps="47"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.11" expl="11. VC for extract_min_tree">
    <proof prover="0"><result status="valid" time="0.02"/></proof>
    <proof prover="1"><result status="valid" time="0.01" steps="18"/></proof>
    </goal>
    <goal name="WP_parameter extract_min_tree.7.12" expl="12. VC for extract_min_tree">
-   <proof prover="0"><result status="valid" time="0.06"/></proof>
+   <proof prover="0"><result status="valid" time="0.02"/></proof>
+   <proof prover="1"><result status="valid" time="0.06" steps="533"/></proof>
    </goal>
   </transf>
   </goal>
@@ -348,16 +361,16 @@
   <proof prover="1"><result status="valid" time="0.01" steps="4"/></proof>
   </goal>
   <goal name="WP_parameter extract_min.4" expl="4. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="30"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="32"/></proof>
   </goal>
   <goal name="WP_parameter extract_min.5" expl="5. precondition">
-  <proof prover="1"><result status="valid" time="0.01" steps="60"/></proof>
+  <proof prover="1"><result status="valid" time="0.00" steps="61"/></proof>
   </goal>
   <goal name="WP_parameter extract_min.6" expl="6. precondition">
   <proof prover="1"><result status="valid" time="0.01" steps="9"/></proof>
   </goal>
   <goal name="WP_parameter extract_min.7" expl="7. precondition">
-  <proof prover="1"><result status="valid" time="0.00" steps="9"/></proof>
+  <proof prover="1"><result status="valid" time="0.01" steps="9"/></proof>
   </goal>
   <goal name="WP_parameter extract_min.8" expl="8. postcondition">
   <transf name="split_goal_wp">
@@ -368,10 +381,10 @@
    <proof prover="1"><result status="valid" time="0.01" steps="13"/></proof>
    </goal>
    <goal name="WP_parameter extract_min.8.3" expl="3. VC for extract_min">
-   <proof prover="1"><result status="valid" time="0.02" steps="103"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="59"/></proof>
    </goal>
    <goal name="WP_parameter extract_min.8.4" expl="4. VC for extract_min">
-   <proof prover="1"><result status="valid" time="0.02" steps="120"/></proof>
+   <proof prover="1"><result status="valid" time="0.02" steps="174"/></proof>
    </goal>
   </transf>
   </goal>
diff --git a/examples/in_progress/binomial_heap/why3shapes.gz b/examples/in_progress/binomial_heap/why3shapes.gz
index 19dfc98cd31ed5dcb5c9d679192859ad1d0f9263..ff575f0cdbb962b4ac645b69d623822b264b58dd 100644
Binary files a/examples/in_progress/binomial_heap/why3shapes.gz and b/examples/in_progress/binomial_heap/why3shapes.gz differ