[ t1 = term; SYMBOL "*"; t2 = term ->
           return_term loc (CicAst.Appl [CicAst.Symbol ("times", 0); t1; t2])
       | t1 = term; SYMBOL "/"; t2 = term ->
-          return_term loc (CicAst.Appl [CicAst.Symbol ("div", 0); t1; t2])
+          return_term loc (CicAst.Appl [CicAst.Symbol ("divide", 0); t1; t2])
       ]
     ];
   term: LEVEL "inv"
         return_term loc (CicAst.Appl [CicAst.Symbol ("uminus", 0); t])
       ]
     ];
+  term: LEVEL "relop"
+    [
+      [ t1 = term; SYMBOL <:unicode<leq>> (* ≤ *); t2 = term ->
+          return_term loc (CicAst.Appl [CicAst.Symbol ("leq", 0); t1; t2])
+      | t1 = term; SYMBOL <:unicode<geq>> (* ≥ *); t2 = term ->
+          return_term loc (CicAst.Appl [CicAst.Symbol ("geq", 0); t1; t2])
+      | t1 = term; SYMBOL "<"; t2 = term ->
+          return_term loc (CicAst.Appl [CicAst.Symbol ("lt", 0); t1; t2])
+      | t1 = term; SYMBOL ">"; t2 = term ->
+          return_term loc (CicAst.Appl [CicAst.Symbol ("gt", 0); t1; t2])
+      | t1 = term; SYMBOL <:unicode<ne>> (* ≠ *); t2 = term ->
+          return_term loc (CicAst.Appl [CicAst.Symbol ("neq", 0); t1; t2])
+      ]
+    ];
 END
 
 let _ =
+  let uri s = UriManager.uri_of_string s in
+  let const s = Cic.Const (uri s, []) in
+  let mutind s = Cic.MutInd (uri s, 0, []) in
+
   DisambiguateChoices.add_num_choice
     ("natural number",
       (fun _ num _ -> HelmLibraryObjects.build_nat (int_of_string num)));
   DisambiguateChoices.add_num_choice
     ("real number",
       (fun _ num _ -> HelmLibraryObjects.build_real (int_of_string num)));
-  DisambiguateChoices.add_symbol_choice "plus"
-    ("natural plus",
-     (fun env _ args ->
-       let t1, t2 =
-         match args with
-         | [t1; t2] -> t1, t2
-         | _ -> raise DisambiguateChoices.Invalid_choice
-       in
-       Cic.Appl [ HelmLibraryObjects.Peano.plus; t1; t2 ]));
-  DisambiguateChoices.add_symbol_choice "plus"
-    ("real plus",
-     (fun env _ args ->
-       let t1, t2 =
-         match args with
-         | [t1; t2] -> t1, t2
-         | _ -> raise DisambiguateChoices.Invalid_choice
-       in
-       Cic.Appl [ HelmLibraryObjects.Reals.rplus; t1; t2 ]));
+
+  DisambiguateChoices.add_binary_op "plus" "natural plus"
+    HelmLibraryObjects.Peano.plus;
+  DisambiguateChoices.add_binary_op "plus" "real plus"
+    HelmLibraryObjects.Reals.rplus;
+  DisambiguateChoices.add_binary_op "minus" "natural minus"
+    (const "cic:/Coq/Arith/Minus/minus.con");
+  DisambiguateChoices.add_binary_op "minus" "real minus"
+    (const "cic:/Coq/Reals/Rdefinitions/Rminus.con");
+  DisambiguateChoices.add_binary_op "times" "natural times"
+    (const "cic:/Coq/Init/Peano/mult.con");
+  DisambiguateChoices.add_binary_op "times" "real times"
+    (const "cic:/Coq/Reals/Rdefinitions/Rmult.con");
+  DisambiguateChoices.add_binary_op "divide" "real divide"
+    (const "cic:/Coq/Reals/Rdefinitions/Rdiv.con");
+  DisambiguateChoices.add_unary_op "uminus" "real unary minus"
+    (const "cic:/Coq/Reals/Rdefinitions/Ropp.con");
+
+  DisambiguateChoices.add_binary_op "leq" "natural 'less or equal to'"
+    (mutind "cic:/Coq/Init/Peano/le.ind");
+  DisambiguateChoices.add_binary_op "leq" "real 'less or equal to'"
+    (const "cic:/Coq/Reals/Rdefinitions/Rle.con");
+  DisambiguateChoices.add_binary_op "geq" "natural 'greater or equal to'"
+    (const "cic:/Coq/Init/Peano/ge.con");
+  DisambiguateChoices.add_binary_op "geq" "real 'greater or equal to'"
+    (const "cic:/Coq/Reals/Rdefinitions/Rge.con");
+  DisambiguateChoices.add_binary_op "lt" "natural 'less than'"
+    (const "cic:/Coq/Init/Peano/lt.con");
+  DisambiguateChoices.add_binary_op "lt" "real 'less than'"
+    (const "cic:/Coq/Reals/Rdefinitions/Rlt.con");
+  DisambiguateChoices.add_binary_op "gt" "natural 'greater than'"
+    (const "cic:/Coq/Init/Peano/gt.con");
+  DisambiguateChoices.add_binary_op "gt" "real 'greater than'"
+    (const "cic:/Coq/Reals/Rdefinitions/Rgt.con");
+  DisambiguateChoices.add_symbol_choice "neq"
+    ("not equal to (leibnitz)",
+      (fun env _ args ->
+        let t1, t2 =
+          match args with 
+          | [t1; t2] -> t1, t2
+          | _ -> raise DisambiguateChoices.Invalid_choice
+        in
+        Cic.Appl [ const "cic:/Coq/Init/Logic/not.con";
+          Cic.Appl [
+            Cic.MutInd (HelmLibraryObjects.Logic.eq_URI, 0, []);
+              Cic.Implicit (Some `Type); t1; t2 ] ]));
+  DisambiguateChoices.add_symbol_choice "neq"
+    ("not equal to (sort Type)",
+      (fun env _ args ->
+        let t1, t2 =
+          match args with
+          | [t1; t2] -> t1, t2
+          | _ -> raise DisambiguateChoices.Invalid_choice
+        in
+        Cic.Appl [ const "cic:/Coq/Init/Logic/not.con";
+          Cic.Appl [
+            Cic.MutInd (HelmLibraryObjects.Logic_Type.eqt_URI, 0, []);
+              Cic.Implicit (Some `Type); t1; t2 ] ]));
 
 
         Ko
 
 let resolve (env: environment) (item: domain_item) ?(num = "") ?(args = []) () =
-  snd (Environment.find item env) env num args
+  try
+    snd (Environment.find item env) env num args
+  with Not_found -> assert false
 
   (* TODO move it to Cic *)
 let find_in_environment name context =
     | CicAst.AttributedTerm (_, term) -> aux loc context term
     | CicAst.Appl terms ->
         List.fold_left (fun dom term -> aux loc context term @ dom) [] terms
-    | CicAst.Binder (_, (var, typ), body) ->
+    | CicAst.Binder (kind, (var, typ), body) ->
+        let kind_dom =
+          match kind with
+          | `Exists -> [ Symbol ("exists", 0) ]
+          | _ -> []
+        in
         let type_dom = aux_option loc context typ in
         let body_dom = aux loc (var :: context) body in
-        body_dom @ type_dom
+        body_dom @ type_dom @ kind_dom
     | CicAst.Case (term, indty_ident, outtype, branches) ->
         let term_dom = aux loc context term in
         let outtype_dom = aux_option loc context outtype in
 
   in
   Hashtbl.replace symbol_choices symbol (codomain_item :: current_choices)
 
+let add_binary_op symbol dsc appl_head =
+  add_symbol_choice symbol
+    (dsc,
+      (fun env _ args ->
+        let t1, t2 =
+          match args with
+          | [t1; t2] -> t1, t2
+          | _ -> raise Invalid_choice
+        in
+        Cic.Appl [ appl_head; t1; t2 ]))
+
+let add_unary_op symbol dsc appl_head =
+  add_symbol_choice symbol
+    (dsc,
+      (fun env _ args ->
+        let t = match args with [t] -> t | _ -> raise Invalid_choice in
+        Cic.Appl [ appl_head; t ]))
+
 let add_num_choice choice = num_choices := choice :: !num_choices
 
 let lookup_symbol_choices symbol =
     List.find (has_description dsc) !num_choices
   with Not_found -> raise (Choice_not_found ("Num with dsc " ^  dsc))
 
+  (** initial table contents *)
+
+let _ =
+  let uri s = UriManager.uri_of_string s in
+  let const s = Cic.Const (uri s, []) in
+  let mutind s = Cic.MutInd (uri s, 0, []) in
+  add_binary_op "exists" "exists" (mutind "cic:/Coq/Init/Logic/ex.ind");
+  add_binary_op "exists" "exists over terms with sort Type"
+    (mutind "cic:/Coq/Init/Logic_Type/exT.ind");
+
 
     ];
 END
 
+(* TODO a lot of hard coded URIs, move them in HelmLibraryObjects *)
+
 let _ =
+    (* TODO cut-and-pasted code: here, in arit_notation.ml and
+     * disambiguateChoices.ml *)
+  let uri s = UriManager.uri_of_string s in
+  let const s = Cic.Const (uri s, []) in
+  let mutind s = Cic.MutInd (uri s, 0, []) in
   DisambiguateChoices.add_symbol_choice "eq"
     ("leibnitz's equality",
      (fun interp _ args ->
        Cic.Appl [
          Cic.MutInd (HelmLibraryObjects.Logic.eq_URI, 0, []);
            Cic.Implicit (Some `Type); t1; t2
-       ]))
+       ]));
+  DisambiguateChoices.add_symbol_choice "eq"
+    ("equality over objects with sort Type",
+     (fun interp _ args ->
+       let t1, t2 =
+         match args with
+         | [t1; t2] -> t1, t2
+         | _ -> raise DisambiguateChoices.Invalid_choice
+       in
+       Cic.Appl [
+         Cic.MutInd (HelmLibraryObjects.Logic_Type.eqt_URI, 0, []);
+           Cic.Implicit (Some `Type); t1; t2
+       ]));
+  DisambiguateChoices.add_binary_op "and" "logical and"
+    (mutind "cic:/Coq/Init/Logic/and.ind");
+  DisambiguateChoices.add_binary_op "or" "logical or"
+    (mutind "cic:/Coq/Init/Logic/or.ind");
+  DisambiguateChoices.add_unary_op "not" "logical not"
+    (const "cic:/Coq/Init/Logic/not.con");