From 81cb773cbc402fc74752fb69a436b25be49489ee Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 11 Sep 2007 16:50:12 +0000
Subject: [PATCH] librarySync - we do not generate the object attributes when
 we publish the xml LambdaDelta-1 - new cast type rule

---
 helm/software/components/cic_acic/cic2Xml.ml  |  15 +-
 helm/software/components/cic_acic/cic2Xml.mli |   1 +
 .../components/library/librarySync.ml         |   9 +-
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma   |  20 +-
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma     |  54 ++--
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma    |   2 +-
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma | 199 ++++++++------
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma     | 242 ++++++++++--------
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma     | 166 ++++++------
 .../LambdaDelta-1/ty3/pr3_props.ma            |  65 ++---
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma   |  41 ++-
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma  | 140 +++++-----
 .../LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma    |  18 +-
 13 files changed, 556 insertions(+), 416 deletions(-)

diff --git a/helm/software/components/cic_acic/cic2Xml.ml b/helm/software/components/cic_acic/cic2Xml.ml
index c90bc348c..eb3b93408 100644
--- a/helm/software/components/cic_acic/cic2Xml.ml
+++ b/helm/software/components/cic_acic/cic2Xml.ml
@@ -267,7 +267,7 @@ let print_term ?ids_to_inner_sorts =
    aux
 ;;
 
-let xml_of_attrs attributes =
+let xml_of_attrs generate_attributes attributes =
   let class_of = function
     | `Coercion n -> 
         Xml.xml_empty "class" [None,"value","coercion";None,"arity",string_of_int n]
@@ -312,9 +312,10 @@ let xml_of_attrs attributes =
    List.fold_right 
     (fun attr res -> [< xml_attr_of attr ; res >]) attributes [<>]
   in
-   Xml.xml_nempty "attributes" [] xml_attrs
+   if generate_attributes then Xml.xml_nempty "attributes" [] xml_attrs else [<>]
 
-let print_object uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter obj =
+let print_object uri 
+  ?ids_to_inner_sorts ?(generate_attributes=true) ~ask_dtd_to_the_getter obj =
  let module C = Cic in
  let module X = Xml in
  let module U = UriManager in
@@ -322,7 +323,7 @@ let print_object uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter obj =
    match obj with
        C.ACurrentProof (id,idbody,n,conjectures,bo,ty,params,obj_attrs) ->
         let params' = param_attribute_of_params params in
-        let xml_attrs = xml_of_attrs obj_attrs in
+        let xml_attrs = xml_of_attrs generate_attributes obj_attrs in
         let xml_for_current_proof_body =
 (*CSC: Should the CurrentProof also have the list of variables it depends on? *)
 (*CSC: I think so. Not implemented yet.                                       *)
@@ -382,7 +383,7 @@ let print_object uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter obj =
          xmlty, Some xmlbo
      | C.AConstant (id,idbody,n,bo,ty,params,obj_attrs) ->
         let params' = param_attribute_of_params params in
-        let xml_attrs = xml_of_attrs obj_attrs in
+        let xml_attrs = xml_of_attrs generate_attributes obj_attrs in
         let xmlbo =
          match bo with
             None -> None
@@ -409,7 +410,7 @@ let print_object uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter obj =
          xmlty, xmlbo
      | C.AVariable (id,n,bo,ty,params,obj_attrs) ->
         let params' = param_attribute_of_params params in
-        let xml_attrs = xml_of_attrs obj_attrs in
+        let xml_attrs = xml_of_attrs generate_attributes obj_attrs in
         let xmlbo =
          match bo with
             None -> [< >]
@@ -429,7 +430,7 @@ let print_object uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter obj =
          aobj, None
      | C.AInductiveDefinition (id,tys,params,nparams,obj_attrs) ->
         let params' = param_attribute_of_params params in
-        let xml_attrs = xml_of_attrs obj_attrs in
+        let xml_attrs = xml_of_attrs generate_attributes obj_attrs in
          [< X.xml_cdata "<?xml version=\"1.0\" encoding=\"ISO-8859-1\"?>\n" ;
             X.xml_cdata
              ("<!DOCTYPE InductiveDefinition SYSTEM \"" ^ dtdname ^ "\">\n") ;
diff --git a/helm/software/components/cic_acic/cic2Xml.mli b/helm/software/components/cic_acic/cic2Xml.mli
index 22c5669df..dcbff3d0f 100644
--- a/helm/software/components/cic_acic/cic2Xml.mli
+++ b/helm/software/components/cic_acic/cic2Xml.mli
@@ -33,6 +33,7 @@ val print_term :
 val print_object :
   UriManager.uri ->
   ?ids_to_inner_sorts: (string, Cic2acic.sort_kind) Hashtbl.t ->
+  ?generate_attributes:bool ->
   ask_dtd_to_the_getter:bool ->
   Cic.annobj ->
     Xml.token Stream.t * Xml.token Stream.t option
diff --git a/helm/software/components/library/librarySync.ml b/helm/software/components/library/librarySync.ml
index 11dccaaa1..9b08ba6dc 100644
--- a/helm/software/components/library/librarySync.ml
+++ b/helm/software/components/library/librarySync.ml
@@ -70,7 +70,7 @@ let save_object_to_disk uri obj ugraph univlist =
     HExtlib.mkdir dir
   in
   (* generate annobj, ids_to_inner_sorts and ids_to_inner_types *)
-  let annobj, innertypes, ids_to_inner_sorts =
+  let annobj, innertypes, ids_to_inner_sorts, generate_attributes =
    if Helm_registry.get_bool "matita.system" then
     let annobj, _, _, ids_to_inner_sorts, ids_to_inner_types, _, _ =
      Cic2acic.acic_object_of_cic_object obj 
@@ -79,15 +79,16 @@ let save_object_to_disk uri obj ugraph univlist =
      Cic2Xml.print_inner_types
       uri ~ids_to_inner_sorts ~ids_to_inner_types ~ask_dtd_to_the_getter:false
     in
-    annobj, Some innertypesxml, Some ids_to_inner_sorts
+    annobj, Some innertypesxml, Some ids_to_inner_sorts, false
    else 
     let annobj = Cic2acic.plain_acic_object_of_cic_object obj in  
-    annobj, None, None 
+    annobj, None, None, true 
   in 
   (* prepare XML *)
   let xml, bodyxml =
    Cic2Xml.print_object
-    uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter:false annobj 
+    uri ?ids_to_inner_sorts ~ask_dtd_to_the_getter:false 
+    ~generate_attributes annobj 
   in    
   let xmlpath, xmlbodypath, innertypespath, bodyuri, innertypesuri, 
       xmlunivgraphpath, univgraphuri = 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
index 25d4ad924..ba58219da 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/arity.ma
@@ -178,13 +178,21 @@ g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
 H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
 T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: 
 A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
-a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (_: (ex2 A 
+a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A 
 (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
 a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) 
 (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
-g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
-a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity 
-g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
-Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x 
-(arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
+g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat 
+Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 
+x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A 
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
+a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) 
+(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 
+t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
+Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) 
+(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g 
+x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 
+(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) 
+H4)))))))))) c t1 t2 H))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
index 4a8ed6c73..ff4a0734e 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/dec.ma
@@ -435,28 +435,36 @@ Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H12: (pc3 c2 x0
 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) 
 t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall 
 (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
-t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H6 t0 x0 H9 H12) x H6)))) 
-(\lambda (H12: (((pc3 c2 x0 t) \to (\forall (P: Prop).P)))).(or_intror (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) 
-(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) 
-t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: 
-(pc3 c2 t t3)).(\lambda (H15: (ty3 g c2 t0 t)).(let H_y \def (ty3_unique g c2 
-t0 t H15 x0 H9) in (H12 (pc3_s c2 x0 t H_y) P)))) (ty3_gen_cast g c2 t0 t t3 
-H13))))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda (H8: 
-((\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))))).(or_intror 
-(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) 
-(\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) 
-t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: 
-(pc3 c2 t t3)).(\lambda (H11: (ty3 g c2 t0 t)).(H8 t H11 P))) (ty3_gen_cast g 
-c2 t0 t t3 H9))))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t 
-t3) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 
-(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
-t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g 
-c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) 
-(ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H8: (ty3 g c2 t0 
-t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x: T).(\lambda 
-(H9: (ty3 g c2 t x)).(H5 x H9 P))) (ty3_correct g c2 t0 t H8)))) 
+t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t x H6 t0 
+x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to (\forall (P: 
+Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead 
+(Flat Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2 
+(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: 
+T).(ty3 g c2 t t4)) P (\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat 
+Cast) x2 t) t3)).(\lambda (H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t 
+x2)).(let H_y \def (ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def 
+(ty3_unique g c2 t0 t H15 x0 H9) in (H12 (pc3_s c2 x0 t H_y0) P))))))) 
+(ty3_gen_cast g c2 t0 t t3 H13))))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) 
+H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat 
+Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2 
+(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: 
+T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat 
+Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 t)).(\lambda (_: (ty3 g c2 t 
+x0)).(H8 t H11 P))))) (ty3_gen_cast g c2 t0 t t3 H9))))))) H7)))) H5)) 
+(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: 
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat 
+Cast) t t0) t3)).(\lambda (P: Prop).(ex3_ind T (\lambda (t4: T).(pc3 c2 
+(THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: 
+T).(ty3 g c2 t t4)) P (\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat 
+Cast) x0 t) t3)).(\lambda (_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t 
+x0)).(ex_ind T (\lambda (t4: T).(ty3 g c2 x0 t4)) P (\lambda (x: T).(\lambda 
+(_: (ty3 g c2 x0 x)).(H5 x0 H9 P))) (ty3_correct g c2 t x0 H9)))))) 
 (ty3_gen_cast g c2 t0 t t3 H6))))))) H4))) f H2))) k H1))))))) t2))) c t1))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
index 8ddb60c91..eff4d8e7b 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/defs.ma
@@ -42,5 +42,5 @@ g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u
 t)))))))))
 | ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) 
 \to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) 
-t2)))))).
+(THead (Flat Cast) t0 t2))))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
index 4bc299e43..1370b3d79 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma
@@ -858,99 +858,130 @@ Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u:
 T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead 
 (Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) 
 (\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u)) \to (ty3 g c0 t5 t3))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u 
-(THead (Flat Cast) t3 t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead 
-e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat 
-Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T (\lambda (t6: 
-T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat 
-Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 
-(THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 
-u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)))) 
-(ty3 g c t5 t3) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat 
-Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda 
+Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: 
+T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: 
+C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda 
 (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 
-u2)) (ty3 g c t5 t3) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Flat 
-Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) 
-x t2) (\lambda (t6: T).(ty3 g c t6 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat 
-Cast) x t2) x (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd 
-i u c t3 x H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c 
-t3) i u c x (fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i 
-u c t3 x H9) e H6)) (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x (fsubst0_snd 
-i u c t3 x H9) e H6)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: 
-T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat 
-Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) 
-t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c t5 t3) 
-(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Flat Cast) t3 x))).(\lambda 
-(H9: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead (Flat Cast) t3 x) 
-(\lambda (t6: T).(ty3 g c t6 t3)) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u 
-c x (fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) 
+u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda 
+(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 
+t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 
+t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq 
+T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g 
+c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead 
+(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat 
+Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind 
+T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead 
+(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 
+x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) 
+(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 
+x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x 
+H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x 
+(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x 
+H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) 
+(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) 
+(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) 
+(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i 
+u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T 
+(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 
+(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq 
+T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 
+x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead 
+(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x 
+(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) 
 (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
 (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
 (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
 t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
 (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
 (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g 
-c t5 t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T t5 (THead 
-(Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 x0)).(\lambda (H10: (subst0 
-(s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda 
-(t6: T).(ty3 g c t6 t3)) (ty3_conv g c t3 t0 H2 (THead (Flat Cast) x0 x1) x0 
-(ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 
-H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat Cast) i) u 
-c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c 
-x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 (fsubst0_snd i u c t3 
-x0 H9) e H6)) (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x0 (fsubst0_snd i u 
-c t3 x0 H9) e H6)) t5 H8)))))) H7)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i 
-H5)))))) (\lambda (c3: C).(\lambda (H5: (csubst0 i u c c3)).(\lambda (e: 
-C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 
-i u c3 t2 (fsubst0_fst i u c t2 c3 H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i 
-u c t3 c3 H5) e H6)))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u (THead 
-(Flat Cast) t3 t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u c 
-c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
-u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) 
-(\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead 
-(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 
-t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c3 t5 t3) 
-(\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) 
-(\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 
-(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 
-t3) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x 
+c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 
+x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead 
+(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) 
+(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 
+x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 
+x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
+(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 
+x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c 
+t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat 
+Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c 
+t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 
+(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) 
+(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead 
+(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat 
+Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct 
+g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) 
+(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda 
+(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e 
+(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 
+H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda 
+(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda 
+(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl 
+i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 
+(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T 
+(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat 
+Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: 
+(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: 
+T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat 
+Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat 
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x 
 t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) 
-(\lambda (t6: T).(ty3 g c3 t6 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 
-(fsubst0_fst i u c t3 c3 H6) e H7) (THead (Flat Cast) x t2) x (ty3_cast g c3 
-t2 x (ty3_conv g c3 x t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e 
-H7) t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x 
-(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 
-H6) e H7))) t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) 
-(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 
-H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 
-(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat 
+Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g 
+c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x 
+t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u 
+c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 
+(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 
+(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i 
+u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat 
+Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) 
+i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) 
+(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e 
+H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e 
+H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) 
-(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 t3) 
-(\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) t3 x))).(\lambda 
-(H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead (Flat Cast) t3 x) 
-(\lambda (t6: T).(ty3 g c3 t6 t3)) (ty3_cast g c3 x t3 (H1 i u c3 x 
-(fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 (H3 i u c3 t3 (fsubst0_fst i u c 
-t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
-T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
-T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 t3) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x0 
-x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: (subst0 (s (Flat 
-Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t6: 
-T).(ty3 g c3 t6 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 
-c3 H6) e H7) (THead (Flat Cast) x0 x1) x0 (ty3_cast g c3 x1 x0 (ty3_conv g c3 
-x0 t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u 
-c3 x1 (fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 
-c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e 
-H7))) t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) 
-(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 
-c3 H6) e H7)) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i 
-H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
+(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead 
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) 
+t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead 
+(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) 
+(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda 
+(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) 
+u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead 
+(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 
+(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: 
+(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) 
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat 
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i 
+u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) 
+(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u 
+c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 
+(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 
+t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 
+(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c 
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat 
+Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead 
+(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 
+i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst 
+i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 
+t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
 
 theorem ty3_csubst0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
index 3e9846516..5267f5c6a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fwd.ma
@@ -87,8 +87,8 @@ t1 (TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
 n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match 
 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H5) in (False_ind (pc3 c0 (TSort (next g n)) t2) H6))))))))))) c y x H0))) 
-H))))).
+H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) 
+H6))))))))))) c y x H0))) H))))).
 
 theorem ty3_gen_lref:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
@@ -383,12 +383,13 @@ Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
 _) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T 
 (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) 
-t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 
-(lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t)))))) H6))))))))))) c y x H0))) H))))).
+(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))))) H6))))))))))) c y x H0))) H))))).
 
 theorem ty3_gen_bind:
  \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
@@ -638,11 +639,11 @@ _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
 \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
 H5) in (False_ind (ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: 
-T).(pc3 c0 (THead (Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t: 
-T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t4: T).(\lambda (_: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: 
-T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5))))) 
-H6))))))))))) c y x H0))) H))))))).
+T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 t2))))) (\lambda (_: 
+T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t4: 
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) 
+(\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) 
+u) t4 t5))))) H6))))))))))) c y x H0))) H))))))).
 
 theorem ty3_gen_appl:
  \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
@@ -810,105 +811,144 @@ Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda
 in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast 
 \Rightarrow True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T 
 T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind 
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) 
-c y x H0))) H)))))).
+Abst) u t)) (THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 
+g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w 
+u)))) H6))))))))))) c y x H0))) H)))))).
 
 theorem ty3_gen_cast:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall 
-(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (land (pc3 c t2 x) (ty3 g c 
-t1 t2)))))))
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: 
+T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
+(\lambda (t0: T).(ty3 g c t2 t0))))))))
 \def
  \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
 (x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T 
-(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (land (pc3 c t2 x) 
-(ty3 g c t1 t2)) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) 
-t2 t1)) \to (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)))))) (\lambda (c0: 
-C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda 
-(_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0 
-t1 t2))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
-t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 
-t3) (ty3 g c0 t1 t2))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u 
-(THead (Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u 
-(THead (Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t4: 
-T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t3) (ty3 g c0 t1 
-t2)))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u (\lambda 
-(t4: T).(ty3 g c0 t4 t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let H10 \def 
-(H8 (refl_equal T (THead (Flat Cast) t2 t1))) in (and_ind (pc3 c0 t2 t3) (ty3 
-g c0 t1 t2) (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)) (\lambda (H11: (pc3 c0 t2 
-t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(conj (pc3 c0 t2 t0) (ty3 g c0 t1 t2) 
-(pc3_t t3 c0 t2 H11 t0 H5) H12))) H10)))))))))))))))) (\lambda (c0: 
-C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 
-t1))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) 
-t2 t1) H1) in (False_ind (land (pc3 c0 t2 (TSort (next g m))) (ty3 g c0 t1 
-t2)) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
-(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to 
-(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead 
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (ex3 T (\lambda (t0: 
+T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
+(\lambda (t0: T).(ty3 g c t2 t0))) (\lambda (y: T).(\lambda (H0: (ty3 g c y 
+x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat 
+Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g 
+c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to 
+(ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 t3)))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: (((eq T u 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat 
+Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g 
+c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u (THead 
+(Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (THead 
+(Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t4: T).((eq T t4 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat 
+Cast) t5 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g 
+c0 t2 t5))))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u 
+(\lambda (t4: T).(ty3 g c0 t4 t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let 
+H10 \def (H8 (refl_equal T (THead (Flat Cast) t2 t1))) in (ex3_ind T (\lambda 
+(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) (ex3 T (\lambda (t4: T).(pc3 c0 
+(THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda 
+(t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: T).(\lambda (H11: (pc3 c0 (THead 
+(Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(\lambda (H13: (ty3 
+g c0 t2 x0)).(ex3_intro T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) 
+t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) x0 
+(pc3_t t3 c0 (THead (Flat Cast) x0 t2) H11 t0 H5) H12 H13))))) 
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
+(TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def (eq_ind T (TSort m) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0: 
+T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u 
+t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
+(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) 
+(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead 
 (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
 (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Flat Cast) t2 t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O t)) (ty3 
-g c0 t1 t2)) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda 
-(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) 
-t2 t1)) \to (land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef 
-n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Cast) t2 t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O 
-u)) (ty3 g c0 t1 t2)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead 
+(Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 t2)) 
+(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: nat).(\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d 
+(Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: 
+(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d 
+(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0: 
+T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2 
+t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 
+t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 
+t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: 
+T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
 (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) 
-t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0 t1 t2))))).(\lambda (b: B).(\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 
-t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead 
-c0 (Bind b) u) t2 t3) (ty3 g (CHead c0 (Bind b) u) t1 t2))))).(\lambda (t4: 
+t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t)) 
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 
+g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 
+t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat 
+Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) 
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (t4: 
 T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3 
-(THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t4) (ty3 g 
-(CHead c0 (Bind b) u) t1 t2))))).(\lambda (H7: (eq T (THead (Bind b) u t0) 
-(THead (Flat Cast) t2 t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) t2 t1) H7) in (False_ind 
-(land (pc3 c0 t2 (THead (Bind b) u t3)) (ty3 g c0 t1 t2)) H8)))))))))))))))) 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 (CHead c0 (Bind 
+b) u) (THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g (CHead c0 (Bind 
+b) u) t1 t2)) (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t2 
+t5)))))).(\lambda (H7: (eq T (THead (Bind b) u t0) (THead (Flat Cast) t2 
+t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) (\lambda (ee: T).(match ee 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) t2 t1) H7) in (False_ind (ex3 T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Bind b) u t3))) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))) H8)))))))))))))))) 
 (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w 
-u)).(\lambda (_: (((eq T w (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 u) 
-(ty3 g c0 t1 t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v 
-(THead (Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) 
-\to (land (pc3 c0 t2 (THead (Bind Abst) u t)) (ty3 g c0 t1 t2))))).(\lambda 
-(H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 t1))).(let H6 \def 
-(eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast 
-\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (land 
-(pc3 c0 t2 (THead (Flat Appl) w (THead (Bind Abst) u t))) (ty3 g c0 t1 t2)) 
-H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) t2 t1)) 
-\to (land (pc3 c0 t2 t3) (ty3 g c0 t1 t2))))).(\lambda (t4: T).(\lambda (H3: 
+u)).(\lambda (_: (((eq T w (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
+(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (v: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat 
+Cast) t0 t2) (THead (Bind Abst) u t))) (\lambda (_: T).(ty3 g c0 t1 t2)) 
+(\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) 
+w v) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w 
+v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: 
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead 
+(Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda 
+(_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H6)))))))))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 
+t3)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
+(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: 
 (ty3 g c0 t3 t4)).(\lambda (H4: (((eq T t3 (THead (Flat Cast) t2 t1)) \to 
-(land (pc3 c0 t2 t4) (ty3 g c0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat 
-Cast) t3 t0) (THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 
-| (TLRef _) \Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat 
-Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq 
-T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat 
-Cast) t2 t1)) \to (land (pc3 c0 t2 t4) (ty3 g c0 t1 t2)))) H4 t2 H8) in (let 
-H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 
-\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to 
-(land (pc3 c0 t2 t) (ty3 g c0 t1 t2)))) H2 t2 H8) in (let H12 \def (eq_ind T 
-t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r T t2 (\lambda (t: 
-T).(land (pc3 c0 t2 t) (ty3 g c0 t1 t2))) (let H13 \def (eq_ind T t0 (\lambda 
-(t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t2) (ty3 g c0 
-t1 t2)))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 
-t t2)) H12 t1 H7) in (conj (pc3 c0 t2 t2) (ty3 g c0 t1 t2) (pc3_refl c0 t2) 
-H14))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
+(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq 
+T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1))).(let H6 \def (f_equal 
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) \Rightarrow t])) 
+(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in ((let H7 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
+\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in 
+(\lambda (H8: (eq T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T 
+t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat 
+Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g 
+c0 t2 t5))))) H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g 
+c0 t t4)) H3 t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0 
+(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat 
+Cast) t5 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 
+t2 t5))))) H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 
+t0 t)) H1 t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T 
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3 
+g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14 
+\def (eq_ind T t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T 
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 
+t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 
+(pc3_refl c0 (THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) 
+c y x H0))) H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
index 99eb3bc22..bc53829ae 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma
@@ -596,53 +596,60 @@ t3 t4) \to (ty3 g c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
 c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let 
 H6 \def (match H5 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda 
 (_: (pr0 t5 t6)).((eq T t5 (THead (Flat Cast) t3 t2)) \to ((eq T t6 t4) \to 
-(ty3 g c2 t4 t3)))))) with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 
-(THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead 
-(Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 t3))) 
-(\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat 
-Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 t3)) (ty3_cast g c2 t2 t3 (H1 c2 
-H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 
-(THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) 
-\Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 
-t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) 
-(THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K 
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in (eq_ind K (Flat Cast) 
-(\lambda (k0: K).((eq T u1 t3) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) 
-t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 t3))))))) (\lambda 
-(H13: (eq T u1 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T 
-(THead (Flat Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 
-t4 t3)))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T 
-(THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 t6) \to (ty3 g c2 
-t4 t3))))) (\lambda (H15: (eq T (THead (Flat Cast) u2 t6) t4)).(eq_ind T 
-(THead (Flat Cast) u2 t6) (\lambda (t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to 
-(ty3 g c2 t7 t3)))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 
-t6)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead (Flat Cast) u2 
-t6) u2 (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 t3 (H1 
-c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 H4 u2 
-H16)) (pc3_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H16))))) t4 H15)) t5 (sym_eq T 
-t5 t2 H14))) u1 (sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) 
-H10)) H9 H6 H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: 
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 
-t2))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow 
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in 
-(False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 
-t6) \to (ty3 g c2 t4 t3)))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 
-u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: 
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) with [(pr0_refl t5) \Rightarrow 
+(\lambda (H6: (eq T t5 (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 
+t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).((eq T t6 t4) \to 
+(ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H8: (eq T (THead (Flat 
+Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t6: T).(ty3 g 
+c2 t6 (THead (Flat Cast) t0 t3))) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl 
+t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat 
+Cast) t3 t2) H6) H7))) | (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda 
+(H8: (eq T (THead k u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T 
+(THead k u2 t6) t4)).((let H10 \def (f_equal T T (\lambda (e: T).(match e in 
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t5 | (TLRef _) 
+\Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
+(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead 
+(Flat Cast) t3 t2) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match 
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) (THead (Flat 
+Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to 
+((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) \to ((pr0 t5 
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda (H13: (eq T u1 
+t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Flat 
+Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead 
+(Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 (\lambda 
+(t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to ((pr0 t7 
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: (eq T 
+(THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) (\lambda 
+(t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat Cast) t0 
+t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T 
+(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat Cast) u2 t6) (THead 
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g c2 t0 x)).(ty3_conv 
+g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 
+(H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 t6) (THead (Flat 
+Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H16) t6 
+t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H16))) t0 (H3 c2 
+H4 u2 H16)) (pc3_pr2_x c2 (THead (Flat Cast) t0 u2) (THead (Flat Cast) t0 t3) 
+(pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast))))) (ty3_correct g c2 
+t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 
+(sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 
+H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda 
+(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast 
+\Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T 
+(THead (Bind Abbr) v2 t6) t4) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 
+H7 u1 u2 H8 t5 t6 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t5)) (THead (Flat Cast) t3 t2))).(\lambda (H11: (eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
@@ -650,40 +657,47 @@ return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
 True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H10) in 
 (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
 t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 
-t5 t6) \to (ty3 g c2 t4 t3)))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 
-H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) 
-(THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) 
-t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T 
-(THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O 
-u2 t6 w) \to (ty3 g c2 t4 t3))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 
-t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) 
-(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def 
-(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to 
-((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 t3)))) H10)) H9 H6 
-H7))) | (pr0_epsilon t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat 
-Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) 
+t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))))) H12)) H11 H6 H7 H8 
+H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) \Rightarrow (\lambda (H9: (eq T 
+(THead (Bind Abbr) u1 t5) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T 
+(THead (Bind Abbr) u2 w) t4)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 
+t5) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 
+t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Flat Cast) 
+t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 t6 H7 u) \Rightarrow 
+(\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) (THead (Flat Cast) 
+t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (eq_ind T (THead (Bind b) 
+u (lift (S O) O t5)) (\lambda (e: T).(match e in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to ((not (eq B b Abst)) \to 
+((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) H10)) H9 H6 H7))) 
+| (pr0_epsilon t5 t6 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) 
+u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) 
 \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in 
 ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
 T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 
 _) \Rightarrow t7])) (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) 
 in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) \to ((eq T t6 t4) \to ((pr0 t5 
-t6) \to (ty3 g c2 t4 t3))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 
-(\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 t3)))) 
-(\lambda (H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to 
-(ty3 g c2 t4 t3))) (\lambda (H13: (pr0 t2 t4)).(H1 c2 H4 t4 H13)) t6 (sym_eq 
-T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10))) H9)) H8 
-H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
-t4))))))))))))))) c1 t1 t H))))).
+t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H11: (eq T t5 
+t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 t4) \to ((pr0 t7 t6) \to (ty3 g 
+c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda (H12: (eq T t6 t4)).(eq_ind T t4 
+(\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) 
+(\lambda (H13: (pr0 t2 t4)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 
+g c2 t4 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H14: (ty3 g c2 
+t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
+(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 
+H13) (pc3_pr2_x c2 t3 (THead (Flat Cast) t0 t3) (pr2_free c2 (THead (Flat 
+Cast) t0 t3) t3 (pr0_epsilon t3 t3 (pr0_refl t3) t0)))))) (ty3_correct g c2 
+t3 t0 (H3 c2 H4 t3 (pr0_refl t3))))) t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T 
+t5 t2 H11))) u (sym_eq T u t3 H10))) H9)) H8 H6)))]) in (H6 (refl_equal T 
+(THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
 
 theorem ty3_sred_pr1:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
index 6cf0c095a..3a17df985 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma
@@ -414,40 +414,45 @@ x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T
 (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h 
 x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 
-t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead (Flat 
-Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 x2))).(\lambda (H9: (eq T t2 
-(lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e t 
-t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall 
-(x5: nat).((eq T t (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) 
-\to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 
-g e0 x4 t4))))))))) H4 (lift h x1 x2) H8) in (let H11 \def (eq_ind T t3 
-(\lambda (t: T).(ty3 g c0 t t0)) H3 (lift h x1 x2) H8) in (let H12 \def 
-(eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t2 
-(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda 
-(t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) 
-H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 
-t2 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t: 
-T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g 
-e (THead (Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: 
-T).(ty3 g c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def 
-(eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t 
-(lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda 
-(t4: T).(pc3 c0 (lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 
-t4))))))))) H12 (lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T 
-(lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) 
-(lift h x1 x2))) (\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: 
-T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda 
+(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq 
+T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 
+x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) 
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead 
+(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind 
+T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 
+x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 
+c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h 
+x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 
+(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall 
+(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: 
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) 
+(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 
+\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in 
+(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead 
+(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g 
+c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) 
+\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 
+(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 
+(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) 
+e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) 
+(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 
+t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead 
 (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 
 x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 
 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 
 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: 
-T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead 
-(Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda (_: (pc3 c0 (lift h x1 x5) 
-t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 
-(lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) 
-x2 x3) t4)) x2 (pc3_refl c0 (lift h x1 x2)) (ty3_cast g e x3 x2 (ty3_conv g e 
+T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda 
+(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda 
+(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 
+x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat 
+Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) 
+(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 
+c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat 
+Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e 
 x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) 
 H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 
 H5))))))))))))))) c y x H0))))) H))))))).
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
index 4fe0d6385..d69a6dc4a 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma
@@ -179,10 +179,14 @@ C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
 nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d 
 t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
 (drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s 
-(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d t3))) (ty3_cast g 
-c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4) 
-(H3 c0 d h H4)) (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) 
-t3 t0 h d)))))))))))))) e t1 t2 H))))).
+(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d (THead (Flat Cast) 
+t4 t3)))) (eq_ind_r T (THead (Flat Cast) (lift h d t4) (lift h (s (Flat Cast) 
+d) t3)) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) (lift h d t3) (lift h (s 
+(Flat Cast) d) t0)) t)) (ty3_cast g c0 (lift h (s (Flat Cast) d) t0) (lift h 
+(s (Flat Cast) d) t3) (H1 c0 (s (Flat Cast) d) h H4) (lift h d t4) (H3 c0 d h 
+H4)) (lift h d (THead (Flat Cast) t4 t3)) (lift_head (Flat Cast) t4 t3 h d)) 
+(lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) t3 t0 h 
+d)))))))))))))) e t1 t2 H))))).
 
 theorem ty3_correct:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
@@ -244,10 +248,14 @@ T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat
 Appl) w (THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u 
 t) x1 (ty3_bind g c0 u x2 H9 Abst t x1 H10 x3 H11)))))))))) (ty3_gen_bind g 
 Abst c0 u t x0 H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (H1: (ex T 
-(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 
-t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) c t1 t2 
-H))))).
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T 
+(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 
+t3 t4)).(\lambda (H3: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def 
+H3 in (ex_ind T (\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g 
+c0 (THead (Flat Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 
+x)).(ex_intro T (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) 
+(THead (Flat Cast) x t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 
+t2 H))))).
 
 theorem ty3_unique:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
@@ -416,11 +424,15 @@ x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0
 w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
 T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0 
 t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 
-t3)).(\lambda (_: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 
+t3)).(\lambda (H3: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3 
 t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0) 
-t4)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5: 
-(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2 
-t4 H4)))))))))))) c u t1 H))))).
+t4)).(ex3_ind T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) 
+(\lambda (_: T).(ty3 g c0 t0 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) (pc3 c0 
+(THead (Flat Cast) t3 t2) t4) (\lambda (x0: T).(\lambda (H5: (pc3 c0 (THead 
+(Flat Cast) x0 t2) t4)).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (H7: (ty3 g 
+c0 t2 x0)).(pc3_t (THead (Flat Cast) x0 t2) c0 (THead (Flat Cast) t3 t2) 
+(pc3_head_1 c0 t3 x0 (H3 x0 H7) (Flat Cast) t2) t4 H5))))) (ty3_gen_cast g c0 
+t0 t2 t4 H4)))))))))))) c u t1 H))))).
 
 theorem ty3_gen_abst_abst:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
@@ -469,6 +481,7 @@ v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
  \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: 
 (ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: 
 T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g 
-c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) v 
-(ty3_cast g c t v H x H0)))) (ty3_correct g c t v H)))))).
+c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) 
+(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v 
+H)))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
index 7a91362cc..60e08dbe1 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
@@ -554,28 +554,34 @@ in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O)
 d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) 
 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
 T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda (_: (subst1 d u 
-t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u 
-d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d 
-u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 
-t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d 
+x0))).(\lambda (H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g 
+a x0 x1)).(let H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d 
 x2))).(\lambda (H13: (subst1 d u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g 
 a x2 x3)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 
 (subst1_confluence_lift t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda 
 (y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
-x0 (eq_ind_r T (THead (Flat Cast) (lift (S O) d x0) (lift (S O) d x2)) 
-(\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) t)) (subst1_head u t4 
-(lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) H12) (lift (S O) d 
-(THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) H8 (ty3_cast g a 
-x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(THead (Flat Cast) x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat 
+Cast) (lift (S O) d x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead 
+(Flat Cast) t4 t3) t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) 
+t3 (lift (S O) d x2) H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat 
+Cast x0 x2 (S O) d)) (eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift 
+(S O) d x0)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) 
+(subst1_head u t0 (lift (S O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) 
+(lift (S O) d (THead (Flat Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) 
+(ty3_cast g a x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 
+t2 H))))).
 
 theorem ty3_gen_cvoid:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
@@ -1103,47 +1109,59 @@ C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
 (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: 
 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
 (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: 
-(eq T t4 (lift (S O) d x0))).(\lambda (H8: (eq T t0 (lift (S O) d 
-x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T t0 (\lambda (t: 
-T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in (let H11 \def (eq_ind T t4 
-(\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) d x0) H7) in 
-(let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: C).(\forall (u0: 
-T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) u0)) \to (\forall 
-(a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
-y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 (\lambda 
-(t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T (lift (S O) d 
-x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
-(Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
-t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
-(let H14 \def (H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
-(lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat 
-Cast) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T (lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T (lift (S 
-O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def 
-(eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x0))) H13 (lift (S O) 
-d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t: T).(ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S O) d x0) 
-t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d 
-x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (let H19 \def (eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 
-(lift_inj x0 x3 (S O) d H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) 
-x0 x2)) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t 
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x0) 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d (THead 
-(Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
-T (lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) x0 (refl_equal T (lift (S O) d 
-(THead (Flat Cast) x0 x2))) (refl_equal T (lift (S O) d x0)) (ty3_cast g a x2 
-x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x0) (lift (S O) d x2)) 
-(lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))))))))) 
-H6)))))))))))))))) c t1 t2 H))))).
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: 
+(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
+(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in 
+(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def 
+(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) 
+d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: 
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) 
+u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 
+(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T 
+(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) 
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def 
+(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d 
+x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S 
+O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda 
+(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 
+x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d 
+x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda 
+(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) 
+(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) 
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def 
+(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d 
+H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: 
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) 
+(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) 
+(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) 
+d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S 
+O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
+(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 
+x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a 
+x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) 
+(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S 
+O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 
+H8))))))) H6)))))))))))))))) c t1 t2 H))))).
 
diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
index 8bc71a82e..a9fb878ea 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/tau0.ma
@@ -622,13 +622,13 @@ v2)).(\lambda (H16: (tau0 g c0 t2 t6)).(let H_y \def (H1 t6 H16) in (let H_y0
 t2) (THead (Flat Cast) v2 t6)) (\lambda (x: T).(\lambda (H18: (ty3 g c0 v2 
 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t6 t)) (ty3 g c0 (THead (Flat Cast) 
 t3 t2) (THead (Flat Cast) v2 t6)) (\lambda (x0: T).(\lambda (H19: (ty3 g c0 
-t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) v2 (ty3_cast g c0 t6 v2 
-(ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead (Flat Cast) t3 t2) t3 
-(ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 (THead (Flat Cast) v2 t6) 
-(pc3_pr2_u c0 t6 (THead (Flat Cast) v2 t6) (pr2_free c0 (THead (Flat Cast) v2 
-t6) t6 (pr0_epsilon t6 t6 (pr0_refl t6) v2)) t3 H17))))) (ty3_correct g c0 t2 
-t6 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H14)) t5 (sym_eq T t5 t2 
-H13))) v1 (sym_eq T v1 t3 H12))) H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 
-H6))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Cast) t3 t2)) 
-(refl_equal T t4))))))))))))) c u t1 H))))).
+t6 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t6) (THead (Flat Cast) x v2) 
+(ty3_cast g c0 t6 v2 (ty3_sconv g c0 t6 x0 H19 t3 v2 H_y0 H17) x H18) (THead 
+(Flat Cast) t3 t2) (THead (Flat Cast) v2 t3) (ty3_cast g c0 t2 t3 H0 v2 H_y0) 
+(pc3_s c0 (THead (Flat Cast) v2 t3) (THead (Flat Cast) v2 t6) (pc3_thin_dx c0 
+t6 t3 H17 v2 Cast))))) (ty3_correct g c0 t2 t6 H_y)))) (ty3_correct g c0 t3 
+v2 H_y0))))))) t4 H14)) t5 (sym_eq T t5 t2 H13))) v1 (sym_eq T v1 t3 H12))) 
+H11))) c1 (sym_eq C c1 c0 H7) H8 H9 H5 H6))))]) in (H5 (refl_equal C c0) 
+(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 
+H))))).
 
-- 
2.39.2